Classical Banach Spaces II ||

Ergebnisse der Mathematik und ihrer Grenzgebiete 97 ASeries of Modern Surveys in Mathematics

Editorial Board: P. R. HaIrnos P. J. Hilton (Chairman) R. Remmert B. Szökefalvi-Nagy

Advisors: L. V. Ahlfors R. Baer F. L. Bauer A. Dold J. L. Doob S. Eilenberg K. W. Gruenberg M. Kneser G. H. Müller M.M. Postnikov B. Segre E. Sperner

Joram Lindenstrauss Lior Tzafriri

Classical Banach Spaces 11 Function Spaces

Springer-Verlag Berlin Heidelberg GmbH 1979

Joram Lindenstrauss Lior Tzafriri

Department of Mathematics, The Hebrew University of Jerusalem J erusalem, Israel

AMS Subject Classification (1970): 46-02, 46 A40, 46Bxx, 46Jxx

ISBN 978-3-662-35349-3 ISBN 978-3-662-35347-9 (eBook) DOI 10.1007/978-3-662-35347-9

Library ofCongress Cataloging in Publication Data. Lindenstrauss, Joram, 1936-. Classical Banach spaces. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 92, 97). Bibliography: v. I, p.; v. 2, p. Inc1udes index. CONTENTS: I. Sequence spaces. 2. Function spaces. 1. Banach spaces. 2. Sequence spaces. 3. Function spaces. I. Tzafriri, Lior. 1936- joint author. 11. Title. 1lI. Series QA322.2.L56. 1977-515'.73.77-23131.

This work is subject to copyright. All rights afe reserved, whether the whole or part ofthe material is concerned, specifical1y those of translation, reprinting, re-use of iHustrations, broadcasting. reproduction by photocopying machine Of similar roeans, and storage in data banks. Under * 54 ofthe German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the ree to be determined by agreement with the publisher.

© by Springer-Verlag Berlin Heidelberg 1979.

Originally published by Springer-Verlag Berlin Heidelberg New York in 1979. Softcover reprint of the hardcover I st edition 1979

2141/3140-543210

To Naomi and Marianne

Preface

This second volume of our book on classical Banach spaces is devoted to the study of Banach lattices. The writing of an entire volume on this subject within the framework of Banach space theory became possible only recently due to the substantial progress made in the seventies.

The structure of Banach lattices is much simpler than that of general Banach spaces and their theory is therefore more complete and satisfactory. Many of the results concerning Banach lattices are not valid (and sometimes even do not make sense) for general Banach spaces. Naturally, the theory of Banach lattices has many tools which are specific to this theory. We would like to draw attention in particular to the notions of p-convexity and p-concavity and their variants which seem to be especially useful in studying Banach lattices. We are convinced that these notions, which play a central role in the present volume, will continue to dominate the theory of Banach lattices and will be also useful in the various applications of lattice theory to other branches of analysis.

The table of contents is quite detailed and should give a clear idea of the material discussed in each section. We would like to make here only a few comments on the contents ofthis volume. The basic standard theory ofBanach lattices is contained in Section l.a and in apart of Section 1. b. The theory of p-convexity and p-concavity in Banach spaces is presented in detail in Sections l.d and 1.f. Chapter 2 is devoted to a detailed study of the structure of rearrangement invariant function spaces on [0, 1] and [0, (0). The usefulness of the notions of p-convexity and p-concavity will become apparent from their various applications in Chapter 2. Three of the sections in this volume are concerned with the general theory of Banach spaces rather than with Banach lattices. Section l.e contains (part 00 the theory of uniform convexity in general Banach spaces. Section l.g deals with the approximation property. It complements (but is independent of) the discussion of this property in Vol. 1. Section 2.g deals with geometrie aspects of interpolation theory in general Banach spaces.

The various sections of this volume vary as far as their degree of difficulty is concerned. The first four sections in Chapter 1 and the first three sections of Chapter 2 are easier than the rest of the volume. The technically most difficult sections are Sections l.g and 2.e. The results of Section l.g are not used elsewhere in this volume and Sections 2.f and 2.g can be read without being acquainted with 2.e.

The prerequisites for the reading of this volume include besides standard material from functional analysis and measure theory only a superficial knowledge

Vlll Preface

of the material presented in Vol. I of this book [79]. The (rather infrequent) references to Vol. I are marked here as follows: I.l.d.6 means for example item 6 in Seetion l.d ofVol. I. In the present volume a much more extensive use is made of ideas and results from probability theory than in Vol.1. For the convenience ofthe reader without a probabilistic background we tried to discuss briefly in the appropriate places the notions and resuIts from probability theory which we apply. The notation used in this volume is essentially the standard one, which is explained for example in the beginning ofVol. I. A few notations will be introduced and explained throughout the text.

The overlap between this volume and existing books on lattice theory is small and consists mostly of the standard material presented in Sections l.a and (partially) l.b. The books ofW. A. J. Luxemburg and A. C. Zaanen [90] and H. H. Schaefer [118] contain much additional material on vector lattices. We do not treat here the theory of positive operators presented in [118]. Further information on isometrie aspects ofBanach lattice theory can be found in E. Lacey [71]. Sections 2.e and 2.f are based almost entirely on material taken from the memoir [58]. This memoir contains more results and details on the subject matter of 2.e and 2.f. The lecture notes of B. Beauzamy [6] contain further material on interpolation spaces in the spirit of the discussion in Section 2.g.

In the writing ofthis volume we benefited very much from long discussions with W. B. Johnson, B. Maurey and G. Pisier. We are very grateful to them for many valuable suggestions. We are also very grateful to J. Arazy who read the entire manuscript of this volume and made many corrections and suggestions. We wish also to thank Z. AItshuler and G. Schechtman for their help in the preparation of the manuscript.

The main part of this volume was written while we both were members of the Institute for Advanced Studies of the Hebrew University. The volume was completed while the first-named author visited the University of Texas at Austin and the second-named author visited the University of Copenhagen (supported in part by the Danish National Science Research Council). In these respective institut ions we both gave lectures based on a preliminary version ofthis volume. We benefited much from comments made by those who attended these lectures. We wish to express our thanks to all these institutions as weil as to the U.S. National Science Foundation (which supported us during the summers of 1977 and 1978 while we stayed at the Ohio State University) for providing us with excellent working conditions.

Finally, we express our indebtedness to Susan Brink and Nita Goldrick who very patiently and expertly typed various versions ofthe manuscript ofthis volume.

August 1978 Joram Lindenstrauss Lior Tzafriri

Table of Contents

1. Banach Lattices

a. Basic Definitions and Results Characterizations of a-completeness and a-order continuity. Ideals, bands and band projections. Boolean algebras of projections and cyclic spaces.

b. Concrete Representation of Banach Lattices . Abstract L p and M spaces. Joint characterizations of L p (J1) and co(r) spaces. The functional representation theorem for order continuous Banach lattices. Köthe function spaces. The Fatou property.

c. The Structure of Banach Lattices and their Subspaces Property (u). Weak completeness and reflexivity. Existence of unconditional basic sequences.

d. p-Convexity in Banach Lattices Functional calculus in general Banach lattices. The definition and basic properties of p-convexity and p-concavity. Duality. The p-convexification and p-concavification procedures. Some connections with p-absolutely summing operators. Factorization through L p spaces.

14

31

40

e. Uniform Convexity in General Banach Spaces and Re1ated Notions. 59 The definition of the moduli of convexity and smoothness. Duality. Asymptotic behavior of the moduli. The moduli of L 2(X). Convergence of series in uniforrnly convex and uniformly smooth spaces. The notions oftype and cotype. Kahane's theorem. Connections between the moduli and type and cotype.

f. Uniform Convexity in Banach Lattices and Related Notions . Uniformly convex and smooth renormings of a (p> I)-convex and (q< IXJ )-concave Banach lattice. The concepts of upper and lower estimates for disjoint elements. The relations between these notions and those of type, co type, p-convexity, q-concavity, p-absolutely summing operators, etc. Examples. Two diagrams summarizing the various connections.

g. The Approximation Property and Banach Lattices. Examples of Banach lattices and of subspaces of Ip ' p # 2, without the B.A.P. The connection between the type and the co type of aspace and the existence of subspaces without the B.A.P. Aspace different from 12 all of whose subspaces have the B.A.P.

2. Rearrangement Invariant Function Spaces

a. Basic Definitions, Examples and Results The definition of r.i. function spaces on [0, I] and [0, IXJ). Conditional expectations. Interpolation between LI and Lw.

79

102

114

114

x Table of Contents

b. The Boyd Indices. The definition of Boyd indices. Duality. The Rademacher functions in an r.i. function space on [0, I]. The characterization of Boyd indices in terms of existence of I;'s for all n on disjoint vectors having the same distribution. Operators of weak type (p, q). Boyd's interpolation theorem.

c. The Haar and the Trigonometrie Systems Basic results on martingales. The unconditionality ofthe Haar system in Lp(O, I) spaces (\ <p < 00) and in more general r.i. function spaces on [0, I]. Reproducibility of bases. The boundedness ofthe Riesz projection and applications.

d. Some Results on Complemented Subspaces . The isomorphism between an r.i. function space X on [0, I] with non-trivial Boyd indices and the spaces X(l2) and Rad X. Complemented subspaces of X with an unconditional basis. Subspaces spanned by a subsequence ofthe Haar system. R.i. spaces with non-trivial Boyd indices are primary.

e. Isomorphisms Between r.i. Function Spaces; Uniqueness of the r.i.

129

150

168

Structure . 181 Isomorphie embeddings. Classification of symmetrie basic sequences in r.i. function spaces of type 2. Uniqueness of the r.i. structure for Lp(O, I) spaces and for (q<2)concave r.i. spaces. Other applications.

f. Applications of the Poisson Process to r.i. Function Spaces The isomorphism between Lp(O, I) and Lp(O, 00)nL2(0, 00) for p>2. R.i. function spaces in [0, 00) isomorphie to a given r.i. function space on [0, I]. Isometrie embeddings of L,(O,I) into Lp(O,I) for I ~p<r<2 and in other r.i. function spaces. The complementation of the spaces X p • 2 in Lv<O, I), p > 2 and generalizations.

g. Interpolation Spaces and their Applications . Interpolation pairs. General interpolation spaces and applications to the construction of r.i. function spaces without unique r.i. structure. The Lions-Peetre method of interpolation. Uniform convexity and type in interpolation spaces.

References

Subject Index

202

215

233

239

1. Banach Lattices

a. Basic Definitions and Results

The function spaces which appear in real analysis are usually ordered in a natural way. This order is related to the norm and is important in the study of the space as a Banach space. In this volume we study partially ordered Banach spaces whose order and norm are related by the following axioms.

Definition La. 1. A partially ordered Banach space X over the reals is called a Banach laltice provided

(i) x ~Y implies x + Z ~y + z, for every x, y, Z E X, (ii) ax ~ 0, for every x ~ 0 in X and every non negative real a.

(iii) for all x, y E X there exists aleast upper bound (I. u. b.) x v y and a greatest lower bound (g.l.b.) x /\ y,

(iv) "xii ~ Ilyll whenever lxi ~ Iyl, where the absolute value lxi of x E X is defined by lxi =x v ( -x).

Observe that in (iii) above it is enough e.g. to require the existence of the I.u.b. The greatest lower bound can then be defined by x /\Y= -« -x) v (- y» (or by x /\ Y = X + Y - x v y). It follows from (i), (ii) and (iii) that, for every X,y,zEX,

Ix-yl=lxvz-yvzl+IX/\z-y/\zl,

and thus, by (iv), the lattice operations are norm continuous. It is perhaps worthwhile to make a comment concerning the proof of the preceding identity. Its deduction from (i), (ii) and (iii), while definitely not hard, is not completely straightforward. On the other hand, it is trivial to check the validity ofthis identity if x, y, and z are real numbers. We shall prove below (cf. l.d.l and the discussion preceding it) a general result which asserts, in particular, that any inequality (and thus also any identity) which involves lattice operations and algebraic operations (i.e. sums and multiplication by scalars) is valid in an arbitrary Banach lattice ifit is valid in the realline.

The continuity of lattice operations implies, in particular, that the set C= {x; x E X, x~O} is norm closed. The set C, which is a convex cone, is called the

2 1. Banach Lattices

positive cone of X. F or an element x in a Banach lattice X we put x + = x v ° and x_ = -(x /\0). Obviously,x=x+ -x_ (and thilsX=C- C)and Ixl=x+ +x_. Two elements x, Y E Xfor which lxi /\ lyl=O are said to be disjoint.

Every space with a basis {Xn }:'= l' whose unconditional constant is equal to 00

one, is a Banach lattice when the order is defined by L anxn ~ ° if and only if n=l

an ~ 0, for all n. This order is called the order induced by the unconditional basis. In the sequel, whenever we consider an abstract space with an unconditional basis as a Banach lattice, the order will be defined as above unless stated otherwise. For a general space with an unconditional basis endowed with the order defined above, axioms (i), (ii) and (iii) of 1.a.l always hold but (iv) has to be replaced by

(iv') there exists a constant M such that Ilxll ~ Mllyll whenever lxi ~ lyl.

As in the case of aspace with an unconditional basis, every partially ordered Banach space satisfying (i), (ii), (iii) and (iv') can be renormed, by putting Ilxllo=sup {lIylI; lyl~lxl}, so that it becomes a Banach lattice.

There are many important lattices which are not induced by an unconditional basis. Clearly, every LiJl.) space, 1 ~p~ 00 and every C(K) space is a Banach lattiee with the pointwise order. Unless Jl. is purely atomic (and u-finite), respectively, K is finite, these lattices are not induced by an unconditional basis. The separable Banach lattices L 1(0, I) and C(O, I) do not have an unconditional basis (in fact, they do not even embed in aspace with an unconditional basis, cf. l.l.d.I). The spaces LiO, 1), I <p< 00, have an unconditional basis, namely the Haar basis (cf. 2.c.5 below), but the natural order in Lp(O, 1) (i.e. the pointwise order) is completely different from the order induced by the basis.

Every Banach lattice X has the so-called decomposition property: if Xl' X2 and y are positive elements in Xandy~xl +x2 then there are O~Yl ~Xl and 0~Y2~X2 such that y = y 1 + Y 2· This property is easily checked if we take y 1 = Xl /\ Y and y 2 = Y - Y 1· The converse is not true in general: there exist partially ordered Banach spaces having the decomposition property which are not lattiees.

A linear operator T from a veetor lattiee X (i.e. a linear space satisfying (i), (ii) and (iii) of l.a.l) into a vector lattice Y is called positive if Tx ~ 0 for every x ~ ° in X. It is clear that a positive operator T from X to Y, whieh is one to one and onto, and whose inverse is also positive, preserves the lattice structure, i.e.

for all Xl' x 2 EX. Such an operator is called an order preserving operator or an order isomorphism. Two vector lattiees X and Yare said to be order isomorphie if there is an order isomorphism from X onto Y. For example, a normalized unconditional basis {xn} ~ 1 in a Banach space Xis equivalent to a permutation of a normalized unconditional basis {Yn}~l in a Banach space Yifand only if X, with the order induced by {xn}~ l' is order isomorphie to Y, with the order induced by {Yn}:'=l. Observe that a positive linear map T between Banach lattices is automatically continuous. Indeed, otherwise there would exist a sequence {xn }:,= 1

such that IIxn 11 = 2 - n and 11 TXn 11 ~ 2n, for all n, but this contradicts the fact that

a. Basic Definitions and Results 3

IITXnll~IITj~IIXjlll, for n=1,2, ... In particular, an order isomorphism be

tween Banach lattices is also an isomorphism from the linear topological point of view. The Banach lattices X and Yare said to be order isometrie if there exists a linear isometry T from X onto Y which is also an order isomorphism.

By a sublattiee of a Banach lattice X we mean a linear subspace Y of X so that x v Y (and thus also x A Y = X + Y - x v y) belongs to Y whenever x, Y E Y. Unless stated explicitly otherwise, we shaB assurne that a sublattice is also norm closed. Among the sublattices of a Banach lattice X we single out the ideals. An ideal in Xis a linear subspace Y for which Y E Y whenever lyl ~ lxi for some x E Y. (Again, unless stated otherwise, we assurne that it is also norm closed.) If Y is an ideal in X then the quotient space X/Y becomes a Banach lattice if we take as its positive cone the image of the po.sitive cone of X. It is easily checked that TX I v TX2 = T(x i v x 2) for every Xl' X2 E X, where T: X ~ X/Y denotes the quotient map. In order to verify that (iv) of 1.a.l holds in X/Y we have to show that inf {llx l - yll; Y E Y} ~ inf {llx2 - yll; Y E Y}, whenever O~XI ~X2. This is done as folIows. Let Y E Yand observe that Xl - Y~(XI - y)+ ~XI + Y _. Consequently, since Y is an ideal, (xI-Y)+=XI-z, for some ZE Y. Since 0~XI-Z~(X2-Y)+ we deduce that Ilx l - zil ~ IIx2 - yll·

If {xa}aEA is a set in a Banach lattice we denote by V xa or by l.u.b.{xa}aEA aEA

the (unique) element x EX which has the following properties: (1) x~ Xa for aB IX E A and (2) whenever Z E X satisfies z~xa for all IX E A then Z~X. Unless the set A is finite, V xa need not always exist in a Banach lattice. An ideal Y in a Banach

aEA lattice Xis called a band if, for every subset {Ya}aEA of Y such that V Ya exists in X,

this element belongs already to Y. __ The dual X* of a Banach lattice X is also a Banach lattice provided that its

positive cone is defined by x* ~ 0 in X* if and only if x*(x) ~ 0, for every x ~ 0 in X. It is easily verified that, for any x*, y* E X* and every x ~ 0 in X, we have

(X* v y*)(X) = sup {x*(u)+ y*(x-u); O~u~x}

and

(X* Ay*)(x)=inf {x*(v)+y*(x-v); O~v~x}.

The Banach lattice X* has the property that every non"empty order bounded set ffP in X* has a l.u.b. In order to prove this fact we first replace ffP by the family qj of aB suprema of finite sub sets of ffP. The set qj is upward directed, order bounded and has a 1.u.b. ifand only if ffP has a 1.u.b. For every x~O in Xwe put fex) = sup {x*(x); x* E qj}. It is easily checked thatfis an additive and positively homogeneous functional on the positive cone of X and thus it extends uniquely to an element of X*. Clearly, this element is the l.u.b. of qj.

Since every x* E X* can be decomposed as a difference of two non-negative elements, it follows that every norm bounded monotone sequence {xn }:."= I in Xis


weak Cauchy. If, in addition, x" ~ x for some x E X then IIx,,- xll--+ 0 as n --+ 00.

This is a consequence of the fact that weak convergence to x implies the existence of convex combinations of the x,,'s which tend strongly to x.

Proposition l.a.2. The canonical embedding i of a Banach lattice X into its second dual X** is an order isometry from X onto a sublattice of X**.

Proof It is obvious that i is a positive operator. What we have to show is that ix v iy= i(x v y), for all x, y E X. We prove this first under the assumption that x I\Y=O. For every u* ~O in X*, we have

(ix v iy)(u*)=sup (ix(v*)+iy(u*-v*); O::::;;v*::::;;u*}

=sup {u*(y)+v*(x-y); O::::;;v*::::;;u*}.

By putting w*(z) = sup u*(z 1\ nx), for each z~ 0 in X, we define a bounded linear

" functional w* E X* (the linearity of w* is a consequence of the identity (a + b) 1\ C

::::;;al\c+bl\c::::;;(a+b)1\2c, which holds for all a, b, c~O in X). The functional w* satisfies O::::;;w*::::;;u*, w*(x)=u*(x) and (since xI\Y=O) w*(y)=O. It follows that (ix v iy)(u*)~u*(y)+w*(x- y)=u*(x+ y)=u*(x v y)=i(x v y)(u*), forevery positive u* E X*. Hence, ix v iy ~ i(x v y) and, by the positivity of i, we deduce that ix v iy = i(x v y).

Assume nowthatx,y are arbitrary elements in X. Put u=x-x I\y, v= y-x I\Y. Then u 1\ v = 0 and hence, iu v iv = iu + iv. Consequently, iu 1\ iv = 0 and thus ix 1\ iy = i(x I\Y), which concludes the proof. 0

In general, iX is not an ideal of X**. We shall present in l.b.16 below a necessary and sufficient condition for iX to be an ideal of X**.

Definition l.a.3. A Banach lattice Xis said to be conditionally order complete (0'order complete) or, briefly, complete (O'-complete) if every order bounded set (sequence) in X has a l.u.b.

The discussion preceding l.a.2 shows that every Banach lattice X, which is the dual of another Banach lattice, is complete. In particular, every reflexive lattice is complete. The simplest examples of concrete complete Banach lattices are the LiJl) spaces with I ~p~ 00 (though L l (0, 1) is not a conjugate space). Banach lattices generated by unconditional bases are also complete; the supremum can be taken coordinatewise. On the other hand, C(O, I) is not O'-complete. In fact, we have the following result, due to H. Nakano [103] and M. H. Stone [122].

Proposition l.a.4. (i) The space eCK) of all continuous functions on a compact Hausdorff topological space K is a O'-complete Banach lauice if and only if K is basically disconnected, i.e. the closure of every open Fa-set in K is open.

(ii) The space C(K) is a complete Banach lauice if and only if K is extremally disconnected, i.e. the closure of every open set in K is open.


Proof The proof of both assertions is similar. We shall present here only the proof of (i).

Assurne that C(K) is a-complete. Let {En}:'= 1 be a sequence of closed sub sets 00

of K so that E= U En is open. For every integer n we construct a function n=l

fn E C(K) such thatfn(t) = 1 for t E En,fn(t) =0 for t ~ E and 0 :0'n(t) ~ 1 whenever t E K. Since the sequence {In} ~ 1 is order bounded by the function identically

00

equa1 to 1 on K there exists f= V In E C(K). It is clear that f(t) = I for tEE n=l

andf(t)=O for t ~ E. Hence, the set E is both open and closed. Conversely, suppose that K is basically disconnected. For every fE eCK)

put Ef ().) = {t J(t) < ).} and observe that Ef ()") is an open Fa-set since Ef ()") = 00

U {t;f(t)~)"-1/n}. Let {gn};:'= 1 be a bounded sequence of elements of C(K). n= 1

00

By our assumption on K the set n (EgJ)..)) is a closed Gb-set. Hence, its comple-n=l

ment is an open Fa-set whose closure ~ust be open. It follows that the set E()") = 00

int n (Egn()")) is both open and closed. n=l

Put

go(t)=sup {A; t ~ E()")}.

Notice that the above supremum exists since {gn}:= 1 is bounded and that go is continuous on K for both sets

{t; go(t)<)..} = U E(Il) 11<).

and

{t; go(t»)..} = U (K~E(Il)) 11>).

are open. The function go is the l.u.b. of {gn}~l' Indeed, since E()")cEgJ)..), n= 1,2, ... we get that gn~go for al1 n and if gn ~ h, n= 1, 2, ... for some h E C(K) then

00 00

Eh()..)c n Egn()")c n EgP)· n=l n=l

In view of the fact that Eh().) is open it follows that Eh ()..) cE()") for every real ).. i.e. h~go' 0

It should be pointed out that no infinite compact metric space K is basically disconnected. The simplest example of an extremally disconnected space is ßN,

6 I. Banach Lattices

the Stone-Cech compactification of the integers. It is, of course, easy to check directly that 100 = C(ßN) is indeed a complete Banach lattice. A simple example of a a-complete C(K) space, which is not complete, is the subspace of loo(r), with r uncountable, spanned by the constant function and the functions with countable support.

The following fact, due to Meyer-Nieberg [98], is very useful in applications.

Theorem 1.a.5. A Banach lattice which is not a-complete contains a sequence of mutually disjoint elements equivalent to the uni! vector basis of co.

Proof Let {xn };:'= 1 cX be an order bounded sequence which does not have a

l.u.b. By replacing {Xn}n~l by the sequence t~l Xi}:l we can assume with no

loss of generality that O~Xl ~X2 ~ ... ~xn ~ ... ~x, for some x EX. If {Xn}:'= 1

converges in norm to an element of X then, obviously, this element is also the l.u.b. of {Xn}~I. Otherwise, there is an 0.:>0 and a subsequence {XnJ~l of

j

{Xn};:'=1 so that the vectors Uj=xnj + 1 -xnj satisfy IIUjll~o.:, Uj~O and I Uk~X k-l ~~j -

We claim now that, for every e>O and every ß>O, there exists a subsequence {vdf=l of {Uj}~1 so that II(vk- ßv1)+11 ~o.:-e for all k> 1. Indeed, if this is not true then there is a subsequence {wk}f= 1 of {Uj}~ 1 such that II(wk - ßwj )+ 11< 0.:- e for all k > j. It follows that, for any k, we have

Ilxll~IIJl w;II=ß-lllkWk+l- itl (Wk+1- ßWJ II

=ß-lllkWk+l- Jl (Wk+1-ßWJ++ Jl (Wk+l-ßW;LII· k

Since kwk+ 1 ~ L (Wk+ 1 - ßw;)+ we get that ;= 1

and this is contradictory for large values of k. Now, fix 0<e<0.:/2 and construct a subsequence {Vk}~1 of {Uj}~1 so that

II(vk - ßv1)+ 11 ~ 0.: - e for all k> 1, where ß = 21IXII/e. Put Yl = ß - l(ßV l - x)+ and h=(vk- ßv1)+ for k> 1. It is clear that YlI\Yk=O for every k> 1. By the choice of the sequence {vk}f=1 we also get that Yk~Vk~X, IIYkll~o.:-e for k>l, and IIYll1 = II(vl - P-lx)+11 ~ Ilv111-ß- 1 1Ixll-ll(vl - ß-lx LII ~o.:-e.

Applying again this argument to the sequence {Ydf= 2' instead of {u j } 1= l' and with e/2, instead of e, we can produce a new subsequence for which the norms of its elements are ~ 0.: - e - e/2, each element is ~ x and the first two elements are mutually disjoint and also disjoint from the rest of the sequence. Continuing by induction we obtain a sequence {Zk }f= l' of mutually disjoint elements of X,


so that IIZk II ~IX- 28 and Zk ~x for all k. This sequence is clearly equivalent to the unit vector basis of CO. 0

The converse of l.a.5 is evidently false since e.g. Co itself is er-complete.

Definition l.a.6. A Banach lattice Xis said to have an order continuous norm (er-order continuous norm) or, briefly, to be order continuous (er-order continuous) if, for every downward directed set (sequence) {x«}«eA in X with /\ x«=O,

«eA lim Ilx«11 = o.

«

A simple example of a er-order continuous Banach lattice, which is not order continuous, is the subspace of 100 er) spanned by co{r) and the function identically equal to one, where r is an uncountable set. Typical examples of order complete Banach lattices, which are not er-order continuous, are '00 and Loo CO, I).

Proposition l.a.7 [85]. A er-complete Banach lattice X, which is not er-order continuous, contains a subspace isomorphic to 100 • Moreover, the uni! vectors 01100

correspond, under this isomorphism, to mutually disjoint elements 01 x.

Proo! Assurne that {Xn }:'=l is a non-convergent decreasing sequence in X with 00

/\ xn = o. The sequence {Xl - Xn }:'= 1 is increasing, order bounded and not n=l

strong Cauchy. It follows from the proof of l.a.5 that there exists a sequence {Zdf=l' ofmutually disjoint elements in X, which is equivalent to the unit vector basis of Co and for which O<Zk~X1> k~1. For a={ak}k"= 1 'Eloo , with ak~O for

00

every k, we put Ta= V akzk (the supremum exists since X'is er-complete and k=l

akzk~xl sup laml for every k). It is easily checked that T is an isomorphism l::5;m<oo

from the positive cone of (X) into that of X which extends uniquely to an isomorphism from '00 into X. 0

The result 1.a.7 implies, in particular, that a separable er-complete Banach lattice is er-order continuous. It is easily seen from the definition that a separable er-order continuous Banach lattice is already order continuous. Thus, every separable er-complete Banach lattice is order continuous. The converse to this assertion is also true even without the separability assumption. Every order continuous Banach lattice is also order complete. This is the main assertion of the following proposition.

Proposition l.a.8. Let X be a Banach lattice. Then the lollowing assertions are equivalent.

Ci) Xis er-complete and er-order continuous. (ii) Every order bounded increasing sequence in X converges in the norm topology

o/x.


(iii) X is order continuous. (iv) X is (order) comp/ete and order continuous.

Proof The equivalence (i) <? (ii) follows direct1y from the definitions (that (ii) <? (i) was used already in the proofs of l.a.5 and l.a.7). We have just to prove that (ii) => (iii) and that (iii) => (iv).

(ii) => (iii): Let {X~}aEA be a downward directed set satisfying /\ x~=O. Ifthe net ~EA

{x" LEA does not converge to 0 there are b > 0 and a decreasing sequence {x"J J:= 1

in this net so that IIX"j -X"j+ ,li ~b, for every j, and this contradicts (ii). (iii) => (iv). Let U be an order bounded set in X. By adding to U the finite

suprema ofits elements we may assume that U={X"}"EA is upward directed. Let Vby the set of all upper bounds of U i.e. V = {y; Y E X, Y~ x for all XE U}. Clearly, V is downward directed and so is also V - U= {y-x; Y E V, XE U}. We claim that 0 is the g.l.b. of V - U. Indeed, let Z ~ 0 satisfy z:::;Y - x, for every Y E Vand x E U. Then Y E V => Y - Z E Vand thus, by induction, Y - nz E V for every integer n. Consequently, IIzll = O. Hence, by (iii), there are, for every B> 0, an rx E A and a Y E V so that lIy-x"II:::;B and therefore IIxp-x"II:::;B for every ß>rx in A. Thus, the net {X"LEA converges to a limit x which is the l.u.b. of U. 0

Weshall encounter below several less trivial and more interesting characterizations of order continuous Banach lattices (see l.a.lI, l.b.14 and l.b.l6). Condition (i) of l.a.8 is however the easiest to check characterization of order continuity in concrete examples.

In the study of spaces with an unconditional basis {xn } ~ 1 the projections P (1

(with a being a subset of the integers), defined by,

playa fundamental role. A natural generalization of these projections is possible in every a-complete Banach lattice.

Let X be a a-complete Banach lattice ; to every x ~ 0 we associate a projection Px in the following way. For z~O in Xwe put

00

Px(z)= V (nxAz) n= 1

and, for a general y= y + - y _ EX, we set

P x (y) = P x (y +) - P x (y -) .

It is easily verified that, for every x i~ the positive cone of X, Px is a norm one positive linear projection. Note also that, for every x, y ~O in X, we have


To understand better the definition above we should point out that in the case when Xis a cr-complete Banach lattice of functions with the pointwise order, the projections {Px}x~o are just "multiplications" by characteristic functions. For instance, in X = LpCp), 1 ~p ~ 00, the projection Px is the multiplication operator by the characteristic function of the support of the function XE LiJl.), i.e. {t; x(t):;eo}.

Using these projections we can decompose Banach lattices into a direct sum of ideals with a weak unit. An element e ~ 0 of a Banach lattice X is said to be a weak unit of X if e A x = 0 for x E X implies x = 0 (there is also a notion of strong unit which will be mentioned in the next section in connection to the study of M spaces). In a cr-complete Banach lattice X an element e is a weak unit if and only if Pix)=xforevery x EX. Indeed, for every x~O in X, we have eA (x-Pe(x»=O. Therefore, if e is a weak unit then x-Pe(x)=O. Conversely, eAx=O clearly implies x=O ifwe assume that x=Pix) for every XEX.

Proposition l.a.9 [63]. Any order continuous Banach lattice X can be decomposed into an unconditional direct sum of a (generally uncountable) family of mutually dis joint ideals {XIX}IXEA' each XIX having a weak unit xlX>O. More precisely, every y E X has a unique representation of the form y = L y IX with y IX E XIX' only countably

IXEA many YIX:;eO and the series converging unconditionally. Moreover, if Z is a separable subspace of X then one of the indices IX, say 1X0, can be chosen so that Z c X IXO '

Proo! By Zorn's lemma there exists a maximal family {XIX}IXEA of mutually disjoint positive elements of X. Let XIX be the set of all x E X such that lxi AY=O whenever XIX A Y = O. It is easily checked that XIX = P x«X' that it is an ideal (even a band) of X and that XIX is a weak unit for XIX'

Fix y~O in X and consider the set {Px«(y)}IXEA' Then, by l.a.8, for any 00

countable sub set Ao={1X1, 1X2 , ... , IXn , ... } of A, the series L Px (y) converges n= 1 IXn

00

strongly to V Px (y). This implies that only countably many ofthe {Px (y)LEA n= 1 Cl n 01:

are different from zero and that L P xJy) converges unconditionally to an IXEA

element Yo of Xwhich clearly satisfies Yo ~y. If y-Yo>O then, by the maximality of the family {XIX}IXEA' there is at least one index IX E A such that XIX A (y- yo):;eO. This is however a contradiction since 0 ~ XIX A (y - Yo) ~ XIX A (y - P xJy» = O.

If Z is a separable subspace of X then Z is contained in the direct sum of at most 00

a countable number of the XIX's, say {X1X1 ' XIX2 , ... }. By taking x lXO = L x lX j2"lIxlXn ll n=1

we can replace the indices {lXn }:'= 1 by the single index 1X0 and get that Z is contained in the ideal XIXO generated by x IXO' D

Proposition l.a.9 shows that, for each ofthe bands XIX' we have X=XIX E9 X;, where XlXl. is the set of all x E X which are disjoint from every y E XIX' A band Y of a Banach lattice X is called a projection band if

X= YE9 Y\

10 I. Banach La ttices

where y.l = {x EX; lxi" I Yl = 0 whenever Y E Y}. The set y.l is also a band of X, called the polar of Y. The (positive) projection P y from X onto Y, which vanishes on y.l, is called a band projection. It is easily seen that in the case when Xis a a-complete Banach lattice and {pJx;'o are the projections associated to X as above then Px is a band projection whose range is the band generated by x i.e. the set of all u E X for which lul" Y = 0 whenever X" Y = O. In general, there exist band projections other than the {Px}x;'o (e.g. in Lp(fl), 1 ~p< 00, with fl being a non a-finite measure). There are also bands which are not projection bands (e.g. the subspace of C(O, 1) consisting of all the functions which vanish on [0, 1/2]). A simple characterization of those bands which are projection bands is given by the following result.

Proposition l.a.l0. A band Y oJ a Banach lattice X is a projection band if and only if, Jor every x ~ 0 in X,

Py(x) = V{y E Y; O~y~x}

exists in X. In this case, x=Py(x)+Py.L(x), where Py(x) E Y, Py.L(x) E y.l and Py.L(x) = V{z E y.l; O~z~x}.

Proo! If Y is a projection band and 0 ~ x = u + v with u E Yand v E y.l then, for every YE Ysatisfying O~y~x, we have x-y=(u-y)+v. Since x-y~O we get that u-y~O, i.e. u= V {y E Y; O~y~x} =Py(x). Similarly, we also get that v= V {z E Y\ O~z~x} =Py.L(x).

Conversely, if Py(x) exists in X for every x~O in X then, since Yis a band, we obtain that Py(x) E Y. Put w=x-Py(x) and take any y~O in Y. The element w"y also belongs to Y and O~w"y~x-Py(x). Therefore, Py(x)+w"y~x which, in view of the definition of Py(x) as a supremum, implies that w"y=O, i.e. w E y.l. 0

It folIo ws, in particular, that in a complete Banach lattice every band is the range of a positive contractive (Le. of norm one) projection. If we require that the same holds for every ideal then we get a property which characterizes order continuous Banach lattices.

Proposition l.a.ll (Ando [3]). A Banach lattice X is order continuous if and only if every ideal oJ Xis the range oJ a positive projection Jrom X.

Proo! Assurne first that X is order continuous and thus also order complete by l.a.8. Since every ideal in Xis, by definition, a closed subspace we get that it is also a band and, by l.a.IO, there is a positive contractive projection on it.

Conversely, let X be a Banach lattice in which every ideal is the range of a positive projection. We prove first that every ideal Y in Xis a projection band. Let P be a positive projection from X onto such an ideal Y and let x ~ 0 be an


element in X. It is enough to show that (x - Px) + E Y J.. since then

x=Px-(Px-x)+ +(x-Px)+ E Y+ yJ...

If (x - Px) + ~ Y J.. then, since Y is an ideal, there would exist a Y E Y with O<y~(x-Px)+. Then y=Py~Px and hence y~(x-Px)+ ~x-Y, i.e. 2y~x. An easy induction argument shows that ny ~ x, for every integer n, and this clearly leads to a contradiction.

Let now {X~}~EA be a downward directed set in X with /\ x~=O. We may ~EA

clearly assurne that x~ ~ x, for some x E X and all cx E A. Let e > 0 and, for cx E A, let P~ be the band projection on the ideal generated by (x~-ex)+ (which is a projection band by the first part of the proof). Since (I - P~)(x~ - ex) = - (XIX - ex) _ ~ 0 it follows that

Thus, in order to prove that Ilx~11 ! 0, it suffices to show that IIP~oxll ~ e for some CXO•

Put y ~ = (1- P ~)x, cx E A. Since x~ ~ P ~x~ ~ eP ~x it follows that /\ P ~x exists ~EA

and is equal to 0= /\ x~ and hence, x= V y~. Consequently, by the first part ~EA ~EA

ofthe proof, x belongs to the ideal generated by {Y~}~EA. Using the fact that this set is directed upward it follows that there is a Z E X, a constant M and an CXo E A so that IIx-zll ~e and O~z~My~o· Since xl\z~xI\My~o=(P ~x+ y~)I\My~o ~y~o we get that

We introduce next a notion which was originally considered in connection with multiplicity theory for spectral operators.

Definition 1.a.12. (i) A family !J4 of commuting bounded linear projections on a Banach space Xis called a Boolean algebra (B.A.) ojprojections if

P, Q E!J4 => PQ, P+Q-PQ E!J4.

A B.A. of projections !J4 will be ordered by P ~ Q ~ P = PQ. With this order !J4 is a lattice with P 1\ Q=PQ and Pv Q=P+Q-PQ.

(ii) A B.A. of projections !J4 is said to be u-complete if, for every sequence 00 00

{Pn }:'=l in!J4, V Pn and /\ Pn exist in!J4 and, moreover, n=l n=l


(iii) With X and f!4 as in (i) and x E X the subspace vfl (x) = span {Px; P E f!J} of X is called the cyclic space genera ted by x. It is the smallest c10sed linear subspace of X which contains the vector x and is invariant under f!J. If there exists a vector x such that X = vfl (x) then x is ca11ed a cyclic vector and X a cyc1ic space (with respect to f!J).

A simple example of a B.A. of projections is the family tff of a11 multiplication operators on X = L,,(p.), 1 ~p ~ 00, by characteristic functions of measurable sets. If pis finite this B.A. is O"-complete. It is also evident that the space Xis cyc1ic with respect to tff if and only if p. is a O"-finite measure. In this case one can choose as a cyc1ic vector in X any functionfE Lp(p.) for which p.({ t;J(t) =O})=O. If Xis aspace with an unconditional basis {xn }:'= 1 the set of a11 natural projections P (1 ; 0" being a subset of the integers, is again a O"-complete B.A. of projections and Xis always a

00

cyc1ic space. As a cyc1ic vector in this case we can take any vector x = L anxn ,

n= 1

with an#- 0 far a11 n. The previous examples are just two particular cases of the following result of general character [44], [63].

Theorem l.a.13. Let X be an order continuous Banach lauice with a weak unit e>O (in particular, X can be any separable O"-complete Banach laltice). Then the family {PJx;'o of projections on X forms a O"-complete B.A. of projections and X = A(e) is a cyclic space with respect to this B.A.

Proof The first part ofthe statement is proved by direct verification ifwe observe that (i) Pe acts as the identity operator on X, (ii) for x, y~O, PXPy=Px/\y (iii) for x~ y~O we have Px - P y = PX-Py(X) and (iv) the O"-completeness of {P xL;.o follows from the O"-completeness and O"-order continuity of X. In order to prove that X = vfl(e), fix x ~O in X and, für any real A ~O, put X(A) = P(Ae-X) + (e). If A ~.rl then ()~e-x)+ ~ (IJe-x)+ which implies that X(A) ~x(!1). This means that x(·) is a nondecreasing map from [0, (0) into the positive cone of X. As in the case üf the

00

scalar Riemann-Stieltjes integral we can consider the integral J A dX(A) with o

respect to the X-valued measure dx(·). This integral should be understood n

as the limit in norm, if it exists, of sums of the form L Ai(X(AJ-X(Ai_J). i= 1

00

We shall prove that x= J A dX(A) and this will imply that XE A(e). For this we o

need the fo11owing lemma.

Lemma 1.a.14. For every element O~z EX which satisfies z = Pz(e) (or equivalently, z /\ (e- z) = 0) the following two assertions are true:

(i) Ifz~x(A)for some A~O then Pz(X)~AZ. (ii) Ifz~e-x(A)for some A~O then AZ~Pz(X).

Proof Observe that z ~ X(A) implies that 00

O~(x-Ae)+ /\z~(x-Ae)+ /\p(Ae-x)+(e)=(X-Ae)+ /\ V (n().e-x)+ /\e)=O. n=l


It follows that Pz«X-Ae)+)=O and, since X~(X-Ae)+ +Ae, we also get that Pz(x)~PZ«x- Ae)+) + APAe) =AZ. The proof of (ii) is similar. 0

In order to complete the prooj oj l.a.l3 we fix e > 0 and A < 00, and let A =

{0=Ao<A1 < ... <An=A} be a partition of the interval [0, A] such that max (Ai-Ai- 1)<e. Put Zi=X().i)-X(Ai_1) and observe that zi=Pz.(e) and

1 :::;i~n 1

zi~e-x(Ai_l)· Since we also have Zi/\Zj=O, whenever i"l=j, it follows from l.a.14(ii) that

n n

s(A)= i~l Ai-1Zi~ i~l Pz,(x)=P f z, (X)=PX(A)(X). i=l

On the other hand, Zi ~ X(Ai) and therefore, by l.a.14(i),

n n

PX(A)(X)= L Pz,(x)~ L AiZi=S(..:1) . i= 1 i= 1

However, S(..:1)-s(..:1)~ex(A)~ee which implies that IIS(..:1)-s(..:1)II~ellell. Thus, A

by fixing A and letting e --+ 0, we get that J A dX(A) = Px(A)(X) for every A. Since X o

is u-complete and x(A) ~ e for every A we obtain that x( (0) = lim x(A) exists A .... oo

in X. But e - x( (0) satisfies the condition of 1.a.14 and, clearly, e - x( (0) ~ e - x(A) for every A. Thus, again by l.a.14(ii), it follows that A(e-x(oo»~p(e_x(oo»(x), Le. Alle-x(oo)II~IIP(e-x(oo»(x)ll. Since A is arbitrary we get that x(oo)=e i.e. x(A) --+ e as A --+ 00. Hence, by the u-completeness of the B.A. of projections {Px}x~o, we conclude that PX(A)(x) --+ Pe(x)=x as A --+ 00. Consequently, 00

J A dX(A)=X. 0 o

Remarks. 1. A variant of Theorem l.a.13 holds for Banach lattices without a weak unit. By combining 1.a.13 with 1.a.9, we get that, for any order continuous Banach lattice X, the projections {PJx~o form a u-complete B.A. of projections on every cyclic subspace of X and Xis an (not necessarily countable) unconditional direct sum of cyclic subspaces. In general, X itself need not be a cyclic space and {Px}x~o need not be a B.A. on X because it could lack the property of complementation (take e.g. an LiJ.l) space, I ~p ~ 00 where J.l is not au-finite measure).

2. If X is complete but not order continuous it is not true in general that {Px}x~o is a u-complete B.A. of projections. For instance, let Pn be the natural projection in 100 whose range is the closed linear span of the first n unit vectors. Then

00

V Pn exists in {Pxlx~o and is equal to the identity operator. However, n=l


To conclude this section we would like to mention without giving the proof that there is also a converse result to l.a.13. Every cyclic space with respect to a a-complete B.A. oJ projections can be ordered and given an equivalent norm wi!h which it becomes an order continuous Banach lattice with a weak uni!. This fact is due essentially to Bade [4] (see H. H. Schaefer [118] Section V.3 for a concise presentation.) In this paper Bade showed that if vII(x) is a cyclic space then, to every element y in vII(x), there corresponds a functionJ(w) on some fixed measure space so thaty is the integral ofJwith respect to some vII(x)-valued measure. The space of all the functions which correspond in this manner to vectors in vII(x) is a vector lattice which satisfies (with respect to the norm induced by vii (x)) axiom (iv') of the beginning of this section. Thus, after a suitable renorming, Y becomes a Banach lattice. That the space Y in Bade's representation is order continuous (i.e. a-order complete and a-order continuous) is easily verified by using standard results from measure theory. As a weak unit for Y we can take, of course, the function which corresponds to x (namely, the function identically equal to one). In the next seetion (cf. l.b.14), we shall prove a representation theorem for order continuous lattices with a weak unit which is essentially equivalent to Bade's representation.

b. Concrete Representation of Banach Lattices

Many Banach lattices which appear in the literature are, in fact, spaces offunctions; e.g. spaces of continuous functions on some compact Hausdorff space or spaces of measurable functions on a measure space (Q, 1:, j1), with the order defined by J~g ~ J(w)~g(w), for j1-a.e. w E Q. There are many cases when abstract Banach lattices can be represented as concrete lattices of functions. Such representation theorems are very convenient since they facilitate, e.g., the application of many results of measure theory to the study of Banach lattices.

The best known theorems in this direction are those of S. Kakutani [63], [64] on the concrete representation of the so-called abstract L p and M spaces. We present first these resuIts. The representation theorems of Kakutani are followed by several results (of an isometrie as weIl as of an isomorphie nature) which give joint characterizations of the abstract L p spaces and M spaces among general Banach lattices. We then prove a functional representation theorem for general order continuous Banach lattices with a weak unit. Such lattices can be represented as suitable spaces of measurable functions on a measure space (Köthe function spaces). The section ends with abrief discussion of general properties of Köthe function spaces.

Definition l.b.1. (i) Let 1 ~p < IX). A Banach lattice X for which Ilx + ylI p = IIx Il P + lIyllp, whenever x, y E X and x l\y=O, is called an abstract L p space. (ii) A Banach lattice X for which IIx+ ylI = max (lIxii, lIyll), whenever x, y EX and x 1\ y = 0, is called an abstract M space.

b. Concrete Representation of Banach Lattices 15

It is obvious that every L/J1) space is an abstract L p space if p < 00 or an abstract M space if p= 00. The converse is also true if p < 00 (cf. S. Kakutani [63]).

Theorem l.b.2. An abstract L p spaee X, 1 ~p< 00, is order isometrie to an L p{J1) spaee over some measure spaee (Q, 1:, J1). lf X has a weak uni! then J1 ean be chosen to be a finite measure.

Proof By l.a.5 and l.a.7, Xis a-complete and a-order continuous. Thus, by l.a.l3 and l.a.9 (see also Remark 1 following l.a.I3), X is an unconditional direct sum of mutually disjoint cyclic spaces XIX' (X E A (with respect to the family f!l of projections {P x} x;' 0). By the p-additivity of the norm we get that Xis actually

a direct sum in the lp sense of cyclic spaces, i.e. X = (L EB XIX) . It is therefore IXEA p

sufficient to show that each XIX is order isometric to some L p (J1 IX ) space. For simplicity of notation we shall assurne that X has a weak unit e> 0 with

Iiell = 1, i.e. that X itself is a cyclic space .I/(e) with respect to the B.A. of projections f!l= {PxL;.o. We want to construct a probability space (Q, 1:, J1) and an order isometry T of X onto Lp(Q, 1:, J1) such that Te is the function identicallY equal to one. This is achieved by using the following well-known representation theorem, due to M. H. Stone [121].

Theorem l.b.3. Every Boolean algebra with a uni! is isomorphie to the Boolean algebra of all simultaneously open and closed subsets of a totally diseonneeted eompaet Hausdorff spaee.

A proof of this classical result can be found, e.g. in [32] 1.2.1. In view of l.b.3 we shall identify f!l with the B.A. 1:0 of all simultaneously

open and closed sub sets of a totally disconnected space and, if a projection Px E [Jl corresponds to a set a E 1:0, we shall write P" instead of Px. By putting, J1(a) = IIP,,(e)lIp for a E 1:0 we clearly define an additive measure on (Q,1:o). Actually, the measure J1 is vacuously a-additive on (Q, 1:0 ) since, by the fact that every set in 1:0 is both open and compact, no set in 1:0 can be expressed as a union of infinitely many mutually disjoint non-void sets from 1:0 . Thus, by the Caratheodory extension theorem (e.g. cf. [32] III.5.8), J1 has a a-additive extension, still denoted by J1, to the a-fie1d 1: of subsets of Q generated by 1:0 . This extension has the additional property that every (j E 1: differs from some set a E 1:0 bya set of J1-measure zero (i.e. that, up to sets of J1-measure zero, 1:0 is a a-field). Indeed, suppose that some set (j E 1: is the union of an increasing sequence {an}:~ 1 of elements of 1: o. Then, by the a-completeness and a-order continuity of X and l.a.13,

00

V P"n=P" exists in f!l and P,,(e) = lim P"n(e). It follows that a~(j (since a~an n=1 n-oo

for all n) and

n .... 00 n .... 00


In order to complete the proof of 1. b.2 we define an order isometry T from X onto Lp(D,];, Ji) in the following way: if {Uj}j=l are arbitrary disjoint sets in];o and {aj}j=l are arbitrary scalars then we put

Since {P"j(e)}j= 1 are mutually disjoint elements of X it follows that

Le. Textends uniquely to an order isometry of X=.ß(e) onto LiD,];, Ji). (Here k

we have used the fact that elements of the form L bjPx.(e) are dense in X since j= 1 J

k

Xis a cyclic space and each expression ofthe form L bjPx.(e) can be written as j= 1 J

m

L ajPyj(e), with {PyJj= 1 being mutually dis joint projections in f!4). 0 j= 1

Corollary lob.4. Any closed sublattiee 01 an Lp{J.l) spaee, l:;;;;"p < 00, is order isometrie to an Lp(v) space,lor a suitable measure v.

Before stating the result on the representation of the abstract M spaces we recall the following classical result of S. Kakutani (and also M. H. Stone) on the structure of sublattices of C(K) spaces (see e.g. the proof of [32] IV.6.16).

Theorem lob.5. Let X be a closed linear subspace 01 a C(K) spaee. Let !F be the eollection 01 all the tri pies (k l , k 2 , ,1) with k l , k 2 E K and ,1 ~O so that

lor all lEX. Then X is a sublattiee 01 C(K) if and only if X eontains every lunetion I E C(K) whieh satisfies (*) lor all triples (k l' k 2 , ,1) in !F.

We also point out that in C(K) spaces the function 1= 1 plays a special role which is clarified by the following definition. An element e > 0 of a Banach lattice X is said to be a strong uni! of X provided that Ilx 11 :;;;;,. 1 if and only if Ix I :;;;;,. e. The space Co is an example of an abstract M space without a strong unit which has, however, a weak unit.

Theorem lob.6. Any abstract M spaee X is order isometrie to a sublattiee 01 a C(K) spaee,for some eompaet HausdorjJ spaee K. Jf, in addition, X has a strong unit then X is order isometrie to a C(K) spaee.


Proof We show first that X* is an abstract Li space. Let x! and x! be two positive elements in X* with x! " x! = O. Fix e > 0 and, for i = 1, 2, let Xi be positive elements of norm one in X so that x1(xi) ~ (1- e)llxill. Since x! "x! =0 there exist ui ' Vi ~ 0 in X so that

Clearly, x!(vi)~(1-2e)llx!ll, x!(u2)~(1-2e)llx!ll. Put wi =(vi -u2 )+, w2 = (u2 - vi )+ and notiee that wi " w2 =0, IIwdl:::::;; 1 and Xi(Wi)~(1- 3e)llxt'll. i= 1, 2. Since X is an M space it follows that Ilwi + w2 11:::::;; 1, and thus

Since e>O was arbitrary we get that X* is an Li spaee. Henee, by I.b.2, X* is order isometrie to Li (}l), for some JI., and X** is therefore isometrie to Loo(JI.). The spaee Loo(JI.) is, in turn, order isometrie to a C(K) spaee, for some eompaet Hausdorff K. This weIl known fact ean be proved in various ways; the most eommon approach is by using that Loo(JI.) is a commutative C* algebra (cf. e.g. [33] IX.3.7). Sinee, by l.a.2, the canonical image of X in X** is a sublattice of X** the first part of the theorem is already proved.

Suppose now that Xhas a strong unit e. Then, for every O:::::;;x* E X*, we have Ilx*ll= sup {x*(x); O:::::;;XEX, Ilxll:::::;;I}=x*(e). This shows that the function fELl (J1.), whieh corresponds to x* (under the order isometry between X* and Li(JI.)), satisfies J f dJl. = x*(e), i.e. e, eonsidered as an element of X**=Loo(JI.)= C(K), is the function identieally equal to one. It follows from 1. b.5 that the sublattiee X of C(K), whieh contains the function identieally equal to one, is obtained in the following way: the set K is divided into a eertain family of equivalence classes {Kp} peH so that X consists exactly of all those functions in C(K) which are constant on each equivalence class. Thus, if H is the compact Hausdorff space obtained from K by identifying each class Kp to a point P E H, X is order isometrie to C(H). 0

Remarks. 1. The original definition of an abstract Li or M space given by S. Kakutani was slightly different from that presented in I.b.1 in the sense that he required that Ilx+ Yll = Ilxll + Ilyll, respectively Ilx v yll = max (IlxII, Ilyll), be valid for all x and y in the positive eone of X.

2. Y. Benyamini has proved in [8] that every separable abstract M spaee is isomorphie to a C(K) spaee though, obviously, not necessarily isometrie to a C(K) space. He has also shown that there exist non-separable abstract M spaces whieh are not isomorphie to any C(K) space [9].

The classes of lattices introduced in 1. b.l, and concretely represented in 1. b.2 and I.b.6, are the most important lattices which appear in analysis. From the point of view of Banach space theory itself their significance sterns also from the fact that these lattices are the only ones whieh admit an abstract characterization similar in spirit to 1. b.l. Weshali present next several results of an isometrie as weIl as of an isomorphie nature whieh clarify this point. The first result proved in


this direction is due to Bohnenblust [13]. This theorem, which was proved by Bohnenblust at about the same time in which Kakutani proved the representation theorems 1. b.2 and 1. b.6, inspired all the subsequent results in this direction as weH as the analogous results in the setting of spaces with an unconditional basis (cf. section I.2.a).

Theorem l.b.7. Let X be a Banach lattice of dimension at least 3 for wh ich there exists a function F(s, t) (defined on {(s, t); s~O, t~O}) such that, for all x, Y EX with Ixl/\lyl=O, we have Ilx+yll=F(llxll,llyll). Then Xis either an abstract L p space,for some 1 ~p< 00, or an abstract M space.

(Another way to express the assumption on X in l.b.7 is the following: If Xl' X2' Yl' Y2 E X satisfy Ixll/\ IYll = IX 21/\ IY21 =0, IIxlll = IIx211 and IIYlll = IIY211 then IIxl +Y111=llx2+Y211.) Proof It is easily verified that the axioms on the norm in a Banach lattice force the function F to have the following properties.

(1) F(O, 1) = 1 (2) F(s, t) = F(t, s), s, t ~ 0, (3) F(rs, rt)=rF(s, t), r, s, t~O, (4) F(r, F(s, t»=F(F(r, s), t), r, s, t~O, (5) F(sp tl)~F(S2' t2), O~Sl ~S2' O~tl ~t2' (6) F(t, 1-t)~1, O~t~l.

We shall show that (l}-(6) imply that F(s, t) is either (sP+ tP)l/p, for some 1 ~p< 00, or max {s, t}. Define numbers An' n= 1, 2, .. , inductively by Al = 1 and An + 1 = F( 1, An)' Evidendy, {An }:'= 1 is a non-decreasing sequence. By induction on m we get that An + m = F(An, Am)' n, m = 1, 2, ... Indeed, by (2) and (4),

An+m+ 1 = F(l, ~+m) = F(l, F(An, Am»= F(F(1, Am)' An) = F(An, Am+ 1)'

Another simple induction on m proves that AnAm=Anm , n, m= 1,2, ... Indeed, by (3) and the identity proved above,

IfA2 =F(1, 1)=1 then, by(1), (3)and(5),F(s, t)= max {s, t}. If A.2>1 weobserve first that the monotonicity of {A.n}~l and the relation AnAm=Am+n imply that log A.n/10g n is independent of n (use the fact that if hand k(h) are such that mk~nh~mk+l then A~~A=~A~+t, and let h - 00). Hence, An=nl /p for some p which, by (6), must satisfy p ~ 1. Thus,

and consequently, by (3) and (5), F(s, t)=(SP+tP)l/p for s, t~O. 0

The next theorem, due to Ando [3], characterizes the L p spaces and some M spaces by an intrinsic Banach space property.


Theorem l.b.8. Let X be a Banach lattice of dimension;?; 3. Then X is order isometric to L/Jl), for some l:!!!;.p < 00 and measure p., or to co(r), for some index set r, if and only if there is a contractive positive projection from X onto any (closed) sublattice ofit.

The proof of 1. b.8 will be broken up into three lemmas.

Lemma 1.b.9. In an Lp(p.) space, l:!!!;.p < 00, there exists a contractive positive projection onto every sublattice.

Lemma l.b.l0. An abstract M space is an order continuous lattice if and only if it is order isometric to co(r), for some index set r.

Lemma l.b.ll. Let X be a Banach lattice of dimension 3 so that there is a contractive projectionfrom X onto any ofits sublattices of dimension 2. Then X is order isometric to I:, for some 1 :!!!;.p:!!!;. 00.

Let us first show that l.b.8 is indeed a consequence of l.b.9, l.b.lO and l.b.ll.

Proof of l.b.B. The "only if" part of l.b.8 for 1 :!!!;.p< 00 is l.b.9. The "only if" part for co(r) is proved as folIows. By l.b.lO, every sublattice Y of co(r) is the closed linear span of a set {y;.LeA ofvectors in co(r) having the form

y;. = L: ayey, A. E A, YET ..

where ay> 0, {ey} yeT are the unit vector basis of co(r) and {r;.} J.eA are mutually disjoint subsets of r. For every A., let Y;. be such that ay;' = Ily;.lI. A positive contractive projection from X onto Y is given by the formula

whenever L: byey E co(r)· YET

For the proof of the "ir' part of 1. b.8 we note first that, by l.a.ll, X must be an order continuous lattice. By 1. b.II, there is, for any three disjoint positive

vectors {xl' x 2 ' x 3 } ofnorm one in X, al :!!!;.p:!!!;.oo so that Ili#1 aixill =(#1 lailP) l/p

ifp< 00 (respectively, max lail ifp=oo). We show that thisp does not depend on 1 :S:;i~3

the tripIe. To this end we notice first, by an easy induction on n, that, for every n-tuple {y;}7= 1 of disjoint positive vectors of norm one in X, there is a 1 :!!!;.p:!!!;. 00

such that 11 i~1 aiYil1 = (~1 lailP Y/P (prove first that all the tripIes h, Yi2' Yi3'

1 :!!!;.i1 :!!!;.i2 :!!!;.i3 :!!!;.n, have acommonp, and then use the induction hypothesis). Ifnow {xj}j= 1 and {uj}j= 1 are any two tripIes of disjoint positive vectors of norm one


in X then, for every e>O, there is a n-tuple {Yi}7=1 of disjoint positive veetors in X so that the distanee of eaeh of the six given veetors from [Yl]?= 1 is less than e. (By l.a.13, we may assurne that Xis a eyclie space vI/(e) and we ean thus take Yi=Pie, I :::;;i:::;;n, where the {PJ7=1 are suitable disjoint projeetions in the underlying Boolean algebra.) This approximation argument clearly shows that the p assoeiated to {xJ;= 1 is equal to the one associated to {uJ;= l' If the p, whieh is eommon to all tripies in X, is finite then, by l.b.2, X is order isometrie to L/J.l). 1fthis eommonp is CIJ then, by I.b.6 and I.b.lO, X is order isometrie to co(n. 0

ProoJoJl.b.9. Let YbeasublatticeofX=L/D,E,J.l), l:::;;p<oo.Foreveryfinite set B= {J;}7= 1 of disjoint positive vectors of norm one in Y, there is a positive contractive projection PB from X onto [/;]7= 1 =1;, defined by

We partially order the set f!J of finite sets of disjoint positive vectors of norm one in Y by {yJ7= 1 -< {Zj }j= 1 if [yJ7= 1 c: [Zj]j= l' Assurne first that I <p < 00. For every JE X and every BE f!J, the vector PBJ belongs to the w compact subset {Y; Ilyll:::;; IIJII} in Y. Hence, by Tychonoff's theorem, the net {PBheal of operators from X to Y, has a subnet which converges to some limit point P (in the topology of pointwise convergence on X taking in Y the w topology). It is clear that P is a positive contractive projection from X onto Y.

If p= I we note first that, by I.b.2, Y is an L 1(v) space for a suitable positive measure v on some compact Hausdorff space K(K is the one point compactification of the union of a set {Ky } y er of disjoint compact sets so that V!K y is finite for every y). We can thus eonsider Yin a canonical way as a subspace of C(K)*. There is a positive contractive projection Po from C(K)* onto Y; wesimply take as Poyf, where yf is a finite Borel measure on K, its absolutely continuous part with respect to v. We return to the argument given above for p> 1. We consider now each PB as an operator from X into C(K)*. Since the unit ball in C(K)* is w* compact there is a subnet of {PB}Beal which eonverges pointwise on X (taking in C(K)* the w* topology) to a map P from X into C(K)*. Clearly, Pis a positive operator of norm one whose restriction to Yis the identity. Hence, PoP is a positive contractive projection from X onto Y. 0

Before proving 1. b.lO let us introduce the following notion. An element x> 0 is called an atom of a Banach lattice X if {y E X; O:::;;y:::;;x} = {h; 0:::;; Je :::;; I}. It is easily verified that in a 0" -complete Banach lattice an element x> 0 is an atom if and only if x cannot be written as x = Y + Z with y, Z # 0 and y /\ z= O.

ProoJoJ l.b.lO. Let Xbe an order continuous M space. Let {xy}yer be the set ofall the atoms of X of norm one. By 1.a.8, X is order complete and thus it is clear that Y = [xy]yer is a band of X which is order isometrie to co(n. We have to prove that Y = X i.e. that y.1 consists of the 0 element only. Assurne that Y> 0 is an element of norm one in y.1. Since no element of y.1 is an atom of X and X is an M


space, every z>o in y.l can be written as z=u+v with ut\v=O, Ilull=llzll, v#O. Hence, if F is a maximal (with respect to inclusion) downward directed chain of elements {ZJ~EA satisfying O~z~~y, Ilz~ll= 1 then F does not have a g.1.b. This contradicts the order continuity of X. . 0

Proof of l.b.lI ([3], cf. also [71]). We start with the trivial observation that, by the convexity of the norm in a Banach space, whenever v and ware two vectors with Ilvll< 1 and w#O, there is a unique positive t so that Ilv+twll= 1.

Let x, y, z be the three disjoint positive vectors of norm one which span X. For every ° ~ a < 1 and ° ~ () < 2n, let Uo = (y cos () + z sin ())/IIY cos () + z sin ()II and let r(a, ()) be the unique positive number so that Ilax + r(a, ())uoll = 1. Let Po be a contractive projection from X onto span {x, uo}. Since the basis {x, y, z} of X has an unconditional constant equal to one we may assurne without loss of generality that poY and Poz are multiples of Uo (otherwise, pass to the "diagonal" of Po cf. l.1.c.8). Thus, there exists a unit vector Vo E span {y, z} in the kernel of Po. We have that Ilax+ r(a, ())uo+ tvoll ~ 1, for every a and () as above and every real t. Geometrically, this means that the line in the y, z plane through r(a, ())uo in the direction of Vo is a li ne of support to the convex planar set which is defined (in polar coordinates) by O~r~r(a, ()), 0~()<2n. In particular, for every () in which the curve K~ = {«(), r); r = r(a, ())} has a tangent (and thus, for all () except possibly for countably many values) this tangent is in the direction of vo. Since Vo does not depend on a the curves K~ are all homothetic to each other, i.e. r(a, ())/r(ß, ()) is independent of () (speaking analytically, the derivative of the absolutely continuous function g«()) = r(a, ())/r(ß, ()) vanishes whenever it exists and thus this function is a constant). Since r(O, ()) == 1 it follows that r(a, ()) = r(a) is independent of (), i.e. Ilax+r(a)ull=l, for every O~a<l and uEspan {y,z} of norm one. In other words, there is a function F(s, t) so that Ilax+ull=F(a, Ilull), for every a~O and every u E span {y, z}. Similarly, there exist functions G(s, t) and H(s, t) so that, for every a ~ 0,

Ilay+ vii = G(a, Ilvll), v E span {x, z} and Ilaz+ wll = H(a, Ilwll), wEspan {x, y} .

Since

F(s, t) = Ilsx + tyll = G(t, s) = Ilty + szll = H(s, t) = Ilsz + txll = F(t, s) .

we get that F==H==G. The desired result follows now by using l.b.7. 0

In the paper [3] Ando proved also a dual version of l.b.8 which has the esthetical advantage that it characterizes all the abstract L p and M spaces simultaneously (and does not single out a special subclass of the M spaces). In order to state this result, let us introduce the following notion. Let X be a Banach space and Ya closed subspace of X. An operator T: Y* - X* is called a simultaneous extension operator if Ty* I Y = y*, for every y* E Y*. If P is a projection from X onto Y then p* is a simultaneous extension operator. The converse need


not be true unless X is reflexive (i.e. not every simultaneous extension operator is necessarily the adjoint of an operator from X to Y).

Theorem l.b.8'. A Banach lattice X oJ dimension ~3 is an abstract L p space, Jor some 1 ~p < 00, or an abstract M spaee if and only if, Jor every sublattiee Y oJ X, there is a positive simultaneous extension operator oJ norm one Jrom Y* to X*.

For a reflexive X, l.b.8' is completely equivalent to l.b.8 by duality. For a nonreflexive X, the derivation of 1.b.8' from 1.b.8 requires some quite simple arguments which we omit however.

We pass now to the isomorphie versions of 1.b.7 and l.b.8 which we state and prove only in the order continuous case (cf. [127], [128], [77]).

Theorem l.b.12. Let X be an order eontinuous Banach lattice. Then the Jollowing assertions are equivalent.

(1) X is order isomorphie to either L/Il),for some 1 ~p < 00 and some measure Il, or to eo(r),jor some set r.

(2) There exist a non-negative valuedJunction F(t p t2 , ... ) (oJinfinitely many real variables) and a constant A so that, Jor every ehoice oJ a sequence

00

{Xn }:'= 1 OJ disjoint elements in X sueh that L: xn converges, we have n=l

(3) Every sublattiee oJ X is eomplemented.

For the proof of l.b.12 we need the following lemma.

Lemma l.b.13. Let X be an order eontinuous Banaeh lattiee and let 1 ~p ~ 00.

Assume that every sequenee oJ disjoint elements oJ X oJ norm one is equivalent to the uni! veetor basis in lp (in eo if p = 00). Then X is order isomorphie to L/Il), Jor some measure Il (to co(r),jor some r, if p = 00).

Proo! Assurne first that 1 ~p < 00 and put

( n )l/P

11 lxIII = sup i~l IIxi IIp ,

where the supremum is taken over all finite sequences {Xi }7= 1 of disjoint elements n

such that lxi = L: Xi' The supremum may a-priori be infinite for so me x but, i= 1

by the decomposition property, it is easily seen that 111·111 satisfies the triangle inequality. Clearly, IIxil ~ IIlxlll, for every x E X. We claim that there is a constant K< 00 so that IIlxill ~ Kllxll, for every x E X. Assurne to the contrary that there exist positive xm E X, m = I, 2, ... so that IIlxmlll > m and IIxm 11 = 1. Let m1 = 2 and let


n1 n1

{X;"'}?;I be disjoint vectors in X so that xm,= LX;'" and L IIx;"'llp~2P. Let i= I i= I

{PI,;}?; I be the band projections on the bands generated by x;"', i= I, 2, ... , nt n1

and let PI,o=I- L PI,i' Since i= I

n, Illxmlll~ L IIIPl,ixmlll

i=O

we may assume without loss of generality that lim IllpI,n,Xmlll-4 00. Hence, there m

n2

L Ilx;"zllp ~ 22p + Ilx:"llp i= I

and thus

n1 - 1 n2

I Ilx;,,'llp+ I Ilx;"zIIP~2P+22P. i= I i= I

Continuing inductively, we construct a set {x~j; 1 ~ i~nj' j= 1, 2, ... } of vectors in X so that {x;"j; 1 ~ i ~nj -1,j= 1, 2, ... } are mutually disjoint,

k nj- I

I I IIx7'jllp~2P + 22p+ ... +2kP_llx~kllp~2kP j= I i= 1

and IIn~~: X;"jll ~ 1. By our assumption, this double indexed sequence is (after

normalization) equivalent to the unit vector basis of lp' That is, for so me constant A and every integer k,

wh ich is clearly impossible. We have thus shown that 111·111 is an equivalent lattice norm on X. Obviously,

Illx+ ylllp~ IllxlllP+ IllylllP, whenever IXI!\ lyl =0. If p= I this already proves that (X, 111·111) is an abstract LI space. If I <p < 00 we pass to the dual which, as easily verified, satisfies the assumption of l.b.13 for q, where I/q+ I/p= 1. Moreover, 111·111 induces a norm on X* (also denoted by 111·111) for which Illx* + y*IW ~ Illx*IW + Illy*IW, whenever Ix*l!\ ly*1 =0. Starting with 111·111 we re norm again X*· by the procedure described above for X (replacing p by q) and arrive at a norm 111·1110 for wh ich IIlx*lIlö + IIly*lIlö = IIlx* + y*lllö, whenever Ix*l!\ ly*1 =0.


If P = 00 we get from the preceding argument, by passing to the dual, that there

is a K< 00 so that II.i Xiii ~K m~x Ilxill, whenever {X;}7=1 are disjoint vectors l;;;;1 l~l.~n

in X. We also get that X* can be renormed so as to become an abstract LI space. It is not however immediately c1ear that the new norm on X* is induced by a norm in X (Le. that the unit ball of (X*, 111·111) is w* c1osed) and thus it is simpler to renorm direct1y X. We put

IIIXIII = inf max Ilxi II ' 1 ~i~n

n

where the inf is taken over all decompositions of X as a finite sum L Xi' with i= I

Ixil!\ Ixjl=O for i#j. Clearly, K-Illxll ~ Illxlll ~ Ilxll, forevery X E X, and Illx+ ylll = max (Illxiii, Illylll), whenever IXI!\ lyl =0. In order to verify that 111·111 satisfies the triangle inequality it suffices to remark that the inf in the definition of 111·111 is actually the limit over the net of all partitions of X into a finite sum of disjoint

n m

elements (a partition x= L ui precedes x= L vj if each ui is a sum of v/s). i=1 j=1

The fact that X is order isomorphie to co(r), for some r, follows now from 1.b.lO. 0

Proof of 1.b.12. It is c1ear that (1) implies (2) and (3) (see 1.b.8). It follows from Zippin's theorem on perfectly homogeneous bases (cf. 1.2.a.9) that if X satisfies (2) then any sequence {xn } ~ 1 of disjoint vectors of norm one in X is equivalent to the unit vector basis of Co or Ip , for some 1 ~P< 00. We arrive at the same conc1usion if we assurne (3) and apply L2.a.1 o. Thus, in order to be able to apply 1. b.13 and therefore to conc1ude the proof, we have just to show that the P does not depend on the particular choiee of {xn } ~ 1. Start with one such sequence, say {Yn },';"= l' which is equivalent to the unit vector basis of 111 for some 1 ~Po ~ 00

(Po = 00 corresponds to co). Let P be the band projection 'trom X on the band spanned by {Y2n}~1. For every sequence of disjoint vectors {xn}~1 of norm one in PX, the sequence {Xn}:'=1U{Y2n+d:'=1 consists of disjoint vectors and thus {xJ:'= 1 must be equivalent to the unit vector basis of Ip . The same is true for a sequence in (1- P)X. Thus, if {un };:'= 1 is an arbitrary 0 sequence of disjoint vectors ofnorm one in Xthen both {PUn/llpUnll}~ 1 and {(I - P)Un/ll(/ - P)Un II};:,= 1

are (we count only those indices for which the denominator is #0) equivalent to the unit vector basis of Ipo • The same is therefore true for {Un }:'= 1. 0

The preceding theorems explain the special role of the L p and M spaces in Banach lattice theory. It follows from these theorems that functional representation theorems like l.b.2 and l.b.6, which involve e.g. two sided estimates ofnorms of sums of disjoint elements, cannot be proved for more general c1asses of Banach lattices. If, however, we are satisfied with weaker estimates of the norms we can obtain representation theorems in a quite general setting. We present now a very useful general representation theorem. This was developed in the work of several authors ([4], [88], [130] and [99]).


Theorem l.b.14. Let X be an order eontinuous (Le. a-eomplete and a-order eontinuous) Banaeh lattiee whieh has a weak unit. Then there exist a probability spaee (Q, 1:, J.1), an (in general not closed) ideal X of LI (Q, 1:, J.1) and a lattiee norm 11'llx on X so that

(i) X is order isometrie to (X, 11·llx). (ii) Xis dense in L 1(Q, 1:, J.1) and Loo(Q, 1:, J.1) is dense inX.

(iii) Ilflll:( Ilfllx :(21IfII00' whenever f E Loo(Q, 1:, J.1). (iv) The dual of the isometry given in (i) maps X* onto the Banaeh lattiee X *

of all J.1 measurable funetions g for whieh

The value taken by the funetional eorresponding to 9 at fEX is S fg dJ.1. Q

The main tool in the proof of 1. b.14 is the following proposition which ensures the existence of strict1y positive functionals in X*.

Proposition l.b.15. For every order eontinuous Banaeh lattiee X with a weak unit e>O there exists afunetional e*>O in X* sueh that e*(lxl)=O implies x=O.

Proof Since in many applications we work with separable lattices we present first a very simple proof which is valid only under the assumption that Xis separable. Let {X"}~1 be a dense sequence in the set {x E X; x~O, Ilxll= I} and choose positive Hahn-Banach functionals x: E X* so that Ilx:11 = 1 and x:(x") = 1 for all n (if a Hahn-Banach functional x: is not positive we replace it by Ix:I). Then

OCJ

e* = L x: /2" is a strict1y positive functional on X. Indeed, if x > 0 is a norm one "=1

vectör in X then we can find an integer n so that Ilx-xn 11< 1/2 wh ich implies that e*(x) ~ x: (x)/2" ~ 1/2"+ 1 > O.

The proof in the non-separable case is longer. We observe first that it suffices to show that there exists a sequence of mutually disjoint norm one positive functionals {e:} ~ 1 which is maximal in the sense that no more functionals can be added to this sequence without losing the disjointness. In this case, e* = L e:/2"

" would be a strict1y positive functional on X. Indeed, otherwise Y = {x E X; e*(lxl)=O} is a non-trivial projection band of X and thus, there exists a positive functional e6 E X* such that IIe611 = land e6 yL = O. This fact, however, contradicts the maximality of {e:}:'=1 since e* 1\ eö=O.

In order to complete the proof, we show that any maximal family {e:}aEA' of disjoint norm one positive functionals in X* is countable. Put Ya = {x EX; e:(lxl)=O} and let Pa be the band projection from X onto Y; i.e. PaYa=O. For each pair rt.,ßEA with rt.#ß, we get that ya.LnYt={O}. Indeed, for every O:(u E y;n Yt, we have

O=(e: 1\ e;)(u) = inf {e:(v)+ep(w); u=v+w; O:(v, w:(u} .


Thus, there are sequences {Vn}:'=1 and {Wn}:'=1 so that u=vn+wn' O~vn' wn~u, co co

e:(vn)~rn and et(wn)~2-n for all n. Put v~= V vn, w~= V Wn and observe n=k n=k

that {va~)= 1 and {W~} ~= 1 are decreasing sequences of positive elements in X. By the a-completeness and a-order continuity of X, these two sequences must have strong limits v' ~ 0, respectively w' ~ 0, which clearly satisfy e:( v') = 0, et( w') = 0 and u~v'+w'. Using the decomposition property we get that there are O~V~V' and O~W~W' such that u=v+w. Since e:(v)=O and et(w)=O we conclude that v E Y", and w E Yp• On the other hand, Y,; and Yj are ideals of X and this implies that v E Y",l. and w E Yf. Thus, u=O.

Put e",=Pie), !XE A, and notice that e",#O for all !X since e is a weak unit. The series L e", converges in X for every countable subset A' c: A since Xis a-complete

12eA'

and a-order continuous. This clearly implies that A is countable. . 0

Remark. In the non-separable case the assumption that X is order continuous cannot be dropped. Consider e.g. X=lco(r) with r uncountable.

Proololl.b.14. Let X be an order continuous Banach lattice having a weak unit eo>O with Ileoll=2. By l.b.l5, X* contains a stricdy positive functional e~ with Ile~II=1. Let u* be a positive element of norm one in X* for which u*(eo)=2. Then e*=(e~+u*)/lle~+u*11 is a stricdy positive functional on X, e=eo/e*(eo) is a weak unit of X and

lIe*ll=e*(e)= 1, Ilell~2.

By l.a.13, Xis a cyclic space .ß(e) with respect to a a-complete Boolean algebra ofprojections f!4 (which, by l.b.3, can be regarded as {Pa}aeEo' where 1:0 is the set of all simultaneously open and closed subsets of a totally disconnected space Q). We define a probability measure Jl on Q by Jl(a) =e*(Pae) (as in the proof of l.b.2 we have that, up to sets of Jl measure zero, 1:0 is already a a-algebra). The map

n n

which assigns to x= L aiPa/ the function x= L ai'Xal in L 1{Jl) is clearly an i= 1 i= 1

order isomorphism and IIxl11 =e*(lxl)~ Ilxll. The map x --. x extends thus to a positive and contractive operator from X to L 1 (Jl). Since e* is strictly positive this operator is one to one and an order isomorphism. The imageX of X under this operator is clearly dense in L 1 (Jl) (it contains all the simple functions). For every simple function/on Q we have 1/1~ll/llco·l; ifwe putl=x with XEXwe get that Ixl~llxllcoe and thus Ilxll~21Ixllco' This proves assertions (i), (ii) and (iii) of the theorem.

In order to show that X is an ideal in L 1{Jl), we observe that every I~O in L 1 (Jl) is the limit in the L 1 norm of an increasing sequence of simple functions {f,,}:'= l' Each /" is of the form Yn for some Y n E X. Thus, if 1 ~ x for some x E X it follows from the order continuity of X that {Yn}:'= 1 converges in norm to some Y in X. Clearly 1= Y.

It remains to prove (iv). For every x* in the dual ofX we consider the (signed)


measure vx*(a)=x*x.,., a E L (the a-additivity of vx• follows from the a-order continuity of X). Since Vx* is absolutely continuous with respect to j). there exists a function gELl (D, L, j).) such that

J 1 dvx*= J Ig dj)., u u

for every 1 which is v x*-integrable. By using approximation by simple functions it is easily checked that if 1 EX then 1 is vx*-integrable and J 1 dvx*=x*(f).

u This implies that x*(f) = J Ig dj). and

u

Since the converse (Le. the fact that every g as in (iv) defines an element of X *) is obvious this completes the proof. 0

Remark. As is easily seen from the proof, the number 2 in I. b.14(iii) can be replaced by 1 +B for any B>O (of course, for different B we obtain different representations). It is also easy to see that we can obtain (iii) with 2 replaced by 1 if and only if there are a weak unit e in X and a strictly positive functional e* in X* with llell = Ile*11 = e*(e) = 1. If, for example, X = Co then such e and e* fail to exist.

As an application of I. b.14 we present a proof of a result of H. Nakano [104J.

Theorem l.b.16. A Banach lattice X is order continuous if and only if the canonical image 01 X into its second dual X** is an ideal 01 X**.

Proof Suppose that X is an order continuous Banach lattice and let i denote the canonical embedding of X into X**. Let x E X and x** E X** be two vectors satisfying 0:::;,; x**:::;,; ix. By the last part of l.a.9, there exists an ideal Xo of X with a weak unit so that XE XO and X=Xo Ef> X~. In view of this decomposition, we can consider x** as an element of X~*. By using the representation theorem l.b.14, Xo can be regarded as an ideal of an L 1(D, 1:, j).)-space (with j).(Q)= I), which has all the properties described in the statement of l.b.14. In particular, for every a E L, 'X.s is an element of X~ and therefore we can put v(a)=x**(X.,.) and A(a)= J x(co) dj).(co), where x(co) is the function in L1(D, 1:, j).) representing

(1

the element XE Xo. The measure A is clearly a-additive and A(a)~ v(a), for all a E L. Hence, also v is a-additive. Since v is also absolutely continuous with respect to j). it follows from the Radon-Nikodym theorem that there exists a function 1 E L1(D, L, J1) so that v(a)= J!(co) dJ1(co) for all a E L. The relation

(1

between v and A. implies that/(co):::;';x(co) for a.e. co E D, i.e./E Xo since Xo is an ideal of L1(D,L,j).). From this fact and assertion (iv) of l.b.14 it follows easily that x** = if, Le. that iX is an ideal of X**.

In order to prove the converse, assume that iX is an ideal of X** and that {xn }:,: 1 is an increasing sequence of positive elements of X which is bounded in


order by an element x E X. Since {Xn}:= 1 is necessarily a weak Cauchy sequence in X there exists an xö* E X** such that iXn ~ xö*. For every positive x* E X* we have xö*(x*)= lim x*(xn)~x*(x), i.e. O~xö*~ix. Since we have assumed

that iX is an ideal of X** we get that xö* = ixo, for some Xo EX, i.e. that xn ~ xo. However, for monotone sequences in a Banach lattice, weak convergence implies strong convergence (use the fact that if xn ~ X o then there exist convex combinations of the xn's which tend strongly to x o). This completes the proof of the order continuity of X, by 1.a.8. 0

It follows from 1.b.16 that a Banach lattice X is order continuous if and only if, Jor every y, ZEX, the order interval [y, z]={x;y~x~z} is weakly compact. Indeed, if i: X ~ X** denotes the canonical embedding and if X is order continuous then, by 1.b.l6, [iy, iz] = i[y, z] for every y, Z EX. Since any order interval in X** is w* compact it follows that [y, z] is a w compact subset of X. This proves the "if" part of the above assertion. The "only if" part is obvious.

Let us also mention here another characterization of order continuous lattices (cf. [133]): A Banach lattice (X, 11·11) is order continuous if and only if there is an equivalent lattice norm 11 111 on X so that

In 1.1. b.ll it was shown that every separable Banach space X admits an equivalent norm so that (*) holds. The point in the result stated here is that in case X is a Banach lattice, the new norm 11 1/1 can be chosen to be again a lattice norm if and only if X is order continuous. The proof of the "only if" part is easy. Indeed, by 1.a.5 and 1.a.7, if X is not order continuous there exists a sequence of disjoint positive vectors {Yn}'% 1 in X which is equivalent to the unit vector basis in Co

and a vector y E X so that Yn~y for all n. Then, for any lattice norm 1/ 1/1 in X, y-Yn~ y, I/y-Ynl/1 ~ I/yl/1' but clearly {Y-Yn};;"=1 does not tend strongly toy. The proof of the "if' part is somewhat long and will not be reproduced here. We just mention that this proofis based on the representation theorem 1.b.l4.

The proof of 1.b.16 shows, in particular, that also the converse to 1.b.14 is true. Every lattice X, which satisfies (i)-(iv) in 1.b.14, is order continuous and has a weak unit. It is however very simple to prove this fact direct1y. We shall now discuss this point and some related questions in a somewhat more general context.

Definition l.b.17. Let (Q, 1.:, J1.) be a complete u-finite measure space. A Banach space X consisting of equivalence classes, modulo equality almost everywhere, of locally integrable real valued functions on 1.: is called a Köthe Junction space if the following conditions hold.

(1) If IJ(w)I~lg(w)1 a.e. on Q, withJmeasurable and gE X, thenJE X and I/JI/~I/gl/.

(2) For every u E 1.: with J1.(u) < CI) the characteristic function X" of u belongs toX.


Recall that a measure space is said to be complete if any subset of a set of measure zero is measurable. The assumption of completeness of the measure space is just a minor technieal convenience; the measure space constructed in 1. b.2 (and thus in 1. b.14) can clearly be taken to be complete. A function f is called locally integrable if it is measurable and J If(w) I dp.< 00, for every u E E

a

with p.(u) < 00.

Every Köthe function space is a Banach lattice in the obvious order ~O if f(w)~O a.e.). This lattiee is u-order complete. Indeed, if {J,,}:'=1 is an order bounded increasing sequence in X then f(w) = limJ,,(w) is the l.u.b. of {J,,}:'=1'

n .... 00

Theorem l.b.14 asserts, in partieular, that every order continuous Banach lattice with a weak unit is order isometrie to a Köthe function space. Thus, a separable Banach lattice is order isometrie to a Köthe function space if and only if it is u-order complete.

The assumption that every fEX is locally integrable implies that, for every u E E, the positive functionalf ~ J f(w)Xa(w) dp. is well defined and thus bounded

!l i.e. it is an element of X*. In general, every measurable function 9 or 0 so that gfE L 1{J1.), for every fE X, defines an element x; in X* by x;(f) = J f(w)g(w) dp.

!l (we shall often identify 9 with x:). Any functional on X ofthe form x: is called an integral and the linear space of all integrals is denoted by X'. It is an immediate consequence of the Radon-Nikodym theorem that a functional x* E X* is an integral ifand only if, for every sequence {J,,}:;'1 in XwithJ,,(w)!O a.e., we have Ix*l(fn)~ O. In partieular, if Xis u-order continuous then X*=X'. The converse is also true. Assurne that X* = X' and let {J,,}:;'1 be an increasing sequence of positive elements in X whieh converges pointwise a.e. to some f in X. Then clearly J" ~ fand hence also IIJ" - fll ~ O. Since every Köthe function space is u-order complete the condition X* = X' is, by 1.a.8, also equivalent to the order continuity ofX.

It is easily verified, by using the characterization of integrals given above, that, for every Köthe function space X, X' is an ideal of X*. In the norm induced on X' by X*, this space is also a Köthe function space on (0, E, p.). The following proposition, due to G. G. Lorentz and W. A. J. Luxemburg (cf. [86]), characterizes those Köthe spaces for which X' is a norming subspace of X* (i.e·llx 11 = sup {lx*(x)1 ; x* E X', Ilx*11 = I}, for every x EX).

Proposition l.b.18. Let X be a Köthe function space. Then X' is a norming subspace 01 X* if and only if, whenever {J,,} ~ 1 andl are non-negative elements 01 X such that

J,,(w) i I(w) a.e., we have IIJ"II ~ 11/11·

Proof Assurne that X' is norming, let J,,(w) i f(w) EX a.e. and let e>O. Pick x* E X' with Ilx*11 = I and x*(f)~ II/II-e. Since x* is an integral x*(f,,) ~ x*(f) and thus lim inf I In 11 ~ 11/11- e. ConsequentIy, 11I11 = lim 11f" 11·

n .... 00 n .... oo

Conversely, suppose thatf,,(w) i I(w) a.e. in 0 implies IIJ"II ~ Ilfll. Consider Y=XnL1(p.) as an, in general not closed, subspace of L 1(p.) with the norm \\·\b

30 1. Danach Lattices

induced by LI (P). The set {/;f e Y, 11/11::s:; I} is c10sed in Y. Indeed, assume that {In}:' 1 and/are in Y, with 11f..11::s:; 1, for an n, and 11f..-/111 -+ O. By passing to a subsequenee if necessary we may assume without loss of generality that f.. (w) -+

/(w) a.e. Hence, gn(w) i 1/(w)1 a.e., where gn(w)= inf 1/,,(w)l, and thus li/li = k~n

11-+ 00 n-+ 00

Let now / eX be an element with 11/11> 1. By our assumption on X, there is a er e E with /l(er) < 00 so that IIfxull> 1. By using the separation theorem for Y, there is an he Loo(P) so that f h(w)/(w)Xu(w) d/l> 1 and I f h(w)g(w) d/ll::s:; 1,

u u for every ge Ywith Ilgll::S:; 1. Consequently, hXu defines an element x* e X' so that Ilx*ll::s:; land x*(f) > 1. 0

Remarks. 1. A typica1 example of a Köthe function space X such that X' =1= X*, but X' is norming, is Loo(P). An example of a Köthe function space X for which X' is not norming is the space 100 with the equiva1ent norm Ilxlln= sup Ix(k) I +

k

n lim sup IX(k) I (here Q is the set of integers with the discrete measure and n is a k

positive integer). The space C~I EB (100' 11'lln»)2 is an examp1e ofa Köthe function

space X for which sup {lx*(x)l; x* e X', Ilx*11 = I} does not even define an equivalent norm on X.

2. For every Köthe function space X, we can define also the space X" = (X')'. If X' is a norming subspace of X* then X is isometrie to a subspace of X". The space X coincides with X" if and only if

(*) f..(w) i f(w) a.e., {f..}~=IC:X, f..(w)~Oa.e. and supllf..ll<oo n

~ / eX and li/li = lim 11f..11 . n

Indeed, it is easily verified directly that, for every Köthe function space Y, the space X = Y' satisfies (*). Henee, if X = X" then (*) holds. Conversely, if (*) holds then, by l.b.l8, Xis isometrie to a subspace of X". Letfe X" be non-negative and let {f..}:'= 1 be a sequence of non-negative simple functions increasing to f a.e. It follows from (*), applied to this sequence {f..} ~= 1> that / e X. Property (*) is called the Fatou property.

3. A very simple but useful fact is the following. If Xis a Köthe funetion spaee then

f..(w) -+ /(w) a.e., {f..}:'= 1 cX and sup 1If..11 < 00 ~ /e X". n

Indeed, for every 9 E X',

f 1/(w)g(w)1 d/l::S:; lim inf f I f..(w)g(w) I d/l::s:;llgllx' sup 1If..llx· U n .... oo U n

c. The Structure of Banach Lattices and their Subspaces 31

c. The Structure of Banach Lattices and their Subspaces

We begin this section by presenting some results concerning weak comp1eteness and reflexivity of Banach lattices as weIl as of their subspaces. Similar theorems have already been proved for spaces with an unconditional basis in Section I.1.c but their extension to Banach lattices requires somewhat different methods of proof (for subspaces of aspace with an unconditional basis we have just stated the results in I.l.c.13 without giving a proof). We also present in this section so me results on complemented subspaces and basic sequences in Banach lattices. An important tool in the study of subspaces of Banach lattices is the so-called property (u), introduced in [110].

Definition l.c.1. A Banach space Xis said to have property (u) if, for every weak Cauchy sequence {Xn}:'= I in X, there exists a sequence {Yn}:'= I in X such that:

00

(i) the series L Yn is weakly unconditionally convergent (W.U.C.), i.e. n=1

00

L IY*(Yn)1 < 00 for every y* E X*, n= I

(ii) the sequence {xn - jtl yj} ~= I converges weakly to zero.

A. Pelczynski [110] proved that every space with an unconditional basis has property (u). A similar result is valid for Banach lattices.

Proposition l.c.2 [l30]. Any order continuous Banach lattice X has property (u).

Proof We first observe that it suffices to prove the assertion for separable lattices. Thus, by l.b.14, we may assurne without 10ss of generality that Xis a Köthe function space on some prob ability measure space (Q, L, Jl) and that every element of X* is an integral (i.e. X* = X').

Let {!,,}~ I be a weak Cauchy sequence offunctions in X. This sequence is also a weak Cauchy sequence in LI (Q, L, Jl). Thus, since LI spaces are weakly sequentially complete (cf. [32] IV.8.6), there exists an fELl (Q, L, Jl) so that S f.h dJl--> S fh dJl as n --> 00, whenever hE Loo(Q, E, Jl). Let gE X*; then Q Q

vn(O") = S fng dJl, 0" E E, is a sequence of O"-additive measures which converges for (J

every 0" E L. It follows from a well-known result ofNikodym (cf. [32] III.7.4) that v( 0") = lim v n( 0"), 0" E E, is also a O"-additive measure wh ich is absolutely continuous

n .... 00

with respect to Jl. Since, for every set 0" E L on which g is a bounded function, we have vn(O") --> S fg dJl as n --> 00 it folIo ws that fg E L l (Q, E, Jl) and

(J

lim S (fn - f)g dJl = 0 n--+oo n

(use the uniqueness of the Radon-Nikodym derivative).


00

Put "ln={WEQ;n-1~lf(w)l<n}, bn= U ."lj and hn=/xqn' n=1,2, ... j=n+ 1

Then, for every g E X*, we get that

and

n

J (J,.- L h)g d/1= J (J,.-f)g d/1+ J fg d/1 ~ 0 as n ~ 00 . f.! j= 1 f.! /in

This comp1etes the proof. 0

The fact that subspaces of an order continuous Banach lattice also have property (u) is a consequence of the following general result from [110].

Proposition 1.c.3. Every closed subspace of a Danach space with property (u) has also property (u).

Proo! Let {Yn}:= 1 be a weak Cauchy sequence in a subspace Y of a Banach space 00

X. If X has property (u) then there exists a w.u.c. series L Xi in X so that the i= 1

n

sequence Un=Yn- LXi' n=l, 2, ... , converges weak1y to zero in X. We wou1d i= 1

like to replace the series L Xi by a w.u.c. se ries consisting of elements of Y. Since i= 1

Pj

Un ~ 0 we can find convex combinations uj = L AnUn, j= 1, 2, ... , with

Pi 00

O=PO<Pl< ... <Pj<"·' L An=l, for all j, and L lIujll< 00. Put yj= n=p.i-,+l j=l

Pj

L AnYn, zo=y~ and Zj= Yj+ 1 -Yj forj~ 1, and observe that the sequence n=Pj-' + 1

n

Yn- I Zj=Yn-Y~+pn=1,2, ... j=O

00

converges weakly to 0 in Y. Since Zj E Y, for allj, it remains to show that L Zj is a j=O

w.u.c. series. A simple computation shows that each vector Zj,j>O, can be written as

Pi + 1

AnYn=Uj+l -uj+ L /1~Xn'


where, for eachj, the coefficients J1~ are suitable numbers between 0 and 1. Thus, for any x* E X*, we have

00 00 00 Pj+l

L Ix*(z)1 ~ 211x*11 L Ilujll + L L J1:IX*(Xn)I j=O j=l j=O n=Pj_l+1

00 00

~21Ix*11 L Ilujll+2 L IX*(Xn)l< 00 , j= 1 n= 1

00

since L X n is a w.U.C. series. 0 n= 1

Remark. An interesting application of the argument used in the proof of l.c.3 was found by M. Feder [134] who showed the following. Let X be a Banach space with an unconditionalfinite dimensional Schauder decomposition (F.D.D. cf I.I.g) {Bn }:'= 1 and let Y be a reflexive subspace of X. Then Y is isomorphie to a complemented subspace of aspace with an unconditional F.D.D. if and only if Y has the approximation property (A.P.).

The "only if" assertion is trivial. The "if" assertion is proved as folIows. For each nIet Qn be the natural projection from X onto Bn· Let {Tn}:'= 1 be a sequence offinite rank operators in L(Y, Y) so that lim IIT"y-yll=O, for every y E Y. Such

n .... oo

a sequence exists since Y is separable and has the M.A.P. (use the reflexivity of Y

and I.l.e.15). The sequence {Tn - itl QiIY}~=l in L(Y, X) tends to zero pointwise

and hence, by the reflexivity of Y, also in the weak topology of L( Y, X). (U se the fact that the map S ~ x* (Sy) defines an isometry from the subspace of L(Y, X) consisting ofthe compact operators into C(B y x Bx ')' where B y is taken in the w topology and Bx• in the w* topology). Consider L(Y, y) as a subspace of L(Y, X) and apply the argument of l.c.3 (with Tn= Yn and QiIY=xi ), It follows that there exists a sequence {Si} i= 1 of finite rank operators in L( Y, Y) so that

tends pointwise to zero and sup sup 11 .. i:. ()iSill<oo. Thus, for every YE Y, n 91= ± 1 ,= 1

00

y= L SnY and the series converges unconditionally. An argument identical to n= 1

that used in the proof of I.I.e.13 now concludes the proof. Indeed, let Wn = Sn Y, n = I, 2, ... , and let W be the completion of the space of all sequences of vectors w = (w 1> W 2' ... ), which are eventually zero, so that

wn EWn ,n=I,2, ... and IIlwlll=sup supll'±OiWill<oo. n 91 = ± 1 ,= 1


Clearly, W has an uneonditional F.D.D. The operators U: Y - Wand R: 00

W- Y, defined by Uy=(SlY' S2Y' ... ), yE Y and R(w1' w2' ... )= L wn ' n= 1

(W 1 ' W 2' ... ) E W, are bounded and satisfy RUy=y for every y E Y. Henee, UYis a eomplemented subspaee of W isomorphie to Y. D

We state now the main result eoneerning weak eompleteness in Banaeh lattiees.

Theorem l.c.4. The Jollowing conditions are equivalent Jor any Banach lattice X. (i) Xis weakly sequentially complete.

(ii) No subspace oJ Xis isomorphic to co' (iii) Every norm-bounded increasing sequence in X has a strong limit. (iv) The canonical image oJ X in X** is a (projection) band oJ X**. The equivalence (i) <=> (ii) remains valid in the case when X is a subspace oJ an

order continuous Banach lattice.

The implieation (iii) => (i) was originally proved in [106] (see also [89]) while (ii) => (i) in [84], [99]. The result on subspaces of Banach lattiees was proved in [129] and [130].

Proof The implication (i) => (ii) is trivial. By l.a.5, l.a.7 and l.a.8, a Banach lattice satisfying (ii) is both u-complete and u-order continuous and thus order continuous. Hence, it suffices to prove that (ii) => (i) only in the case when Xis a subspace of an order continuous Banach lattice. By l.c.2 and l.c.3, such a subspace X has property (u). Thus, if there exists in X a weak Cauchy sequence which does not converge weakly to any element of X then there is in X also a w.u.c. series whieh does not eonverge. It folIo ws from 1.2.e.4 that X eontains a subspaee isomorphic to co'

We show next that (ii) => (iii). Assurne that 0::::; Xl ::::;X2::::; ... is an inereasing non eonvergent sequence in Xwith Ilxnll::::; 1 for all n. Then there are a 15>0 and an inereasing sequence {nk}:'= 1 of integers so that if Yk = X nk + 1 - X nk thenllYkl1 ~ 15, k= 1,2, .... The sequenee {YkHx~l tends weakly to 0 and henee, by I. l.a. 12, has a subsequenee {Yk).f:= 1 whieh is abasie sequenee. Since, for every choice of sealars

{Aj}j= l' we also have that

II .I AjYkill::::; m~x IAjll1 f Ykill::::; m~x IAjl sup Ilxnll::::; m~x IAjl )=1 ) )=1 ) n )

it folIo ws that {YkJi= 1 is equivalent to the unit veetor basis of co' in contradiction to (ii).

Assurne now that condition (iii) holds in X. Then Xis a u-eomplete and u-order continuous Banach lattice and thus, by l.b.l6, the canonical image iX of X into X** is an ideal of X**. Let {x"JaEA be an upward direeted set in X so that V iXa = x** exists in X**. Then x** = lim iXa (in the strong topology of X**),

a:eA Cl

i.e. x** E iX. Indeed, otherwise there would exist an increasing subsequenee {Xa)i= 1 of {Xa}aEA such that i~f Ilxai + 1 -xaill >0. Sinee this fact eontradicts (iii) we

)


conclude that iX is a band of X**. Since X** is order complete it follows from l.a.lO that this band is actually a projection band. Hence, condition (iv) holds.

It remains to show that (iv) ~ (i). If iX is a band of X** it follows from 1.b.16 that X is order continuous. Since every' separable subspace of Xis contained in a band of Xhaving weak unit we may assurne without loss of generality that Xhas a weak unit. Hence, we can apply the functional representation theorem l.b.14 to X. Let Un}::'= 1 be a weak Cauchy sequence in X. By arguing as in the proof of l.c.2, we can construct a function fE Li (Q, E, Ji) such that, for every gE X*, fg E Li (Q, E, Ji) and J (fn - f)g dJi ~ 0 as n= 00. This means thatfis the w*-limit

n in X** of the sequenee {ifn}:'= 1 . On the other hand, Ifl is the l. u. b. in X** of the sequence {lflxO"J:'= l' where O"n= {w E Q, If(w)l~n}. This implies thatfE iX sinee IflxO"" E X for all n and iX is assumed to be a band of X**. 0

Remark. The proof of 1.a.5 ean be used as an alternative proof of (ii) ~ (iii) in 1.cA. This proof is more complieated than the simple argument presented here. It has however the advantage that it produces a sublattice order isomorphie to Co

(and not only a subspaee isomorphic to co), We ean deduee thus that if a Banaeh lattiee has a subspaee isomorphie to Co then it has also a sublattice order isomorphic to co'

We pass now to the eharaeterization of reflexivity in Banaeh lattices and their subspaees. In this context, reflexivity is usually proved by using the well known result of Eberlein which asserts that a Banach space X is reflexive if and only if it is weakly sequentially complete and its unit ball Bx is conditionally weakly compact (the latter means that every bounded sequence contains a weak Cauchy subsequence).

Theorem l.c.5. The following properties are equivalent for every Banach lattice X. (i) X is reflexive.

(ii) No subspace of Xis isomorphic to 11 or to co' (iii) Every norm bounded increasing sequence in X has a strong limit and X* is

O"-order continuous. The equivalence between (i) and (ii) remains valid also in the case when X is a

subspace of an order continuous Banach lattice.

The proof for (i) <0> (iii) was given in [106] while (i) <0> (ii) was proved first in [84], [99] for generallattiees and in [130] for subspaces.

Proo! The implication (i) ~ (ii) holds trivially in every Banach space. Assurne now that Xis either a Banach lattiee or a subspaee of an order eontinuous Banach lattiee and that X does not have any subspaee isomorphie to 11 or to co. Then, by l.cA, X is weakly sequentially complete. Furthermore, by 1.2.e.5, Bx is also eonditionally weakly eompaet. Thus, X is reflexive i.e. (ii) ~ (i). The fact that (i) ~ (iii) follows easily from l.eA and l.a.7. Therefore, it remains to prove that (iii) ~ (i). Using once more l.eA, we conclude that it suffiees to prove that (iii) implies the conditional weak compactness of Bx . Let {xn }::"= 1 be a norm bounded


sequence in X. By l.a.9, X is an unconditional direct sum of a family of mutually disjoint ideals having a weak unit so that [Xn],~)~ 1 is entirely contained in one of these ideals, say X o. Since X = X 0 EB X t we get that XJ' is order isometric to a sublattice of X* and, therefore, also a-order continuous.

We use now the functional representation of Xo as a Köthe function space on some probability space (Q,E,/1). Put via)= J xn d/1, aEE, n=1,2, ... , and

a

observe that the measures {Vn}:~1 have the following properties: (1) {Vn}:~l are uniformly bounded since Ivn(a)1 ~ sup Ilxnllxo < 00, for all n and a E E, (2) the

n

a-additivity of {Vn}:~l is uniform since Ivia)I~IIXallx~· sup Ilxnllxo and xt is n

a-order continuous. It follows that these measures form a conditionally weakly compact set in the Banach space of all bounded measures on (Q, E) (see e.g. [32] IV.9.l) and, therefore, there is a subsequence {vn}~l of {vn}:~l such that v(a)= lim vn;(a) exists for all a E E. By the a-order ~ontinuity of XJ', the /1-simple i-+ 00

functions are norm dense in Xti. Hence, the limit, as i ~ 00, of

g(Xn)= J gxn;d/1= J gdvn;, Q Q

exists for every 9 E Xti. 0

It is clear that, in general, 1.cA and 1.c.5 fail to be true for subspaces of arbitrary Banach lattices. For instance, the space J of R. C. James presented in I.l.d.2, whieh is, as any separable space, isometric to a subspace of C(O, 1) or of 100 , is not weakly sequentially complete despite of the fact that it contains no subspace isomorphie to eo.

It is however possible to extend I.cA and 1.c.5 to complemented subspaces of a general Banach lattice. For this purpose we need first the following result from [42], [59].

Proposition l.c.6. Let Y be a eomplemented subspaee 0/ a Banaeh lattiee X. If Y eontains no subspaee isomorphie to eo then there exists an order eontinuous Banaeh lattiee Xl whieh eontains a eomplemented subspaee YI isomorphie to Y.

Proo! Let 11·11 denote the norm in X and let P be a projection from X onto Y. Define a new semi-norm on X, by putting,

(the triangle inequality for 11·111 is proved by using the decomposition property). Let I denote the ideal of X consisting of all x EX for which IlxIII = O. The quo

tient space X/ f becomes a vector lattice when its positive cone is taken to be the

image, under the quotient map Q: X ~ X/I, ofthe positive cone of X. We norm

X/fby putting IIQxll, = Ilxlll. The completion Xl of X/fis a Banach lattiee and the


map Q, restrieted to Y, is an isomorphism from Y onto a subspace Y1 of Xl since

Observe also that if, for some Xl' X2 EX, we have QX1 =Qx2 then PX1 =Px2 •

Thus, by putting P1(Qx)=Q(Px), we define a map P1 from X/Ionto Y1 which extends uniquely to a bounded projection from Xl onto Y1 since

for all X E X. It remains to show that Xl is order continuous. If this were not true then, by

l.a.5 and l.a. 7, Xl would contain a sequence ofpositive vectors ofnorm one which is equivalent to the unit vector basis of Co' Evidently, there is no loss of generality in assuming that the elements of this sequence belong to X/I, i.e. that there exist a constant C< 00 and a sequence {xn }:'= 1 of positive elements of X so that IIxnl1 1 = 1 for all n and

for every choice of (a 1 , a2 , ... ) E Co' Choose now vectors Un E X for which IUnl ~ X n

and 1 ~ IIPunl1 ~ 1/2 for all n. Then, for (al' a2 , ... ) E Co' we have that

IIJ1 anP1QUnt ~IIP1111In~1 lanllQUnll11

~IIP111 t~l lanlXnl11 ~IIP11Icm:x lanl·

On the other hand, QUn ~ 0 in Xl (use the fact that QXn ~ 0 in Xl and that IQunl ~ Q(lunl) ~ QXn for all n) and therefore also P 1 QUn ~ 0 in this space. Thus, by I.l.a.12, we can ass urne without loss of generality that {P 1 Qun },% 1 is a basic sequence in Y1. Since IIp1Qun l1 1 ~llpunll ~ 1/2 for aHn it follows that {P1QUn}:'=1 is equivalent to the unit vector basis of Co and this contradicts our assumption on Y (which is isomorphie to Y1). 0

The following result is an immediate consequence of l.c.4, l.c.5 and l.c.6.

Theorem l.c.7. Let Y be a complemented subspace of a Banach lauice. Then (i) Y is weakly sequentially complete if and only if no subspace of Y is isomorphic

to co. (ii) Y is reflexive if and only if no subspace of Y is isomorphic to 11 or to co·

We conc1ude this section by presenting some results concerning the existence of unconditional basic sequences in subspaces of Banach lattices. We begin with a


generalization to Banach lattices ofa result ofM.1. Kadec and A. Pelczynski [61] (proved originally only for Lp spaces).

Proposition 1.c.8 [42]. Let X be an order continuous Banach lattice with a weak unit (in particular, a separable u-complete lattice). Any closed subspace of Y of Xis either isomorphie to a subspace of some L1 space or there exist a sequence of normalized vectors {Yn}:'=1 in Yanda sequence ofmutually dis joint elements {Xn}:'=1 ofXsuch that {Yn}:'=1 is equivalent 10 the (unconditional) basic sequence {Xn}:'= l'

Proof We assume, as we may, that Xis a Köthe function space over some probability measure space (Q,E,Jl). For XEX and e>O, put u(x,e)={WEQ; Ix(w)l~ellxllx} and consider the set M(e)={XEX; Jl(u(x, e»~e}. If YcM(e) for some e> 0 then

Ilyllx~ IIyI11 = f ly(w)1 dJl~ f I Y(W) I dJl~e21Iyllx' Y E Y Q a(y,e)

i.e. Yis isomorphie to a subspace of L 1 (Q, E, Jl). Otherwise, we can find a sequence {Zn}:'=1 c Y with Ilznllx=l and Zn ~ M(2- n) for all n. For m>n, put un,m=

00

u(zn' r n)_ U U(Zk' r~. Then, for any fixed n, lim /l(Un,m)=/l(u(zn' 2-n», k=m m-+oo

which implies that lim IlznXa(Zn,2-n)-ZnXan,JX=0 for all n (use the fact that m-+oo

znXan,m' znXa(zn,2-n) EX, since X is an ideal in L1(Q, E, Jl), and the u-order con-tinuity-of X).

We choose now, inductively, a subsequence {Zn.}~1 of {zn}:'=1 and a sequence ofmutually disjoint sets {oJ~1cE such that, for each i, uicu(zn.' r ni ) and IIzniXa(zni' 2 -ni)-zniXa,llx:::::;2-i. Put Yi=zni' Xi =Zn,Xai , i= 1,2, ... and observe that

Hence, by the perturbation result 1.1.a.9(i), {y;} ~ 1 is equivalent to the sequence of disjoint elements {Xi} ~ 1 . 0

Remark. The assumption in l.c.8 that Xhas a weak unit is redundant: in the general case we get that either Y contains an unconditional basic sequence or every separable subspace of Y is isomorphie to a subspace of some L1 space. As it will be shown in Vol. III, the latter condition already implies that Y itself is isomorphie to a subspace of an L 1 space.

A deep result of H. P. Rosenthal [115], to be proved in Vol. IV, states that every infinite dimensional subspace of an L 1 space contains a subspace with an unconditional basis. Combining this fact with l.c.8 and the remark thereafter we obtain a positive solution to I.I.d.5 for subspaces of order eontinuous Banaeh lattices.

Theorem 1.c.9. Every infinite dimensional subspace of an order continuous Banach lattice contains a subspace with an unconditional basis.


We already mentioned above that it follows from l.c.4 and l.c.5 that there exist spaces (e.g. the space 1) which do not embed isomorphically in an order continuous Banach lattice. We present next a result of a different nature which also allows us to deduce that certain Banach spaces do not embed in such a lattice.

Proposition l.c.l0. Let X be an order continuous Banach lattice and let {xn} ~ 1 be a sequence of elements of norm one in X. Then either there exists a constant c > 0 such that, for every choice of scalars {an}:'= l' we have

or [Xn}:'= 1 contains a subsequence {xn) ~ 1 which is an unconditional basic sequence equivalent to a sequence of dis joint elements of X.

Proof We may assurne without 10ss of generality that X has a weak unit and we can thus represent X as in 1. b.I4. U sing the same notation as in the proof of I.c.8, if {Xn }:'=1 CM(8) for some 8>0 then

By Khintchine's inequality 1.2.b.3 and the triangle inequality in 12 , we get, for arbitrary functions {J;}r;.l in an LI (Q, E, Jl)-space,

where {r;}~= 1 denote the first n Rademacher functions. Hence,

for every choice of {a;}~= 1. On the other hand, if there is no 8>0 so that {xn}:'= 1 CM(8) then we proceed exactly as in the proof of l.c.8 and choose a subsequence {Xn) J= 1 of {Xn}:'= 1 which is equivalent to a sequence of disjoint elements in X and thus, unconditional. 0

In connection with I.c.IO, consider the space E of Maurey and Rosenthai which was presented in I.I.d.6. It follows from the properties of the sequence


{mi} ~ 1 of integers constructed there that, for every 1 > 1'f > 0, it is possible to find integers hand j so that

j-l (m.)1 /2 h 2 L ---.! ~-~.1'f. i=l mj mj

j-l

Therefore, for k = L mi + h, the unit vector basis {en}:'= 1 of E satisfies. i= 1

i.e.

k- 1/2 max II.i Bieill~2!1' e;=±l,=l

This fact and the property of E that no subsequence of {en}:'= 1 is unconditional, imply, by l.c.l0, that Eis not isomorphie to a subspace of an order continuous Banach lattice.

d. p-Convexity in Banach Lattices

In this section we introduce and study the mutually dual notions of p-convexity and q-concavity in Banach lattices. These two notions turn out to be an important tool in the study of isomorphie properties of lattices. For example, they playa crucial role in the study ofuniform convexity in Banach lattices (in Section fbelow) and in the study of rearrangement invariant function spaces (in Seetion 2.e. below).

In the definition of these notions there enter expressions of the form (tl IXiIP) l1P, wherep~ 1 and the {xi }f=l are elements ofa latticeX. IncaseXis ordercontinuous such an expression can be easily defined by using the representation theorem 1. b.14. We need however a proper definition ofthis expression for general Banach lattices. We start this section by presenting a method which allows us to define even more

complicated expressions than (~1 IxilP) IIp in general Banach lattices.

Let Jrn be the family of all functions l(t l' ... , tn): Rn ~ R which are obtained from the functions q>i(t l' ... , tn) = ti, i = 1, ... , n, by applying finitely many

operations of addition, multiplication by scalars and finite suprema and infima. It is easily seen that eachl E Jf',. is continuous on R" and homogeneous of degree one i.e., for every A ~ 0, I(At l' ... , Atn) = J.f( t 1 , ••• , tn). Therefore, for any 1 E Jf',., there exists a constant M< 00 so that II(t 1> ••• , tn)1 ~ M(lt 11 v ... v Ilnl) for all (t l' ... , tJ E Rn.

d. p-Convexity in Banach Lattices 41

Let f E JIl'n and let {x;}? = 1 be a finite set of elements of a Banach lattice X. By replacing formally each of the variables t i with the corresponding vector Xi' we can give a meaning to the expression f(x l' ... , Xn) E X. This procedure defines the element f(x l' ... , xn) in a unique manner in the sense that if f(t l' ... , tn) = g(t l' ... , tn) for some g E JIl'n and every (t l' ... , tn) E Rn then also f(x l' ... , Xn) = g(X l' ... , Xn). The proof of this fact is trivial when X consists of functions on so me set and the order in Xis defined pointwise. The case of a general Banach lattice can be reduced to the function case in the following way. Put x o= Ixll'v .,. v Ixnl and observe that both f(x l' ... , Xn) and g(X l' ... , Xn) belong to the (in general nonclosed) ideal I(xo) ofall XE Xforwhich there exists somd ~O so that lxi::::; Axo/llxoll. In the idealI(xo) we define the norm Ilxoll oo = inf {A ~O; lxi ::::;.,ho/llxoll} and observe that the completion of I(xo), endowed with the norm 11·1100' is an abstract M space with a strong unit. Thus, by I. b.6, the completion of (l(xo), 11·1100) is order isometrie to aspace of continuous functions. The observation above shows that in this M space we havef(x l , ... , xn)=g(x l , ... , xn) i.e.llf(x l , ... , xn)-g(x l , ... , xn)lloo=O. This implies thatf(x1, ... , xn)=g(x1, ... , xn) also in X since Ilxll ::::;llxlloo for every XE I(xo).

The fact that f(x 1 , ... , xn) is uniquely defined for every fE JIl',. implies that the map r: JIl'n - X, defined by rf(t l' ... , tn) = f(x l' ... , Xn), is linear and preserves the lattice operations (e.g. iff(t1' ... , tn)=g(t1 , .•• , tn)Vh(tl' ... , tn) in JIl'n then one possibility, and therefore the only possibility to define f(x l' ... , xn) is to put f(x 1, ... , Xn)=g(Xl' ... , Xn) v h(x1, ... , Xn».

The map :r: JIl'n - X can be made into a continuous map in the following way. Let Bn be the subset of Rn of all n-tuples (t l' ... , tn) for which It 11 v ... v Itnl = I (i.e. the unit sphere of the real space l~) and consider JIl',. as a sublattice of C(Bn), the space of all continuous functions on Bn • The map r: JIl'n - Xis continuous when JIl'n is endowed with the norm induced by C(Bn). Indeed, iffor somef E JIl'n we have

then

for every (t l' ... , tn) ERn. Since r is order preserving it follows that

Hence, for every f E JIl' n ,

where Xo = Ix l l v··· v Ixnl. Observe also that JIl'n' as a sublattice of C(Bn), separates the points of Bn and


contains the function identically equal to one (since

Thus, by l.b.5, Jt'n is den se in C(Bn). The closure Jt'n of Jt'n, when the elements of Jt'n are considered as functions defined on all of Rn, consists of all the functions f: Rn ~ R, which are continuous and homogeneous of degree one on Rn. This fact enables us to extend r, in a unique manner, to a map from Jt'n into X which is linear, continuous (when Jt'n is identified with C(Bn)) and preserves the lattice operations.

We collect the observations made above in the following theorem (of. Yudin [131] and Krivine [66]).

Theorem l.d.1. Let X be a Banach tattice and J!..t {Xi}? = 1 be a finite sub set of X. Then there is a unique map r from the tauice Jt' n' of all the functions which are continuous and homogeneous of degree one on Rn, into X such that:

(i) r({>i = xJor I ~ i ~n, where ({>;Ct 1> ... , tn) = ti . (ii) T is linear and preserves the taUice operations. The map T satisfies

for every f E Jt'n'

The element T(f) will be usually denoted by fex l' ... , xn). In many cases we shall work with functions having the form

( n )l/P

for some p ~ 1. It is worthwhile to remark that the vector i ~1 Ix;j P ~an be also

represented by the following formula: if l/p+ l/q= I then

the 1.u.b. being taken (in the sense of order in a Banach lattice X containing the n

vectors {x;}?= 1) over all (al> ... , an) E Rn for which L la;jq ~ 1. Indeed, for every

n (n i=)\IP such (al' ... , an) ERn, we have that i~l aiti~ i~l Itil P for (t1' ... , tn) E Rn and,

n ( n )l/P therefore, by l.d.l, also that i~ aixi~ i~l Ixil p , for {X;}?=l in X. Since the

statement we want to prove for vectors in X is true for scalars we can find an increasing sequence of functions fk(t l' ... , tn) E Jt'n = C(Bn) , k = 1, 2, ... , which are


n n

finite suprema of linear combinations of the form L ai t i with L lail q ~ I, so that i= 1 i= 1

{fk}~= 1 converges pointwise and thus, by Dini's theorem, in the norm of C(BJ to

(tl ItiIPYIP. The continuity ofthe map.: Jf'n -4 X, described in l.d.l, shows that

fk(X l' "', Xn) -4 (~1 lXii p) 1/p as k -4 00 and this completes the proof.

A special ca se of I.d.1 can be used to define the notion of a complex Banach lattice. Let X be areal Banach lattice and let X be the linear space X EB X which is made into a complex linear space by setting (a+ ib)(xl' x 2 ) =(ax1 -bx2 , aX2 +bx1). We define the notions of absolute value and norm in X by putting

The space (X, 11 Ilx) is said to be a complex Banach lattice or, more precisely, the complexification of the real Banach lattice X. As expected, the complex L/J.1) or C(K) spaces are the complexifications of the real L/J.1) or C(K) with the same J.1, respectively K. Since, by definition, every complex Banach lattice is the complexification of a reallattice all the notions and results of this volume can be carried over in a straightforward manner from the real case to the complex case. We shall however continue to assurne in the sequel that, unless stated otherwise, the Banach lattices which we consider are real.

In connection with the functional calculus established in l.d.1 it is often useful to apply the following set of Hölder type inequalities (cf. [66]).

Proposition l.d.2. Let X be a Banach lattice.

(i) For every 0< 0 < land every X, y E X,

(ii) For every choice of I ~p<r<q~ 00, {x;}?= 1 in X and positive scalars

{a i}?=l'

where 0<0< 1 is defined by l/r=O/p+(I -O)/q. (iii) Forevery 1 ~p, q~ 00 with I/p+ l/q= 1 andevery choiceof{xJ7=1 in X and

{ *}n . X* Xi i=l zn ,


( n ~l/q n

As usual, if q= 00 an expression of the form i~l IUil) means i'f.1

Iuil.

Proof (i) Since Is181t11-0 is a homogeneous expression of degree one on R 2 and Isl8W -o:::::;Olsl +(1- O)ltl, for every (s, t) E R 2 , we get, by 1.d.l, that

Illx181Y11-811 = Illcl/8XI8Ic-I/(1-8)yll-811 :::::; OCI/81Ixll + (1- 0)C- I/(l-8)lIyll '

for every c~O. The proof of (i) is then completed by taking c=(llyll/llxll)/J(1-0). Assertion (ii) follows from 1.d.l and the usual Hölder inequality. Also (iii) is an immediate consequence of Hölder's inequality when X and X* are lattices of

functions. In order to prove (iii) for general Banach lattices, put x~ = (tl Ixtlq) l/q

and notice that

defines a seminorm on X which is additive on the positive cone of X. Hence, the completion Xl of X endowed with 11·111 (modulo those elements x E Xl for which Ilxlll =0) forms an abstract LI space which, by l.b.2, is order isometrie to an L 1(Q, E, v) space. Moreover, since Ilxlll :::::;llx~llllxll, for all XEX, the formal identity mappingj from X into Xl is bounded. Observe also that, for each 1 :::::; i:::::; n and x E X, we have

i.e. xt extends to an element gi of Loo(Q, E, v) with IIgdloo:::::; 1. Leth E L 1(Q, E, v), 1 :::::; i:::::; n, be the functions corresponding to the elements jXi E Xl' 1:::::; i:::::; n. Then, by the usual Hölder inequality, we get that

n

wh ich means that L xt(xi ) is :::::; than the number obtained by applying the func-i= 1

tional (~l Igilqy/q E L 1(Q, E, v)* to the element (~1 I/;IPY/P of L 1(Q, E, v).

However, by the uniqueness of the map 1" of l.d.l from the lattice JIt'" of all functions which are continuous and homogeneous of degree one into LI(Q, E, v),

we conclude that the elements (~1 1/;1 p) l/p and j(t1 lXii p) l/p are actually the

same. Similarly, it follows that also the elements (~l Ixtlqy/q andj*(~l Igilqy/q


eoincide. Thus,

and this, of course, completes the proof of (iii). ,0

We turn now to the main topie of this seetion, namely that of p-convexity and p-eoneavity. The starting point is the observation that, for any sequenee Lt;}~= 1 of funetions in Lip,), l:::;"p:::;" 00, we have the equality

or

max 11/;1100' if p = 00 . l~i~n

Ifwe replace in the formulas above the equality sign by an equivalence sign (i.e. we

assume the existenee of a two sided estimate of IICtl IXilPYIPl1 in terms of

( n )llP

i~l Ilxill P , for any n-tuple ofveetors {xJ~= 1 in a Banaeh lattice X) then we get

just a eharaeterization of spaees isomorphie to Lp(p,) spaees (use l.b.13). The notions of p-eonvexity and p-eoneavity arise if we replaee in the formulas above the equality sign by one sided estimates. These two notions were introdueed, under different names, in [31] and [41], for spaees with an uneonditional basis, and in [66], for general Banaeh lattiees as weIl as for operators from and into a Banaeh lattiee.

Definition l.d.3. Let X be a Banaeh lattiee, V an arbitrary Banaeh space and let 1 :::;"p:::;"oo.

(i) A linear operator T: V - Xis ealled p-convex if there exists a eonstant M< 00 so that

or

II .v 1 Tvil II :::;"M m~x Ilvill, if p= 00 , 1=1 l::S;l~n

for every choice of veetors {Vi}? = 1 in V. The smallest possible value of M is denoted by M( p)( T).

46 l. Banach Lattices

(ii) A linear operator T: X - V is called p-concave if there exists a constant M<oo so that

or

for every choice of vectors {x;}~ = 1 in X. The smallest possible value of M is denoted by M(p)(T).

(iii) We say that Xis p-convex or p-concave if the identity operator Ion Xis p-convex, respectively, p-concave. In this case, we write M(P)(X) and M(p)(X) instead of M(P)(I), respectively, M(p)(I).

The constants M(p)(X) and M(p)(X) are called the p-convexity, respectively, the p-concavity constant of X.

Obviously, an operator T, which is p-convex or p-concave, for some 1 ~p~ 00,

is necessarily bounded and M(P)(T) ~ IITII, respectively, M(p)(T) ~ IITII. The cases p = 1 and p = 00 are interesting only in part since every bounded operator T: V - X is l-convex with M(1)(T) = IITII and every bounded operator T: X - V is oo-concave with M(oo)(T) = IITII. In particular, every Banach lattice is both l-convex and oo-concave.

There is a simple and sometimes useful way of interpreting the notions of p-convex and p-concave operators by using some auxiliary spaces. We recall first the definition of the spaces co( V) and li V), 1 ~p ~ 00. These spaces consist of all the sequences v=(v p v2 ' ••• ) of elements of the Banach space V so that a=<llvtll, Ilv2 11, ... ) belongs to co, respectively lp' and the norm of v in co(V) or liV) is, by definition, the norm of a in the respective sequence space.

For a Banach lattice X and 1 ~p ~ 00, we let X(lp) be the space of all sequences x = (x l' x 2 .•• ) of elements of X for which

or

The closed subspace of X~), spanned by the sequences x = (x 1 , X 2' ••. ) which are eventually zero, is denoted by X(lp). The space X(loo)' which is always a proper subspace of XUoo), is denoted also, for obvious reasons, by X(co). For 1 ~p< 00, the


space X(/p) coincides with XTlr,) if and only if every norm bounded increasing sequence in Xis convergent i.e. (in view of l.c.4) if and only if X is weakly sequentially complete. In order to verify this statement we have to show that the sequence

{(tl Ix;j p) I/P} ~= 1 converges in norm if and only if II(tm lXii p) I/Pli ~ 0, as m and

n tend to 00. Both parts of this assertion are easy consequences ofl.d.l and l.d.2. The "only if" assertion, for example, is proved as follows. Letf(s, t) be the continuous function on R 2 satisfying

fes, t)lltl-lsll = Iitl P -Isl pi '

for every (s, t) E R2 • Then, by I.d.l and I.d.2(i), we get, for every y, z E x, that

1IIIyI P -Izipil/PII ~ Illyl-lzllll/pIIJ(lyl, Izl)q/Plll/q

~ CpIIIYI-lzllll/plllyl v Izlill/q ,

where l/p+ l/q= 1 and Cp= max {fes, t)l/p; Isl~ 1, Itl~ I}. A linear operator T from a Banach space V to a Banach lattice Xis p-convex for

some 1 ~p<oo ifand only ifthe map 1': Il V) ~ X(lp)' defined by t(v" v2 ' •.• )= (Tv l , Tv 2 , ..• ), is a bounded operator. Moreover, we have Iltll=M(P)(T). The operator T is oo-convex if and only if l' is a bounded linear operator from co(V) into X(lcx,)=X(co) or, alternative1y, if l' defines a bounded linear operator from loo(V) into X(loo) (and, aga in, we have that 111'11 = M(oo)(T». Similarly, a linear operator T: X ~ V is p-concave for some 1 ~p< 00 if and only if the map t: X(/) ~ IlV), defined by t(xl , x 2 , ••• )=(Txl , Tx2 , ••. ), is bounded. Moreover, Irtll = ~p)(T). Note that if Tis p-concave then t can be actually defined as a map from X(/p) into Il V).

In order to study the behavior of the notions of p-convexity and p-concavity under duality we have to characterize the duals ofthe spaces introduced above. We note first the obvious fact that Ip(V)* is isometric to liV*), where l/p+ l/q= 1 (for p = 00 we have that co( V)* = 11 (V*». We also claim that, Jor every Banach lattice X and every 1 ~p ~ 00, the space X(lp)* is order isometric to X*(lq), where again l/p + l/q = 1.

We shall prove this claim only for 1 <p< 00. For p= 1 andp= 00 the proofis similar and simpler but the notation is somewhat different.

For every (xT, xi, ... ) E X*(lq) and every sequence (Xl' x 2 ' ••• ) E X(lp) which 00

is eventually zero put q>(xl , x 2 ' •.• )= I xi(xJ By l.d.2(iii), we have i= I

Iq>(x l , x 2 ' ···)1 ~ II(x!, xi, ... )11X(lq)ll(x l , x 2 , ••• )llx(lp) .

Hence, q> E X(lp)* and 11q>11 ~ II(x!, xL ... )11X(lq)· Conversely, let ljJ E X(lp)* and, for any integer i, let yt E X* be defined by,

y;*(x) = 1jJ(0, ... ,0, x, 0, ... ) .


The proof of our assertion will be completed once we show that

By the remark fonowing I.d.l, IIC~1 IYTlqy,qll is the supremum of an the

expressions ofthe form 11 ~ .. i:. ai,jYTII, where {ai,j}7= 1 ~= 1 are arbitrary reals with J= 1 ,= 1

n k

L laiJp~ 1 for an 1 ~j~ k, The definition of V in X* implies that, for every i= 1 j= 1 X;;?!: 0 in X,

(.v .f ai,jYt)(X) = sup {.i (.f ai,jYt)(Xj); xj;;?!:O for 1 ~j~k, .i xj=X} )=1,=1 )=1 ,=1 )=1

= sup {",r.i a1 ,jXj' ... , .i an,jXj' 0, 0, ... ); xj;;?!:O for 1 ~j~k, .i Xj=X} ~=1 )=1 )=1

from which we deduce that

where the supremum is taken again over an {Xj}~= 1 as above. By the triangle inequality in lp and l.d.1, we get that

n

since L lai, jl p ~ 1 for an1 ~j ~ k. This completes the proof of our claim concerning i=1

X(lp) * . In particular, we get that X(lp) is reflexive if and only if X is reflexive and 1<p<00.

Suppose now that T: V ~ Xis a linear operator and, for 1 <p< 00, consider the corresponding operator t: I/V) ~ X(lp). It is easily checked that, by the

duality relations established above, (t)* coincides with the operator (T*): X*(lq) ~ Iq(V*), where l/p+ l/q= 1. Hence, Tisp-convex ifand only if T* is q-concave and M(P)(T)=M(q)(T*). Similarly, an operator T: X ~ V is p-concave if and only if T* is q-convex. We collect these facts in the following proposition (cf, [66]).

Proposition l.d.4. Let X be a Banaeh lattiee, Va Banach space and let 1 ~p, q ~ 00 be so that l/p+ 1/q= 1.

(i) A linear operator T: V ~ Xis p-convex if and only if T* is q-coneave and, in this ease M (T*) = M(P)(T) , (q) •


(ii) A linear operator T: X - V is p-concave if and only if T* is q-convex and, in this case, M(q)(T*)=M(plT).

(iii) X is p-convex (concave) if and only if X* is q-concave (convex) and M(q)(X*) = M(P)(X) (M(q)(X*) = M(p)(X».

We study next the dependence of p-convexity and p-concavity on p. For simplicity of notations, we put M(P)(T) = 00 or M(p)(T) = 00 if T is not p-convex, respectively, not p-concave.

Proposition l.d.5. Let X be a Banach lattice and let V be a Banach space. Let T: V - X and S: X - V be linear operators. Then thefunctions cp(rx) = log M<1/a)(T) and t/J(ß) = log M(l/P)(S) are convex. Consequently, M(P)(T) and M(p)(S) are nondecreasing, respectively, non-increasing continuous functions of p on any interval on which they are finite.

Proo! Observe that, by the duality result I.d.4, it suffices to prove that cp is convex. The definition of p-convexity shows that, whenever M(P)(T) is finite, we have

M(P)(T)= sup {iletl /Tvi/pY/PII; Vi E Vfor I ~i~n, itl /lvi/l P= I}

= sup {iletl ai/Tw;jp) I/Pli; wi E V, /lwill = I, ai?;O

for I ~i~n, and itl ai = I} .

But, by the Hölder type inequalities I.d.2(i) and (ii), we get that

whenever I/r=8/p+(1-8)/q and 0<8< 1. It follows that M(r)(T)~M(P)(T)9. M(q)(T)I- 9 and this, of course, implies that cp is convex. By takingp = land using the fact that M(l)(T) = /lT/I ~M(q)(T), we also get that M(r)(T) ~M(q)(T), i.e. that M(P)(T) is a non-decreasing function. That M(p)(S) is non-increasing follows by duality. 0

Before presenting some exampIes, we want to illustrate the manner in which the properties introduced in I.d.3 are gene rally used. A simple and nice application is the following generalization, due to B. Maurey [94], of the c1assical inequality of Khintchine 1.2.b.3.

Theorem l.d.6. (i) Let X be a q-concave Banach latticefor some q< 00. Then there exists a constant C< 00 such that, Jor every sequence {x;}7 = I oJ elements of X, we


have

(ii) Let {xJ ~ I be an unconditional basis oJ a Banach lattice X. Then there is a constant D so that, Jor every choice oJ scalars {aJ7= l' we have

Proo! (i) Let {X;}?=l be an arbitrary sequence of elements of X. Then, by the l-convexity and q-concavity of X, we get that

II i IJl r;(u)x;1 dull ~ i IIJl r;(u)x;11 du ~O II;tl r;(u)x;llq du )I/q

~M(q)(X)IIO l;tI r;(u)X;r du y/qll· Furthermore, by l.d.l and Khintchine's inequality for scalars 1.2.b.3, we have

and

where Al and Bq are the constants appearing in 1.2.b.3. The desired result follows from the monotonicity of the norm in X.

(ii) Let K be the unconditional constant of {x;} ~ I. The left hand side inequality in (i) was proved in every Banach lattice with the absolute constant Al instead of C- 1 . Hence, for all {a;}? = 1 '

Let {xi} ~ I be the functionals in X* biorthogonal to the {x;} ~ I . By the preceding remark we also get that, fur all {bJ?= l'


We prove now the right hand side inequality of (ii). Given {a;}7= l' there are

{bi}7=1 so that lIitl biXill=l and lit aibil;>;llitl aiXilllK. Hence, by 1.d.2(iii),

Ilit aiXill~KIICtl laiXil2Y/21111Ctl IbiXil2Y/211

~AI1K211(tl laixil2Y/211· 0

Remarks. 1. The assumption that Xis q-concave for some q< 00 is essential in (i). Also, we cannot replace in (ii) the assumption that {x;} ~ 1 is an unconditional basis by the assumption that it is an unconditional basic sequence. In order to see this,

consider the Rademacher system {r;}~l in Loo(O, 1). We have Ittl alit = i~l lail

while II(tl lalil2 Y/2t = (~1 at Y/2. It is however clear from the proof that (ii)

remains valid if {xj } ~ 1 is an unconditional basis of a complemented subspace of X. 2. As already remarked in the proof of (ii), the left hand side inequality of (i)

is valid in every lattice X with Al instead of C- 1. The precise value of Al was computed by Szarek [126] (cf. also [49]) who showed that Al = 1/)2.

If Y is a sublattice of X then we obviously have M(P)(Y)~M(P)(X) and M(p)( Y) ~ M(p)(X) for every 1 ~p ~ 00. The situation is more involved if we merely assurne that Y is a subspace of X which is itself a lattice endowed with an order unrelated to that of X. It turns out that under some natural restrietions the properties of p-convexity and p-concavity are still inherited from X to Y.

Theorem l.d.7 [58]. Let X and Y be two Banaeh lattiees and assume that Y is linearly isomorphie to a subspaee of X.

(i) If X is p-eonvex and q-eoneave for some 1 <p~2 and q< 00 then Y is p-eonvex, too.

(ii) If X is p-eoneave for some p;>; 2 then so is Y.

Proof The proofs of (i) and (ii) are entirely similar so we prove only part (i). We first consider the case when Yis aspace with an unconditional basis {Yj}j= 1 of finite length, whose unconditional constant is equal to one. Let Tbe an isomorphism

n

from Y into X and, for l~j~n, put xj=TYj' Let Zi= L aj,jYj' l~i~k, be j= 1

arbitrary elements in Yand let {rj}j= 1 denote, as usual, Rademacher functions.


Then, by 1.d.6(i) applied in X, there exists a constant C< 00 so that

Thus, by the triangle inequality in f21P and the p-convexity of X, we get that

We consider now the case when Y is a general Banach lattice. There is clearly no loss of generality in assuming that Y is separable. Since Xis q-concave for q< 00 it does not contain any subspaces isomorphie to Co (apply 1.cA and the remark following it) and, thus, the same is true for Y. Consequently, we may use the representation theorem 1.b.14 for Y. Since any finite set of disjointly supported measurable functions in Y forms an unconditional basic sequence whose unconditional constant is one, it follows from the first part ofthe proofthat if {j;}~= 1 is a sequence of simple functions in Y then

Since the simple functions are dense in Y the same inequality will hold for every choice of {/;}~=1 in Y. 0

Remark. The restrictions imposed on p and on q in (i) and (ii) of l.d.7 are obviously necessary since, for instance, f2 embeds in any LiO, 1) space, 1 ~p< 00,

and every separable space embeds in C(O, 1). However, as in l.d.6(ii), it is easily


verified that l.d.7(i) holds without the assumption that Xis q-concave for so me q< 00 if Y is isomorphie to a complemented subspace of X.

We present now a general procedure for constructing p-convex and p-concave lattices starting with an arbitrary Banach lattice (cf. [41] and [66]). This procedure isjust an abstract description ofthemapf ---* Ifls signfwhich maps Lr{J.L), I ~ r< 00,

onto Lrs(Ji) (if rs?; 1). In a generallattice X there is no meaning to the symbol x'. We overcome this difficulty by introducing new algebraic operations in X and applying l.d.l.

Let X be a Banach lattice in which the algebraic operations and the norm are, as usual, denoted by +, . and 11 . 11 and let p > 1. For x and Y in X and for a scalar IX, we define

where (x1/p + yl/py is the element in X corresponding, by the procedure described in l.d.l, to the function

and IXP is IlXlp sign IX. The set X, endowed with the operation EB, 8 and the order x @ 0 ~ x?; 0 is, as easily verified, a vector lattice denoted by X(p). Put 11 lxII I = IIxW/P for x E X and observe that 111· 111 defines a lattice norm in X(p). Indeed, for x E X and a real IX,

Also, if x, y E X and IX and ß are positive reals with IXq + ßq = 1, where l/p + I/q= 1, we have by l.d.l and Hölder's inequality

(lxI1/p + Iyll/py~ Ixl/IXP + lyl/ßP .

Hence, by ta king IX P = IIxll1/q/y, ßP= IIyW/q/y and y= (IIxll1iP + IIylll/P)P/q, we get

IIlx EB yIII ~ II(lxI1/p + lyll/PYlil/p ~(IIxil/IXP + IIyII/ßP)l/P = (IIxll 1/p + IIyW/P)l/Pyl/p= IIxW/P + IIylll/p= IIlxill + IIlyIII·

It is also evident that (X(P), 111· 111) is complete (since (X, 11 11) is complete). The lattice (X(P), 111·111) will be called the p-convexification of X. As its name indieates it is p- convex. Indeed, if x, y E X(p)

III(lxIP EB lyIP)l/plliP = 111 lxi + lyl IIIP = lI/xi + lyl 11 ~ IIxil + IIyII = IIlxillp + IIlyll/P '

and hence M(P)(X(p») = 1. More generally, it is easily verified that if X is r-convex


and s-concave for some 1:::::; r:::::; s:::::; 00 then X(p) is pr-convex and ps-concave with

In case Xis a Banach lattice of functions, X(p) can be obviously identified with the space of all the functionsfso thatfP=lflP signfEX endowed with the norm IIlflll=11 IfIPIII,p.

There is also a p-concavification procedure for a Banach lattice X which, however, can be applied only if it is known in advance that Xis p-convex. Let X be a Banach lattice which is r-convex and s-concave for some I <p:::::;r:::::;s:::::; 00.

For vectors x and y in X and a scalar rx put

and IIlxillo = IIxilp. Again, it can be easily verified that X, endowed with the above operations, forms a vector lattice X(P) provided that the order in X(P) is defined as in X. The "norm" 111·1110 is clearly homogeneous but, instead of the triangle inequality, we can only prove that, for {X;}7=1 in X(P) ,

Illxl EB ... EB xnlllo:::::; 11(lxll P + ... + IxnIP)l!pIIP n n

:::::;M(P)(XY L IIxiIIP=M(P)(X)P L IIlxilllo · i=l i= 1

If M(P)(X) were equal to one the "norm" 111· 1110 would have satisfied the triangle inequality but, in general, we have to replace 111·1110 by a different expression, namely

It is easily seen that 111·111 is a lattice norm on X(P) such that

IIlxillo/M(P)(XY:::;IIlxlll:::;IIlxlllo, XE X(p).

The Banach lattice (X(P) , 111·111) is called the p-concavification of X. A simple computation shows that X(p) is r/p-convex with M(r!p)(X(p»:::::;(M(P)(X)M(r)(x»p and s/p-concave with M(s/p)(X(P» :::;(M(P)(X)M(S)(X»p.

The concavification procedure described above can be used to prove the following renorming result from [41].

Proposition l.d.8. A Banach lattice X, which is r-convex and s-concave for some 1 :::::; r:::::; s:::; 00 , can be renormed equivalently so that X, endowed with the new norm and the same order, is a Banach lattice whose r-convexity and s-concavity constants are both equal 10 one.


Proo! We actually prove the theorem only in the case when l<r<s<oo. The cases r= 1 or s= 00 are simpler since the l-convexity and oo-concavity constants of any lattice are always equal to one. The case r=s follows from l.b.13 (in this case Xis isomorphic to L/J.1)}.

Notice now that the r-concavification Y =X(r) of Xis a Banach lattice which is q=s/r-concave. By l.d.4(iii}, the dual Y* of Y is q'-convex, where l/q' + l/q= 1. Hence, by successive q'-concavification and q'-convexification, one can find a new and equivalent lattice norm on Y* such that, endowed with this new norm, Y* has q'-convexity constant equal to one. By using l.d.4(iii} again, it follows that there exist an equivalent lattice renorming on Y** so that the q-concavity constant of Y** becomes one and the same, of course, is true for Y since it is a sublattice of Y**. In order to complete the proof, one just takes the r-convexification of Y endowed with the new norm. 0

The following simple proposition will enable us to describe some classes of p-convex and p-concave operators.

Proposition l.d.9 [66]. Let X and Y be two Banach lattices and let T: X ~ Y be a positive operator. Then,lor every I~p~oo and every choice 01 {X;}?~l in X, we have

Iletl ITXiIPY/PII~IITlllletllxiIPY/l ifp<oo

and IliYl ITXilll~IITlllliYl Ixdll, ifp=oo.

( n )l/P n n

Proo! Since i~l Ixil p ~ i~l aixp whenever i~l lailq~ I and I/p+ I/q= 1, it

follows that

Hence,

and the proof is readily completed by using the monotonicity of the norm. 0

There are some immediate consequences of I.d.9. Let X and Y be two Banach lattices and let T: X ~ Y be a positive operator.

(i) If X is p-convex then T is p-convex and M(P)(T)~ IITII M(P)(X}.

(ii) If Yisp-concave then Tisp-concave and M(p)(T}~IITII M(plY).


From I.d.9 we can also derive some connections between the notions of p-concave operators and p-absolutely summing operators defined in I.2. b.I. Every p-absolutely summing operator T from a Banach lattice X into a Banach space V is p-concave and M(p)(T)~1fiT). Indeed, we have just to observe that for all {XJ~=l in X and x* E X*

A closely related fact is the following result (cf. B. Maurey [94]).

Theorem l.d.l0. Let V be a Banach space, X a Banach lattice, 1 ~p < 00 and M < 00.

An operator T: X-V is p-concave with M(p)(T)~M if and only if, for every positive operator S from a C(K) space into X, the composition TS is p-absolutely summing and niTS) ~ IISIIM. In particular, X is p-concave if and only if every positive operator S: C(K) - Xis p-absolutely summing.

Proof We observe first that, by Hölder's inequality, for every sequence {/;}i= 1

in C(K) and every Jl E C(K)* with IIJlII = 1, we have

and, hence,

(+) sup {(tl IJl(/;WYIP; Jl E C(K)*, IIJlII= I}

= sup {(tllf;(kW)lIP; k E K}.

Suppose now that T: X - V is a p-concave operator, S: C(K) - X a positive operator and {/;}f= 1 are arbitrary functions in C(K). Then, by l.d.9, we get that

(t IITS/;IIPYIP ~M(p)(T)II(t IS/;IPYIPII

~ IISIIM(p)(T)IICtl I/;IP) llPII·

Thus, in view of (+), TS is p-absolutely summing and niTS)~ IISIIM(p)(T). Conversely, assume that np(TS) ~ IISIIM for every positive operator

(m )llP

S: C(K) - X. Let {xj}j= 1 be vectors in X and put X o = '~l Ix jlP . Let I(xo) be

the (in general, non-closed) ideal generated by X o i.e. the set of all x E X for which


Ixl:::;;AxO' for some A.~O. For x E X, set

and notiee that the eompletion of I(xo), endowed with the norm 11·1100' is order isometrie to a C(K) spaee. Let J denote the formal identity mapping from I(xo) into X. Then, by ( + ) applied in I(xo), we have that

i.e. T is p-eoneave with M(p)(T) ~ 1tp(T J):::;; IIJIIM = M. ·0

Remarks. 1. It is easily verified that a lattice X is p-eoneave if every positive operator from Co (instead of an arbitrary C(K) spaee) into X is p-absolutely summing.

2. The faet that a lattiee is p-eoneave for some p > 2 does not neeessarily imply that every bounded linear operator S from Co into it is p-absolutely summing (see [68] or 1.2.b.8). Later on, in Seetion l.f, we shall see that, for p= 2, we ean drop the positivity assumption on S in l.d.lO.

We present now some faetorization theorems for p-eonvex and p-eoneave operators, due to J. L. Krivine [66], which were inspired by results of H. P. Rosenthai [115] and B. Maurey [95].

Theorem l.d.ll. Let X be a Banach taltice, V, W two Banach spaces and fix 1 :::;;p< 00. Let T be a p-convex operator from V into X and S a p-concave operator from X into W. Then the operator ST can be jactorized through an L/Jl) space in the sense that ST= SI Tl' where Tl is an operatorfrom Vinto Lp(p,) with 11 Tl 11 :::;;M(P)(T) and SI is an operator jrom LiJ.1) into W with IIslll ~M(p)(S).

Proo! Let I T be the (in general non-closed) ideal of X generated by the range of T. We define new operations on I T as in the p-eoneavifieation proeedure deseribed above. For x, y E I T and areal 0(, put

and let IT denote the veetor lattice obtained when I T is endowed with the original order and the operations defined above. Set

F l = eonv {x E IT ; lxi:::;; ITvl, for some v E Vwith Ilvll < I/M(p)(T}},

F2 = eonv {x E IT; x>O and IISyll~M(p)(S), for some y with Iyl:::;;x},

where both convex huBs are taken in the sense of 1 T' i.e. by using the new operations.


n

If x=etl 8 Xl EB ... EB etn 8 xn is an element of /T with L et j= I, et j ~O, IxJ:::; ITvjl j= I

and Ilvjll < I/M(pl(T) then

Ilxll ~ 11(let~/Px IIp + ... + let~/Pxnlp)I/PII ~ 11(let~/PTvllp + ... + let~/PTvnlp)I/PII

~M(Pl(T{tl IletflPVjllp rp < I .

On the other hand, if X = ß I 8 Xl EB··· EB ßn 8 xn is an element of J T with n

L ßi= 1, ßi ~O, Xi ~IYil and IISYil1 ~M(p)(S) for all 1 ~i~n then j= I

Hence, FI nF2 = 0 and since 0 is an internal point of FI it follows from the separation theorem that there exists a linear functional cp on IT such that cp(x) ~ I for xEFl and cp(x)~l for xEF2 • Observe that, for every et>O, o<xEIT and every Xo E F2 , we have etcp(x) + cp(xo) ~ 1 since et 8 X EB Xo E F2 . It follows that cp(x)~O whenever x>O and, thus, we can define a semi-norm on I T by putting

Using the linearity of cp with respect to the operations EB and 8, it is readily verified that, with respect to the original multiplication by scalars and addition, 11·110 is homogeneous and satisfies the triangle inequality (the latter fact is proved by arguments similar to those used in the p-convexification procedure).

Observe now that, for any x, Y E IT , we have

since these inequalities are valid for reals. By the fact that cp is non-negative, we get that

Illxl + Iylllg= cp(lxl + Iyl) ~cp«lxlp + lylp)l/P)= cp(lxl EB Iyl)= IIxllg + Ilyllg ~ cp(lxl v Iyl) = Illxl v Iyliig .

This inequality concerning 11·110 clearly remains valid in the completion Z of I T

modulo the ideal ofall XE I T for which Ilxllo=O. Therefore, if lxi" lyl=O for some x and y in the lattice Z then

Le. Z is an abstract Lp space. It follows from l.b.2 that Z is order isometric to an

e. Uniform Convexity in General Banach Spaces and Related Notions 59

LiJ1) space, for a suitable measure J1. Let Tl: V~ Z be defined by T 1v=Tv, VE Vwhen Tv is regarded as an element of I T • If Ilvll<I/M(p)(T) then T 1vEFl wh ich implies that IIT1Vllo~ I i.e. that IIT111~M(P)(T). Let SI be defined by SI X = Sx, X E I T' Then, in a similar way, it ean be shown that SI extends uniquely to an operator from Z into W such that IIS111 ~M(p)(S). This eompletes the proof sinee we clearly have SI Tl = ST. 0

Corollary l.d.12. Let V be a Banach space and fix 1 ~p < 00.

(i) Every p-convex operator T from V into a p-concave Banach lattice X can be factorized through an L p{J1) space in the sense that T= T1T2, where Tl is a positive operator from LiJ1) into X with IIT111~M(p)(X) and T2 is an operator from V into L p(J1) with IIT211~M(Pl(T).

(ii) Every p-concave operator S from a p-convex Banach lattice X into V can be factorized through an L p{J1) space in the sense that S=SlS2' where SI is an operator from L p(J1) into V with Ils111~M(plS) and S2 a positive operator from X into LiJ1) with IIS211 ~M(P)(X).

Proo! Take in l.d.ll S = identity of X, respeetively, T= identity of X. 0

Note that it folIo ws from the proof of I.d.II that if TV is a sublattiee of X then Tl Vis denseLp(J1). Henee, by takingin l.d.ll V= W=Xand S= T=identityof X, we recover a fact which was proved al ready in Section b: a Banach lattiee X whieh is p-eonvex and p-eoneave is order isomorphie to an L p {J1) spaee. Moreover, d(X, L p{J1» ~ M(P)(X)M(p)(X).

e. Uniform Convexity in General Banach Spaces and Related Notions

In the present seetion we investigate so me eoneepts like uniform eonvexity, uniform smoothness, type and eotype, in the eontext of the theory of general Banach spaees. Those aspeets whieh are eharaeteristie to Banach lattiees will be presented in the following seetion. The notions eonsidered here have also a "Ioeal" eharaeter and, therefore, some results, which ean be better understood within the framework of the local theory, will be diseussed only in Vol. III.

We start with some definitions.

Definition 1.e.1. Let X be a Banach space with dirn X~ 2. (i) The modulus of convexity bit:), 0< t: ~ 2, of Xis defined by

bx(t:) = inf {l-llx+ yll/2; x, y E X, Ilxll = Ilyll= 1, Ilx- yll =t:} .

(ii) The modulus ofsmoothness px(r) , r>O, of Xis defined by

px(r)=sup {(llx+YII+llx-yll)/2-I; X,YEX, \lx\\=l,lIy\\=r}.


(iii) Xis said to be uniformly convex if <>x(e) > 0 for every e > 0, and uniformly smooth if lim px(T)/r=O.

, .... 0

In the definition of <>x(e) we can as weIl take the infimum over all vectors x, y E X with Ilxll, lIyll:::;; land IIx-YIi ~e. In order to verify this statement let us note first that we may clearly consider only those pairs x, y with IIxll = I, lIyll:::;; I and Ilx - yll = e. Fix such a pair x, y and let u and v be two norm one vectors in the two-dimensional space containing x and y so that u - v = x - y and, in addition, y, u and v are all contained in one of the half planes determined by the line joining x with -x. Let A~ I and ß~O be such that

A(X+ y)/2=ßu+(l-ß)x .

A quick computation shows that consequently,

A(u+v)/2=(ß+A)U+(I-ß-A)X.

Since ß+A~ max (I, ß) it follows from the triangle inequality that

IIßu+(l- ß)xll:::;; II(ß +A)u+(I- ß -A)xll

(consider separately the cases ß:::;; 1 and ß> 1). Hence,

lI(x+ y)/211:::;; lI(u+ v)/211

and this proves our assertion. Similarly, in the definition of Px(T) we mayas weIl take the supremum over all

x,YEXwith IIxll:::;;1 and lIyll:::;;T. In order to motivate Definition l.e.l, let us recall that a Banach space is called

strictly convex if the equality Ilxll = lIyll = 11 (x + y)/211 = I, for some pair of vectors x and y, implies that x = y. The modulus of convexity of aspace measures in a certain sense its degree of strict convexity. A simple compactness argument shows that a finite dimensional Banach space is strict1y convex if and only if it is uniformly convex. However, there exist many examples of infinite dimensional Banach spaces which are strict1y convex but not uniformly convex (see the remark following l.e.3).

Consider now the notion of smoothness. A Banach space Xis called smooth if, forevery XE Xwith IIxll = I, thereexistsa uniquex* E X* such that IIx*1I =x*(x)= 1. Suppose that X is not smooth and let x E X and u*, v* E X* be so that IIxll = lIu* 11 = IIv*1I = u*(x) = v*(x) = land u* # v*. Let y E X be a norm one vector for which a=u*(y»O and b= -v*(y»O. Then, for every t>O, we have that IIx+tyll~ u*(x+ ty)= I + ta while Ilx- tyll ~ v*(x- ty)= I + tb. Hence, px(T)~(a+b)T/2 and, therefore, X is not uniformly smooth according to Definition l.e.l. An easy compactness argument shows that, for finite dimensional spaces, smoothness is equivalent to uniform smoothness (this fact can be also deduced from l.e.2 below


since it is evident from the definitions that a reflexive space and, in particu1ar, a finite dimensional space is smooth if and only if its dual is strictly convex).

The notions of smoothness and uniform smoothness are closely related also to the question of differentiability of the norm in a Banach space. Since we shall not use this connection in this volume we only discuss it here briefly. It is not hard to see that a Banach space Xis smooth ifand only iflim <llx+tYII-llxll)/t exists for

/ .... 0

every x#O in X and every Y EX. This limit, which is necessarily ofthe form <Px(Y), where <Px E X*, is called the Gateaux derivative ofthe norm in X. A Banach space is uniformly smooth if and only if the limit above exists uniformly in the set {(x, y); Ilxll=IIyII=l}, i.e. if, for every s>O, there exists a 15>0 such that Illx+tyII-llxll-t<Px(y)l<sltl, whenever Itl<b, Ilxll=llyll=1 (if the limit above existsjust uniformly in {y; IIyll = I}, for every x with IIxil = 1, the norm is said to be Frechet differentiable; a uniformly smooth norm is therefore called sometimes a uniformly Frechet differentiable norm).

For a detailed study of the notions of strict convexity, smoothness and differentiability of the norm we refer the reader to [27], [32] and [65].

We start the study of uniform convexity and uniform smoothness by proving a simple duality result.

Proposition l.e.2 [26], [74]. For every Banach space X we have (i) px.(r) = sup {rs/2-bx(S), 0::::;s::::;2}, r>O.

(ii) Xis uniformly convex if and only if X* is uniformly smooth.

Proof (i) We have for r>O,

2Px.(r) = sup {IIx* +ry*II + IIx* -ry*II -2; x*, y* E Bx.}

=sup {x*(x)+ry*(x)+x*(y)-ry*(y)-2; x, y E Bx ' x*, y* E Bx.}

=sup {llx+ yII+rllx- yII -2; x, y E Bx}

=sup {IIx+yll+rs-2; x, Y E Bx ' Ilx-yll=s, 0:::;;s:::;;2} =sup {rs-2bx(s); 0::::;s::::;2} .

The second assertion follows easily from the first. For example, if X* is uniformly smooth then, for every 0<s<2, there exists a r>O so that px.(r):::;;rs/4. Hence, by (i), we get that rs/2 - bx(S)::::; rs/4 i.e. bx(s);?; rs/4. This proves that Xis uniformly convex. The converse is proved in a similar way. 0

Proposition l.e.3 [100], [113]. Every uniformly convex (and thus also every uniformly smooth) Banach space is reflexive.

Proof Assume first that Xis uniformly convex and let x** be a norm one element of X**. Let {Xa}aEA be a directed set in the canonical image iX of X in X** such that xa ~ x** and IIxall::::; 1 for all Ci E A. Since xa+xfJ ~ 2x** it follows that lim IIxa + x fJII = 2. Thus, by uniform convexity, we get that lim IIxa - x fJII = 0 i.e. that a,fJ a,fJ {xa} aEA is norm convergent to some element of iX. This proves that X is reflexive.


If X is uniformly smooth then X** is also uniformly smooth since Px(1:)= Px •• (1:) (use the definition of Px •• (1:) and the w* density ofthe unit ball of iX in that of X**). Hence, by 1.e.2, X* is uniformly convex and, thus, by the first part ofthe proof, X is reflexive. . 0

Remarks. 1. There are simple examples of strictly convex non-reflexive Banach spaces. For instance, if 11·11 denotes the usual norm in C(O, 1) then

(1 )1/2

111/111 = li/li + ! 1/(t)i2 dt

defines an equivalent norm in C(O, 1) which is stricdy convex. Since C(O, 1) is a universal space it follows that every separable space can be given an equivalent strictly convex norm.

2. The converse to l.e.3 is false in a trivial manner since there exist even finite dimensional spaces which are not strictly convex. There are also reflexive spaces which are not uniformly convexifiable, Le. cannot be given an equivalent norm in

which they become uniformly convex. Such aspace is e.g. (~1 Ei1/i) 2 (cf. [25]).

Actually, the following more general fact is true: a Banach space X which contains uniformly isomorphie copies of li for all n (Le., for some constant M< 00 and every integer n, there exists a subspace Bn of X such that d(li, Bn) ~ M) is not uniformly convexifiable. This assertion is an immediate consequence of the following lemma which is a local version of 1.2.e.3.

Lemma 1.e.4 [47]. Let B be afinite dimensional spaee sueh that d(B, 1i')~M2,Jor some n and /or some eonstant M. Then B eontains an n-dimensional subspaee C /or whieh d(C, m~M. Consequently, every infinite dimensional Banaeh spaee X, whieh eontains uniformly isomorphie eopies 0/ I'; for all n, also eontains nearly isometrie eopies 0/ li /or all n (and, henee, bx(e)~bl~(e)=O/or every 0<e<2).

Proo! Let {e i }7: 1 be the unit vector basis of It and let T: li2 - B be an isomorphism such that

for all uE/i2 • If, for some l~k~n, d([Tea~:!'(k-l)n+1,li~M then the proof is already finished. Otherwise, for every I ~k~n, there exists a vector

,ukE[ear:!,(k-l)n+l so that 11 TUkl1 < Ilukll = 1. It follows that

e. Unifonn Convexity in General Banach Spaces and Re\ated Notions 63

The simplest example of a spaee, whieh is both uniformly eonvex and uniformly smooth, is the Hilbert spaee. The moduli of eonvexity and of smoothness of the Hilbert spaee, denoted by b2(e), respeetively P2(e), ean be eomputed easily by using the parallelogram identity and one obtains that

b2(e) = 1-(1-e2/4)1 /2=e2/8+0(e4 ), 0<e<2,

p2(r)=(l +r2)1 /2_1 =r2/2+0(r4), r>O.

Sinee, by a well-known theorem of A. Dvoretzky [35] to be presented in Vol. III, every infinite-dimensional Banaeh spaee eontains nearly isometrie eopies of l~ for all n it follows that the Hilbert spaee is the "most" uniformly eonvex and also the "most" uniformly smooth spaee in the sense that, for any Banaeh spaee X,

Dvoretzky's theorem proves these relations only if dirn X= 00. Aetually, these inequalities are true for every Xwith dirn X~2. This faet is due to G. Nördlander [105] who proved it by using an elegant geometrie argument (see also l.e.5 below whieh implies that bx(e)~Ce2, for every X and for some eonstant C). ather examples of spaees, being in the same time uniformly eonvex and uniformly smooth, are the L p spaees, 1 <p < CIJ. The exaet value of the moduli of eonvexity bp(e) and of smoothness pir) of Lp will be eomputed in Vol. IV: their asymptotical behavior is the following:

b (e) ={(P -1)e2 /8 + 0(e2) , 1 <p< 2 p eP/p2 P+o(eP), 2~p< 00

{r p/ p + o(rP) , 1 <p~2

pir)= (p-I)r2/2 +0(r2), 2~p< 00.

In applieations, we do not use often the preeise value of the moduli of eonvexity or smoothness but only a power type estimate from below or above. We say that a uniformly eonvex (smooth) spaee X has modulus of eonvexity (smoothness) of power type p if, for some O<K< 00, bx(e)~KeP(px(r)~KrP). Using l.e.2, it is easily seen that bx(e) is ofpower type p ifand only if px.(r) is ofpower type q, where l/p+ I/q= 1.

For instanee, it follows from the above formulas that, in this terminology, L p

spaees have modulus of eonvexity of power type 2, for I 2. This faet will be also proved directly in the next seetion (ef. l.f.l below).

We present now a few results which deseribe the behavior ofthe funetions bx(e) and px(r). These results, whose proofs are somewhat teehnieal, are evident whenever bx(e) and px(r) behave like eP, respectively rq• Since this is the situation for L p

spaees and sinee, even for more general spaees, our interest will mostly be in power type estimates for bx(e) and px(r) the readers may choose to skip l.e.5-l.e.8 when first reading this section and then eonsider l.e.9 only in the case where bx(e) and px(r) behave like eP, respeetively rq•


Proposition l.e.5 [74], [40]. The modulus of smoothness px(r) of a Banach space X is an Orliczfunction satisfying the iJ 2-condition at zero and px(r)/r2 is equivalent to a decreasing function. More precisely, Px( r) is a non-decreasing convex function with Px(O)=O, lim sup px(2r)/px(r)~4 and there exists an absolute constant C<oo so

<--+0

that px(rf)/yt2 ~ CPx( r)/r 2 , whenever yt > r > o.

Proo! The assertion that px(r) is a non-decreasing convex function is a direct consequence of l.e.2(i) and of the fact that px •• (r) = px(r) , r >0. In order to prove that px(r) satisfies the iJ 2-condition at zero, we fix r >0 and let x, y E X be vectors with Ilxll = 1 and lIyll = r. A simple computation shows that

and

Hence,

(1Ix+ 2yll + Ilx- 2yll)/2-1

~ Ilx+ yl Ipx(r/I Ix + yll)+ Ilx- yllpx(r/llx-yll)+ Ilx+ yll + Ilx-YII-2

~ Ilx+ yllpx(r/llx+ yll) + Ilx- yllpx(r/llx-yll)+2px(r)

and, by taking the supremum over all possible choices of x and y, we get that

If 0 < r < 1/5 then, by the convexity of px(r), we get that

which further implies that

i.e. there exist a O<ro < 1/5 so that

(*) px(2r)~(4+ l5r)px(r), whenever O<r<ro .

This inequality, which already shows that px(r) satisfies the iJ 2-condition at 0, will also be used to prove that px(r)/r2 is equivalent to a decreasing function. Let 0< r < yt and distinguish between the following cases:

Case I: r~ro. By convexity of px(r), we get that


and, since px(rf)~y/, we have

This completes the proof in Case I. Case 11: O<y/~To' Choose an integer m such that y//2m~T<y//2m-l and ob

serve that, by (*),

m p (y//2i-1) m PX(y/)/y/2 ~(pX(T)1Y/2).TI x ( /2 i) ~(4mpx(T)/y/2) .TI (I + 15y//4.2j )

J;1 Px y/ J;1

00

~4(Px(T)/T2) TI (l + 15To/4·2 i ) . j; 1

By combining the results obtained in the first two ca ses, we obtain, in Case 111: T<To<Y/, that Px(y/)fy/2~CPx(T)/T2, where

00

C=(4TO/P2(T O)) TI (l+15To/4·2 i )<00. 0 i; 1

Proposition l.e.6 [40]. The modulus of convexity Dx(e) of a Banach space X is equivalent to the Orliczfunction bx(e) = sup {eT/2- Px.(T); T~O}. Thefunction bx(e) is the maximal convex function majorated by Dx(e) and ~x(e)/e2 is equivalent to an increasing function.

Proo! The fact that the function ~x(e), as defined in the statement of l.e.6, is nondecreasing and convex is obvious while ~x(e) ~ Dx(e) follows from l.e.2. In order to prove that bx(e) is the maximal convex function majorated by Dx(e), it suffices to note that if a linear function ae+b with a>O is majorated by Dx(e), for every 0<e<2, then it is also majorated by ~x(e). Indeed, if bx(e)~ae+b, 0<e<2 then, for any 0< Y/ < 2, we get that

We next show that bx(e)/e2 is equivalent to an increasing function. By l.e.5, there exists a constant C< 00 so that Px.(Tl)/Ti ~ CPx.(T2)/TL whenever TI> T2 >0. Fix 0< e1 < e2 < 2 and consider first the case when e1 C ~ e2. Then, by the convexity of bx(e), it follows that

In the other case, i.e., when e1 C< e2, we put IY. = e2/Ce1 > I. Then,

- 2 } DX(e1) ~ sup {eI T/2 - Px.(IY.T)/CIY. ; T ~O

=C- 11Y.- 2 sup {e1TCIY. 2/2-px.(IY.T); T~O}

= C- 11Y.- 2 ;)x(e1 CIY.) = C- 11Y.- 2 bX(e2) = CbX(e2)eile~ .


It remains to show that Dx(e) is equivalent at zero to bx(e). This fact is proved by the following two lemmas.

Lemma l.e.7. Let 15 be a non-negatjve function on some interval [0, T] such that D(e)/e is equivalent to an increasing function i.e. D(e}/e~ C [)('1}/'1,jor some constant C< OCJ andfor everyO<e<'1 ~ T. Let;5 be the maximal convexfunctionmajorated by D. Then b(e}?D(e/2)/C,jor all O<e~ T.

Proo! For each 0 ~ '1 ~ T, let !F~ denote the collection of all linear functions fee) which intersect the graph of D(e} in at least two distinct points ('11' D('11}) and ('12,15('12» with 0~'11 ~'1~'12~T. By the definition of b, we have ;)('1)= inf {j('1};J E !F~}. Any functionf E !F~ is of the form

with '11 ~ '1 ~ '12 ~ T. Suppose now that '11 ~ '1/2. Then,

In the other case, i.e. when '11 >'1/2, we get that

Consequently, also ;5('1} exceeds D('1/2)/e. 0

Lemma l.e.8. For every Banach space X, Dx(e}/e is a non-decreasing function on (0,2].

Proo! Fix 0< '1 < e < 2 and vectors x, y E X such that Ilxll = Ilyll = 1 and Ilx - yll = e. In the two-dimensional subspace of X generated by x and y (see the illustration opposite) let OA, OB and OC represent the vectors x, y, respectively (x+ y)/llx+ yll.

Since AB=e we can find points A' and B' on AC, respectively BC, so that A'B'='1 and A'B' is parallel to AB. It is easily seen that the midpoint D' of A'B' is situated on Oe.

Denote now the vectors OA' and OB' by x', respectively y', and notice that CD'/A'B' = CD/AB. Since CD' = 1-II(x' + y')/211 and CD= I-II(x+ y)/211 we have

Dx('1)/'1~ (1 -1I(x' + y'}/211)/'1 = (I-II(x + y)/211)/e

from which, by taking the infimum over all x, y E X as above, we get that


A

0,,-__

c

B

Remarks. The modulus of convexity of a Banach space need not be itself a convex function. An example of aspace whose modulus of convexity is a nonconvex function was given in [80].

Except for a few special cases it is in general quite difficult to compute precisely or even up to an equivalence constant the modulus of convexity of a given space. Such computations were made for Orlicz spaces in [91] (see also [40]) and for some Lorentz sequence spaces in [I]. For instance, the modulus of convexity bM(E) ofthe Orlicz sequence space IM with M(t) = tP/(l + Ilog tl),p ~2, is equivalent at zero to the function EP/(l + lIog EI). More generally, if (i) M is a super-multiplicative function (in the sense that M(st)~cM(s)M(t), for some c>O and every O~s, t~ I) and (ii) M(t 1/2 ) is equivalent to a convex function, then the modulus of convexity bM(E) of IM is equivalent to M(E) itself. Results of a similar nature were obtained for Lorentz sequence spaces. Let d( w, p), p ~ 2, be a Lorentz sequence space for which the sequence w = { w n}:= 1 has the following property: there is a constant c > 0 such that S(kn) ~ cS(k)S(n) for all integers k and n, where

n

S(n)= L Wio The modulus of convexity bx(E) of X=d(w,p) with w as above is i= 1

then equivalent to the function I/S -1(l/EP), where Set) is a strictly increasing function on [0, CD) coinciding with Sen) for t = n.

We prove now a result ofT. Figiel and G. Pisier [43] and T. Figiel [40] which shows that the modul i of convexity and of smoothness of a Banach space X and those of L 2(X), the space of all measurable X-valued functions f on [0, I] such that

(1 )1/2

IIfIIL2(Xj= ! IIf(t)IIi dt < 00, are equivalent functions (a vector valued

function is called measurable if it is the limit in norm of a sequence of simple measurable functions; for details see [32] 111.2.10). The significance of this seemingly special result will become c1ear later on in this seetion (cf. l.e.16).


Theorem 1.e.9. For every Banach space X there exist constants a, b>O and C< 00

so that

(i) <>x(e);;:>; <>L2(X)(e) ;;:>;a<>x(be) , O~ e~ 2, (ii) PX('r)~PL2(X)('r) ~ Cpx(,r) , O~'r.

For the proof of 1.e.9 we need first the following lemma.

Lemma 1.e.l0. For all x, Y E X which satisfy IlxI12+IIYIl2=2, we have Ilx+yI12~ 4-4<>x(llx- yll/2).

Proof Suppose that Ilxll;;:>;llyll. We mayaIso assume that (1Ix ll-llyll)2< 4<>x(lIx-YIi/2) since, otherwise, the desired result follows from

Ilx+ yI12 ~ (lIxii + IlylI)2 = 2(IIx l12 + Ily112)

-(llxll-llylI)2~4-4<5x(llx-yli/2) .

(1Ix ll-llyll)2 <4(1-(I-llx-yI12/16)1/2)

~4(1-(I-llx-yI12/16»= Ilx-YI12/4.

Put z=xllyll/llxll and observe that, by the preceding inequality,

IIY - z 11;;:>; Ilx - YII-llx - zll = Ilx - yll- (1Ixll-llyll);;:>; Ilx - yll/2 .

Hence,

from which it follows that

By using the fact that <>x(e)/e is a non-decreasing function (cf. l.e.8) and the fact that Ilyll ~ I, we get

Ilx+ yI12~(llxll+ Ilyll-2<>x(llx- ylI/2»2

~4(l-<>x(llx-yll/2»2~4-4<>x(llx-yll/2). 0

Proof of 1.e.9. The left hand inequalities in (i) and (ii) are immediate since Xis isometrie to a subspace of LiX). We prove now the right hand side inequality of (i). Letfand g be two elements of LiX) such that Ilfll= Ilgll= 1. There is no loss of generality in assuming that q>(t) = «lIf(t)lli + IIg(t)lli)/2)1/2 does not vanish


(otherwise, we integrate only over {t; cp(t»O}). By l.e.lO, we get that

1

Ilf+gl1 2 ~ S cp2(t)(4-4bx (lIf(t)-g(t)lIx/2cp(t))) dt. o

Put lX(e) = bx (e 1/2 ) and observe that, by the last part of l.e.6, lX(e)/e is equivalent to an inereasing funetion. Thus, by l.e.7, aCe), the maximal eonvex funetion majorated by IX, is equivalent to lX(e). This implies that there exist eonstants K 1 , K 2 >0 so that

1

Sinee S cp2(t) dt= land a is eonvex it follows that o

1 1

S cp2(t)bx(lIf(t) - g(t)lIx/2cp(t)) dt ~ S cp2(t)a(lIf(t) - g(t)lIi /4cp2(t)) dt o 0

Thus, IIf+gll 2 ~4-4a(lIf-gIl2/4) whieh implies that

and henee,

Assertion (ii) follows from (i) by duality. Observe that, for eomputing PL2(X)' it is enough to eonsider expressions of the form IIf+TglI+llf-TglI, withfand g

n times ,

being simple measurable funetions, and that (X EB X EB ... EB X)~ is isometrie to n times ,

(X* EB X* EB ... EB X*)2' 0

Remark. In the last step of the proof above we did not use the duality between L 2(X) and L 2(X*) sinee this faet is less elementary than the duality between (X EB X EB ... EB Xh and (X* EB X* EB ... EB X*}z. It is true that L2(X)* = L2(X*) if X is reflexive (and, in partieular, if X is uniformly eonvex) but this is no longer true for general X. It turns out that L2(X)* = L 2(X*) if and only if X* has the Radon-Nikodym property (see [28] Chapter IV for a detailed diseussion).

We present now some results on uneonditionally eonvergent series in uniformly eonvex and uniformly smooth Banaeh spaees.

Theorem 1.e.1l. Let {xJ0= 1 be a sequence of elements of a Banach space X.


(i) If 8~a:l Ilj~l OjXl~2,Jor some n, then j~l (jx(IIXjll)~1. Consequently, if 00 00

L Xj is an unconditionally convergent series then .L (jx(llxjll)< 00. j= 1 J= 1

(ii) For every .A.>O there exists a choice oJsigns Oj= ± 1, j= 1, 2, ... so that

00

Jor every integer n. Consequently, if aseries L OjXj diverges Jor every j=l

00

choiceoJsignsOj=±1 then L Px(IIXjll) = 00. j= 1

Part (i) was first proved in [60] and Part (ii) in [74].

Proof (i) Suppose that 8~a±xl Iljtl OjXj11 ~2. Clearly, there is no 10ss of generality

in assuming that IISn 11 = flil Xjll~llj~l OjXjl/ for every choice ofOj=±1. Then,

for each 1 ~~n, we have

which implies that

jtl (jx(IIXjll)~ jtl (l-'ISn-Xj'I/"Snll)~n-lIjtl (Sn-X)II/IISnll

=n-ll(n-l)SnII/IISnll= 1 .

00

If the series L x j converges unconditiona11y then, by removing some terms if j=l

needed, we can assume that 8jS~~l IIj~l OjXjll~2 (cf. L1.c.l) and then, by the

00

first part ofthe proof, we get that L (jx(IIXjll)~ 1. j= 1

(ii) We will choose the signs {O j } i= 1 inductively. Fix .A. > 0 and assume tha t the first n signs have already been chosen so that

n

for every 1 ~k~n. Put Sn= L OjXj and observe that j= 1


Thus, there exists a choiee of (}n+1 = ± 1 such that IISn/IISnll+A(}n+1xn+111~ 1 + Px(Allxn + 111) i.e.

IISn + AllSn II(}n+ 1 Xn + 111 ~ IISn 11(1 + Px (AIIXn + 111)) . Using this inequality and the identity

Sn+1 =Sn+(}n+1 X n+1 = AII~nll (Sn+ AIISnll(}n+1 X n+1)+( 1- AII~nll) Sn

we get that if AllSnil > 1,

IISn+ 111 ~ ).II~n 1IIISn 110 + Px().IIXn + 111)) + (1- AII~n 11) IISn 11

~ IISn 110 + Px(A·IIXn + 111))

If, on the other hand, AllSn 11 ~ 1 then every choiee for (}n+ 1 is acceptable since

This proves the first part of (ii). 00

Suppose now that L: Px(llxjll)< 00. Since Px satisfies the Llz-condition at j= 1 00

zero, by l.e.5, it follows that, for every integer k, L: Px(2kllxjll)< 00. Observe j=1

also that lim Ilxjll=O for lim Px(llxjll)=O and j--+a:J j-"oo

These two facts imply the existence of an increasing sequence {nd~ 1 so that 00

f1 (1 +Px(2kIIXjll))~2 and sup IIXjll~ 1/2k for all k. Thus, by the first part of j=nk j?!.nk

(ii), we can find signs {(}j}j~+n~-1, k= 1, 2, ... such that, for nk~n<nk+1' we have

that IljEk (}jXjll ~ 1/2k - z. Hence, j~1 (}jXj is a convergent series. 0

The moduli of convexity and of smoothness of a Banach space are only isometrie invariants and they may obviously change considerably under an equivalent renorming. Therefore, it is natural to ask whether the corresponding moduli for lp spaces or, e.g., for the Orlicz and Lorentz sequence spaces considered in the remarks following 1.e.8, are the best ones up to equivalence, or it is possible


to improve them by a suitable renorming. The key to the study of this question is usually l.e.ll which, for instance, can be used to conclude that in aspace X having a normalized unconditional basis {Xn }:'= 1 with unconditional constant equal to K,

for all n. In particular, for a Banach space X so that d(X, lp) = K for some p > 2, we get that (jx(e)~K2Pep, whenever e= I/K 2n1/p, n= 1, 2, ... Hence, up to a constant, (jie) (or ep) is the best modulus of convexity that 11" p~2 can be given by an equivalent renorming. For p ~ 2 the modulus of convexity of 11' cannot be improved asymptotically by renorming since always (jX(e)~(j2(e)~e2.

It follows from a deep result of Krivine [67], to be presented in Vol. III, that in the case of an infinite dimensional L/J1) space, I< p< 00, the modulus of convexity or smoothness cannot be improved at all (i.e. not only asymptotically) by any renorming.

By computing the value of lIi~l Xi 11 in the Orlicz and Lorentz sequence spaces

discussed before l.e.9 we conclude again that M (e), respectively 1/ S - 1 (e - 1'), are asymptotically the largest moduli of convexity which can be achieved by an equivalent renorming.

00

The two cases encountered in l.e.ll are quite extreme; the series L {)jXj j= 1

is required to converge or to diverge for every choice of signs {)j= ± 1. In many situations, it is very useful to study the behavior of the series for most choices

00

of {{)J~ 1 or, more precisely, to study the series L rit)xj for almost all tE [0, 1], j= 1

where {r j };;'l denotes the sequence of the Rademacher functions. In order to study this question we consider the expressions

and introduce the following two important notions (cf. J. Hoffmann-J0rgensen [53]).

Definition l.e.12. A Banach space Xis said to be of type p for some 1 <p~2, respectively, of cotype q for some q~2, if there exists a constant M< 00 so that, for every finite set ofvectors {Xj}j=l in X, we have


respectively,

Any constant M satisfying (*) or (!) is called a type p, respectively cotype q, constant of X.

The cases p = 1 and q = 00 are not interesting since every Banach space is both

( ( n )l/q

oftype 1 and cotype 00 with 1rr:::n Ilxill replacing the expression i~l Ilxillq

when q = 00 ). The conditions imposed on p, respectively q, are explained by the

following remark. If all the vectors {XJj=l coincide with some vector x E Xwith

IIxll = 1 then j 11.i: ri(t)xill dt = j I.i: rit)1 dt and, by Khintchine's inequality o J= 1 0 J= 1

1.2.b.3 in L 1(O, I), it follows that

where At = 2 - 1/2 • This evidently shows that no Banach space can be of type p > 2 or of cotype q< 2.

The spaces Lip), 1 ~p< 00 are of type min (2, p) and cotype max (2,p). Assume, for example, that 1 ~p~2. Then, since LiJ1.) isp-convex andp (and thus also 2)-concave, we get by Khintchine's inequality (cf. l.d.6(i» that for every choice of Uj}j=l cLiJ1.)

A1Ct1 IIJJI12 Y/2 ~A111Ct1 IJJI2 Y/2 11 ~ i lIit1 ri(t)JJII dt

~O lIitt ri t)JJlI 2 dtY/2 ~IICt1 IJJ1 2Y/2

11

~IICt 'JJ'PY'PII=Ct IIJJIIPYIP

Our assertion for 2p and not of cotype r for any r<p. It is also evident that an infinite dimensional La;,{J1.) space (or more generally, an M space) is not of type p for any p > 1 and not of cotype p for any p < 00.

It follows from the preceding remarks that Hilbert spaces have the "best possible" type and cotype, i.e. are simultaneously of type 2 and cotype 2. The converse of this assertion is also true. Weshall present in V 01. III the proof of


the fact (due to Kwapien [69]) that every space, which is simultaneously oftype 2 and cotype 2, is isomorphic to a Hilbert space. Since the natural context for the study of type and cotype is the local theory of Banach spaces we postpone to Vol. III also the presentation of some other important results concerning these notions. For instance, it will be shown there that a Banach space X has some type p > 1, respectively, some cotype q< 00 if and only if X contains no uniformly isomorphic copies of I;, respectively I:', for all n. (For Banach lattices this result as weH as Kwapien's result will be proved already in the next section).

The L 1 average! Iljt rj(t)xjll dt can be replaced in l.e.12 by any other L r

average, 1 <r<oo, without affecting the definition. More precisely, a Banach space Xis 0/ type p or cotype q if and only if there exists an 1 < r < 00 and a constant 0< M r < 00 so thatJor every finite set {xj }j= 1 in X, we have

respectively,

This assertion is evident for lattices which are q-concave for some q< 00.

Indeed, by 1.d.6(i) and its proof, in such a lattice aH the Lr averages

are equivalent to IIC~l Ixj l2 }/211 and are therefore mutually equivalent. It turns

out that these L r averages are mutually equivalent also if the lattice is not q_ concave for any q< 00 and, what is more interesting, this is true even in an arbitrary Banach space. This fact is due to Kahane [62].

Theorem l.e.13. For every 1 < r < 00 there exists on constant Kr < 00 so that,Jor any Banach space X and every finite subset {xj }j= 1 0/ X, we have

The left-hand side inequality is trivial. The proof of the right-hand side inequality which we present is due to C. Borell [14]. This proof is based on the following inequality from [7].

e. Uniform Convexity in General Banach Spaces and Re1ated Notions 75

Lemma l.e.14. Let 1 <p~q<oo and y=J(p-l)/(q-l). Then, Jor every u~o, we have

Proo! Assume first that 1 <p~ q~ 2. Then al1 the binomial coefficients (2i) and

(Jk) are positive and c1early

Therefore, since (1 + w y ~ 1 + rw whenever 0< r ~ 1 and w > 0, we get

(0 + yu)q + (I- YU)q)P/q =(1 + I (q) y2kU2k)P/q 2 k= 1 2k

~ 1 + I: (p) U2k (1 +u}p+{l-u)p k=l 2k 2

which completes the proof if u~ 1. If u>1 then since 11±yul~lu±yl and since we have already verified the

desired inequality for l/u we deduce that

(li +YU lq ; 11- YU lq) l/q ~(Iu+ Ylq; lu- ylq) l/q

(11 + l/ul p + 11-1/uIP)1/P ~u 2

=(/1 + ulP ; 11- uIPY/P

This shows that l.e.14 is true for l<p~q~2. It is easily verified that l.e.14 is equivalent to the statement that the operator T, defined by

1 1

TJ(s) = J J(t) dt + Y J J(t)r 1 (t) dt· r 1 (s), o 0

is an operator of norm one from L/O, 1) into Lq(O, 1). Since y is also equal to

J(q'-I)/(p'-I), where l/p+l/p'=1 and l/q+l/q'=I, we get by considering T* that l.e.14 is also valid for 2 ~p ~ q < 00. The general case 1 <p ~ q < 00 is then deduced without difficulty. 0


Corollary l.e.IS. Let 1 <p";;'q< 00 anti y=J(p-I)/(q-l). Then,/or every choice 0/ vectors Y1 and Y2 in an arbitrary Banach space Y, we have

(11Y1 +YY2Iq~"Y1 -YY21Iqytq,.;;,("Y1 +Y2Ip;"Y1-Y2IPytP .

Proof Put Z1=Y1+Y2' Z2=Y1-Yl, u1=(lIz111+llz211)/2 and u2=lllz111-llz2111/2. Then, since O<y";;' 1, it follows from 1.e.14 that

("Y1 +YYllq;"Y1 -YY2W)1tQ

,.;;,((1 + y)IIZ111/2 + (1- Y)IIZ2 11/2)q; «(1- y)IIZ111/2+ (1 + Y)IIZ2 11/2)qytq

=(lu1 +yu2IQ;lu1-YU2IQytq ,.;;,('u1 +u2IP;lu1-U2IPytP

=(IZ1IP;"z2IPytp. 0

Pro%/ l.e.J3. Let 1 <p";;'q<oo and y=J(p-I)f(q-I), as before. The main step of the proof consists of showing that, for every choice of vectors {x j }j=o in an arbitrary Banach space X, we have

The proof of 1.e.13 follows then from (*) by taking xo=O and by using Hölder's inequality. Indeed, for any 1 <r< 00,

o IIit1 rj(t)Xjll

r dt ytr ,.;;,0 II jt1 rj(t)xjll dt yt(2r-1l 0 IIjt1 rj(t)Xir dt r-1lt(2r2-rl

from which, by taking p=r and q=2r in (*), we get thai 1.e.I3 is valid with Kr = «2r-I)/(r-l))'-1.

Assertion (*) for n = 1 is just I.e.I5. Suppose now that (*) has been already proved for n and let {Xj}j~~ be a system of vectors in X. Then, by the induction hypothesis, it follows that


where

Hence, by the fact that Lp(O, 1) has q-concavity constant equal to one and l.e.15 n

(used with Yl =Xo + L rj(t)xj and Y2 =xn + l ), we get that i=l .

Remark. The inequality (*), obtained in the proof of l.e.13, yields actually a <Xl

stronger result (due to Kwapien [70]), namely that if aseries L rj(t)xj conj=l

verges in LI (X) then

for every C>O. Indeed, by using it with q=2k andp=2, we get that

<Xl

In view of l.e.13, this shows that if L rj(t)xj converges in LI (X) then j= I

for sufficiently small values of C. To prove the convergence of the integral for arbitrary values of C one uses the fact that

I cllJI rJ(t)xj I1

2 (rl 2cl11t rJ(t)xJI12d ) (rl 2C IIJ+ 1 rj (t)XJI1 2 d.) re dt ~ e t e t

o 0 0

and that j 11. r. r j (t)xj I12

dt can be made as small as we please if n is chosen o J=n+l

large enough. The notions of type and cotype are related to those of uniform convexity and

uniform smoothness (cf. T. Figiel [40] and T. Figiel and G. Pisier [43]).


Theorem l.e.16. (i) A Banach space X which has modulus of convexity of power type q,jor some q ;;:::2, is also of cotype q.

(ii) A Banach space X which has modulus of smoothness of power type p, for some I <p ~2, is also of type p.

Proof (i) The key point in the proof consists of the fact that, for any finite set {xj }j= 1 in X, the elements {rj(t)xj }j= 1 form an unconditional basic sequence (with unconditional constant equal to one) in L 2(X). Thus, it follows from l.e.11 (i) applied in L 2(X) that if

n

then L: c5 L2(X)(llxj 11) ~ I . j= 1

By l.e.9(i), the modulus of convexity of X and L2(X) are of the same power type. Therefore, we can find a constant C< 00 such that c5 L2(X) ;;::: eq / cq for every 0 ~ e ~ 2.

( n )11q (1 11 n 112 )1/2 Consequently, j!:1 Ilxjllq ~c, whenever ! j!:1 rj(t)xj dt ~2, and this

completes the proof of (i) in view of the remarks following Definition l.e.12. The proof of (ii) is quite similar. By l.e.9 (ii), there exists a constant D< 00

such that PL2(xlr)~D,P for every ,;;:::0. Thus, byapplying l.e.ll (ii) to L 2 (X) with Je = D - 11p and by using the identity 1 + u ~ eU , u;;::: 0, we get that, for some choice of signs Oj= ± I,

( n )11P

Hence, if j~1 Ilxjll P ~ I then

The converse to l.e.16 is false. We have already verified that the non-reflexive space L 1 (0, I) is of cotype 2. There is also a non-reflexive space of type 2 (cf. [55], this will be discussed in V 01. III). Also, it is not true that a Banach space X is of type p, for some p > I, if its dual X* is of cotype q = p/(P - I). For instance, X = Co

is of no type p> I while X* = /1 is of cotype 2. (It is an open problem whether there exist such counter-examp1es if we restriet ourselves to uniformly convex spaces). However, the following is true (cf. [53], [94]).

f. Uniform Convexity in Banach Lattices and Related Notions 79

Proposition l.e.17. Let X be a Banach space 0/ type p /or some p> 1. Then its dual X* is 0/ cotype q=p/(p -I).

Proof For every e>O and every choice of {Xt}~=l in X* we can find vectors {X;}~=l inXso that IIxtll«1 +e)xt(x;) and IIxdl= 1 for an 1 ~i~n. It follows that

(tl IIxtllq)l/q ~(1 +e)(tl Xt (X;)q)l/

q

=(1 +e) sup {tl a;xt(x;); ;tl la;lp~ I}

=(1 +e) sup {J (.t r;(u)xi) (t rj(u)ajxj ) du; .± lajlP~ I} . o .-1 J=l J=l

Hence, by Hölder's inequality, we get that

( n )l/q (1 11 n Ilq )l/q

;~1 IIxtllQ ~(I +e)MKp t ;~1 r;(u)xi du ,

where M is a type p constant for X. 0

f. Uniform Convexity in Banach Lattices and Related Notions

In this section we study the relation between the type and cotype of aspace and the moduli of smoothness and convexity of renormings of the space, in the context of Banach lattices. In this framework it is very useful to compare these notions also with those of p-convexity and p-concavity, which were studied in Section d, as wen as with a variant of these notions called upper, respectively, lower pestimate, which will be defined below. The relations among these notions in a Banach lattice turn out to be very elose and lead to a beautiful and useful theory. There is, for instance, a nice duality between the type and the cotype of Banach lattices (cf. I.f.18 below, for a precise formulation) and, under some assumptions, the cotype of a Banach lattice determines the power type estimate for the modulus of convexity of a suitable renorming (cf. 1.f.1O below). We also present some examples which show that the theorems proved in this section are sharp. As happens frequently in Banach space theory, the exponent 2 plays a special role in this section.

The section ends with two diagrams which summarize the various results and examples conceming the relations among the fOUT pairs of mutually dual notions mentioned above.

We begin with a result ofT. Figiel [40] (see also [41]).


Theorem I.f.1. Let 1 <p ~ 2 ~ q < 00 and let X be a p-convex and q-concave Banach lattice. Then X can be renormed equivalently so that X, endowed with the new norm and the same order, becomes a Banach lattice wh ich is uniformly convex, with modulus of convexity of power type q, and uniformly smooth, with modulus of smoothness of power type p.

Jf, in addition, M(P)(X) = M(q)(X) = 1 then X itselj is uniformly convex and uniformly smooth with both moduli being of power type, as above.

We need first a simple lemma.

Lemma I.f.2. Let q ~2; then,Jor any 1 <p< 00, there exists a constant C= C(p, q) such that

for every choice ofreals sand t.

Proof We may assurne without Ioss of generality that s = 1 > t ~ - 1. The function

(1 + ItIP)q/P (1 + t)q cp(t)= -- --2 2

is clearly positive on the interval [-1, 1) and, since cp"(1»O, it follows that cp(t)/(l- t)2 (and thus also cp(t)/(l- t)q) is bounded from below. 0

Proof of 1 f.1. We consider here only the case when M(P)(X) = M(q)(X) = 1. The general case can be reduced to this one by I.d.8. We also observe that, by the duality results I.d.4 and I.e.2 (ii) (see, in addition, the subsequent remarks about the duality of moduli of power type), it is enough to prove that the modulus of convexity of X is of power type q.

Fix e>O and let x, YEXbe such that IIxll=llyll=I and IIX-yll=e. By l.f.2 and I.d.1, it follows that there exists a constant C = C(p, q) such that

Thus, in view of the fact that M(P)(X) = M(q)(X) = 1, we obtain

l.e.


Remarks. 1. Despite its general character, I.f.l seems to provide one of the simplest ways to prove that the modulus of convexity of L p spaces, I <p< 00,

is ofpower type equal to max {2,p}. 2. T. Figiel and W. B. Johnson [41] used I.f.l in order to construct an example

of an infinite dimensional uniformly convex Banach space with an unconditional basis whieh eontains no isomorphie eopies of Ip for I <p < 00. Their approach is the following: let T be the Tsirelson type space presented in 1.2.e.l and let I <p, r< 00 be arbitrary numbers. Denote by Y the dual spaee of the p-convexification T(p) of T and put q = pl(p - 1). Then Y is q-concave and, thus, its r-convexieation y(r) is r-convex and rq-concave. Hence, by I.f.1, y(r) is uniformly eonvex and uniformly smooth (sinee M(r)(y(r») = M(rq)(y(r») = I). The relation (*) of 1.2.e.1, satisfied by T, easily implies that y(r) contains no isomorphic eopy of Is for s# rq. The spaee y(r) contains also no isomorphie copy of Irq. Indeed, if y(r) contains such a subspace there is, by l.1.a.12, a normalized block basis,

mk+l

X k= L Ajej, k=I,2, ... , j=mk+ 1

of the unit vector basis {ej}~l of y(r), whieh is equivalent to the unit vector basis of Irq. By abuse oflanguage, we shall denote by {ej}~l also the unit vector basis of Y. By definition, the norms in y(r) and Yare related by

mk+l

Hence, the normalized sequence Yk= L IAJej, k= 1,2, ... , in y=(T(p»)* is j=mk + 1

equivalent to the unit vector basis in lq. In particular, Ilk~l bkhll y ~~k~l Ibklq) l/q

for some constant K. Let {t j } ~ 1 denote the unit vector basis of T and (again, by abuse oflanguage) also of T(p). Let {Vk}~ 1 be elements ofnorm one in T(p) so that

Yk(Vk)= 1 and vk= L rJjtj , k= 1,2, .... Then, for every choice of scalars

mk+l

Consequently, Wk = L IrJltj' k= I, 2, ... , is a normalized block basis of {t)i= 1 j=mk + 1

(in T) whieh satisfies


and is thus equivalent to the unit veetor basis of 11• This, however, eontradiets the faet proved in 1.2.e.l that 11 is not isomorphie to a subspace of T.

By eombining l.f.l with l.e.l6 we deduee that a Banaeh lattice which is p-convex and q-concave for some I <p:::;;2:::;;q< oo,must be oftypep and cotype q. It is however trivial to prove direct1y the following somewhat stronger assertion.

Proposition 1.f.3. (i) A q-concave Danach lattice X with q~2 is of co type q. (ii) A p-convex Danach lattice with I <p :::;;2, which is also q-concave for some

q< 00, is of type p.

Proof For every choice of {x;}7= 1 in X we have, by l.d.6 (i), that

Ctl IIxd1qy,q :::;;M(qlX) 11 Ctl Ixi1qy,qll:::;;M(q)(X)IICtl IXi 12Y'211

:::;;M(q)(X)A1l IllJ1 ri(t)Xili dt.

This proves part (i) of the assertion. The proof of part (ii) is the same. ,0

Without the assumption of q-concavity for some q< 00, p-convexity for I <p:::;;2 does not imply that Xis of type p. Consider, for examp1e, the space X = Loo(O, 1) which is even oo-convex.

Notice that if in the definitions of type p or of p-convexity we consider only veetors {Xi}~=l with mutually disjoint supports both definitions reduee to exact1y the same condition. A similar situation occurs with the definitions of co type p and p-concavity. It turns out that the notions obtained by considering only disjoint elements in those definitions are useful in studying deeper the connection between type and eonvexity, respectively, cotype and concavity (e.g. in investigating to what extent the converse to 1.f.3 is true). We therefore introduce formally these notions.

Definition 1.f.4 [120], [41]. Let 1 <p< 00. A Banach lattice Xis said to satisfy an upper, respeetively, lower p-estimate (for disjoint elements) ifthere exists a constant M< 00 such that, for every choice of pairwise disjoint elements {xi }7= 1 in X, we have

respective1y,


The smallest constant M satisfying (*) or (:) is called the upper, respectively, lower p-estimate constant of X.

Proposition 1.f.5. Let 1 <p < 00. A Banach lattice X satisfies an upper, respectively, lower p-estimate if and only if its dual X* satisfies a lower, respectively, upper q-estimate, where l/p + l/q = 1.

Proo! The proof is completely straightforward in case X is order continuous. In this case (since the proposition reduces immediately to the separable case) we may assurne that Xis a Köthe function space satisfying X* = X'. Assurne now, e.g. that X satisfies a lower p-estimate with some constant M. Let {gi }~= 1 be positive disjoint functions in X* and letfbe a positive function in X ofnorm ~ 1.

n

Then we can decompose f into a sum L /; of disjoint functions such that i=O

S /;gj dJ,!=O if i#j(O~i~n, 1 ~j~n). Hence, Q

and thus, by taking the supremum over f, we deduce that X* satisfies an upper q-estimate. The other assertions of 1.f.5 for function spaces are just as easy.

If a Banach lattice satisfies a lower p-estimate then, by 1.a.5 and l.a.7, it is already a-complete and a-order continuous and ,thus, the trivial argument above shows that in the general case

(X satisfies a lower p-estimate) = (X* satisfies an upper q-estimate).

Similarly, if X* satisfies a lower p-estimate then X**, and thus also X, satisfy an upper q-estimate. The proofs of the converse assertions in the general case do however require an additional argument which is a direct generalization of the argument used in the proof of 1.b.6.

Assurne that X satisfies an upper p-estimate with constant M and let {xn~= 1

be disjoint positive elements in X*. Fix 8>0 and choose positive elements {u;}~= 1

in X so that

for all i. Since, for every 1 ~j ~ n, we have xj /\ C~j xi) = 0 it follows from the

definition of /\ in X* that, for alll ~~n, there exist Vj' x j ~O in Xwith uj =vj+Xj '


Clearly, Ilxjll ~ 1 and xj(x) ~(1- 2B)llxjll, for allj. Put

Then O~Wi':::;Xj (and, in particular, IIWjll~l) and, since Wj~Xj- .L. Xi' we.get '#-J

that

xj(W) ~(1-2B)llxjll-B(n-l)llxjll =(l-B(n+ l)llxjll.

Moreover, ifj=l=k then

O~WjAWk~(Xj-Xk)+ A(Xk-Xj )+ =0.

We have thus produced a sequence {wj }j= 1 of disjoint positive elements of norm ~ I in X such that x1 almost attains its norm on wj • Since

for every choice of scalars {a] }j= l' and since B is arbitrary, we get, bya straight

forward computation, that IIit1 xtl/ ~M-1Ct1 Ilxtllqy,q, i.e. X* satisfies a lower

q-estimate. Similarly, if X* satisfies an upper p-estimate then X**, and thus also X,

satisfy a lower q-estimate. 0

There are instances when it is preferable to express the existence of an upper or lower p-estimate without the explicit use of disjoint vectors.

Proposition 1.1.6. Let 1 <p < 00 and M< 00.

(i) A Banach lattice X satisfies (*) if and only if

holds for every choice of {Xi }~= 1 in X. (ii) A Banach lattice X satisfies (:) if and only if

holdSfor every choice of{XJ~=1 in X.

Proof Obviously, (*') => (*) and (:') => (:). To prove that (*) => (*') assume first

f. Uniform Convexity in Danach Lattices and Related Notions 85

that X is an order complete lattice. Let {Xi }~= I be vectors in X and consider the n

band projections Pi=PC~i IXjI-!Xd)/ 1 ~i~n, on X. Let z= j'!l IXjl and

Then

n

L Yi=Z, O~Yi~IXil, l~i~n and Yi"Yk=O fori#k. i= I

Hence, by (*),

i.e. (*') holds. If X is a general lattice the implication (*) => (*') is proved by passing to X**, which is order complete and which also satisfies (*) (by l.f.5). This proves (i). Assertion (ii) follows by duality. Indeed, a lattice X satisfies (*') if and only if the identity map from I/X) into X( co) has norm ~ M. Thus (ii) follows from (i), I.f.5 and the fact that (X*(co))*=X**(lI)' 0

It is evident that a Banach lattice X satisfies an upper, respectively lower p-estimate, whenever it is p-convex, respectively p-concave. The converse is false (cf. l.f.20 below) but we have the following result ofB. Maurey [94] and B. Maurey and G. Pisier [96].

Theorem I.f.7. If a Banach lattice X satisfies an upper, respectively, lower restimate for some 1 < r < 00 then it is p-CQnvex, respectively q-concave, for every 1 <p<r<q< 00.

The proof is based on a probabilistic lemma. Before stating the lemma let us recall some elementary notions. By a random variable we understand a measurable function defined on a probability space (0,1:, p.). (Here we are going to use realvalued random variables but later on we shall consider also random variables whose range space is a Banach space X.) A set {hLEA of random variables is said to be independent if, for every finite subset {ocJ7= I cA and every choice of open sets {DJ~= I in X, we have

n

p.({w E O;hi(w) E D j , 1 ~i~n})= n p.({w E O;f~i(W) E D j }).

j=l

For example, the Rademacher functions {ri(t)}~ I form a sequence ofindependent random variables on [0, 1]. In IJ.8 we shall use a sequence ofindependent random variables {j;}~l so that P.({WEO; lJ;(w)l>il})=I/,1,P for every il~l. A simple way to construct such {/;}~1 is to take 0=[0, l]No with the usual product measure and put/;(W)= t i- l /p , where w= (tl> t2 , ••• ) E [0, 1] No.


Lemma 1.f.8. Let 1 Je})= l/JeP,

for every i and every Je ~ 1. Then, for every r > p, there exists a constant K = K{p, r) < 00 such that,for every finite sequence {a;}~= 1 of scalars, we have

n

Proof Let {a;}~l be a sequence of positive reals so that L af= 1. Then, by i= 1

the independence of theJ;'s and the trivial fact that 1- t~e-t, whenever t~O, we obtain

a l:$;i~n l~i~n

n n n

=1- fl Jl.{{WEQ; lJ;(w)I~1/ai})=1- fl (1-af)~1- fl e-af i=l i=l i=l

1 -1 = -e

and this proves the right-hand side inequality of 1.f.8. Put

cp{Je) = Jl.({w E Q; m~x laiJ;(w)I>Je}). 1 ~l::!S:n

Since 1-t~e-2t for O~t~ 1/2 we get that for Je~21/p

n

cp(Je)=1- fl Jl.({WEQ; 1J;(w)I~Je/ai}) i= 1

n n

=1- fl (1-(aJJe)P)~I- fl e- 2a f/).P=I_e- 2/.l.P. i= 1 i= 1

Hence, by integration, it follows that

0() 0()

J max laiJ;(w)1 dJl.= J cp(Je) dJe~21/p+ J (l-e- 2/).P) dJe=K1 , U 1 =:;;i~n 0 21/p

where K 1 < 00 since p > 1. In order to prove the 1eft-hand side inequality of l.f.8, we fix r>p and take

a sequence {gi};':,l of independent random variables on a probability space


(Q', L', 11') so that

1l({W' E Q; Igi(W')1 >A})= I/Ar,

for every i and every A ~ 1. Then, by applying the right-hand side inequality to the funetions {gJ ~ 1 instead of the U;} ~ l' we obtain

for every W E Q. Thus, by integration and the definition of K 1, it follows that

But, for every i, we have

1

J Igi(W')jP dll'(W') = J t- plr dt=r/(r-p). Q' 0

Consequently,

o

Remark. Lemma 1.f.8 implies in partieular that, for every r>p, the spaee lp is isomorphie to a subspaee of Lt(lr).

Proof of 1 f 7. By duality, it suffiees to show that if X satisfies an upper r-estimate forsome I <r< 00 thenitisp-eonvex,forevery 1 <p<r. Fixp< r and let U;}~l bea sequenee of independent random variables on some probability measure spaee (Q,L,Il) so that 1l({WEQ; 1.t;(w»A})=I/,P, for every i and every A~1. By eondition (*'), there is a eonstant M< 00 so that, for every W E Q and every ehoice of {XJ?=l in X, we have


Hence, by I.f.8, there is a constant K =K(p, r) < <Xl such that,

II(t1 IXiIP)lIPII~Klll iY1 Ih(w)xil dJlII~K r 11 iY1 Ih(W)Xilll dJl

~KM J (tl Ilh(w)xiWY/r dJl~K2M(tl Ilxi11 P)11P D

Corollary I.f.9. Let 1 < r < <Xl and let X be a Banach lattice oJ type r, respectively, co type r. Then X is p-convex, respectively, q-concave Jor every 1 <p < r < q.

A Banach lattice satisfying an upper p-estimate for some 1 <p < 2 need not be of type p (take e.g. L",(O, 1)). G. Pisier observed that, among the Lorentz spaces, there are examples of lattices which satisfy a lower 2-estimate without being of cotype 2 (cf. I.f.19 below). However, the following assertion is true (cf. B. Maurey [94]): A Banach lattice, which satisfies a lower q-estimate Jor some q > 2, is oJ cotype q. We omit the proof of this result since a slightly weaker version of it folIo ws from the next theorem due to T. Figiel [40] and T. Figiel and W. B. Johnson [41].

Theorem I.f.l0. Let 1 <p<2<q and suppose that a Banach lattice X satisfies an upper p-estimate and a lower q-estimate. Then there exist two norms 11·111 and 11·112

on X which are equivalent to the original norm so that X, with the norm 11·111 and the original order, is a uniformly convex Banach lattice having modulus oJ convexity oJ power type q while X, with the norm 11·112 and the original order, is a uniformly smooth Banach lattice having modulus oJ smoothness oJ power type p. In particular, Xis oJ type p and oJ co type q.

We need first a renorming lemma which is very similar to l.d.8.

Lemma I.f.11. Let X be an r-convex Banach lattice satisJying a lower q-estimate Jor some 1 <r<q. Then X can be renormed equivalently so that X, endowed with the new norm and the same order, becomes a Banach lattice Jor wh ich the r-convexity constant and the lower q-estimate constant are both equal to one.

Proof Let X(r) be the r-concavification of X and let 111·111 denote the norm in X(r). Since X satisfies a lower q-estimate it follows easily from the concavification procedure that X(r) satisfies a lower qjr-estimate. For x E X(r!' put

where the supremum is taken over all possible decompositions of x as a sum (in X(r)) of pairwise disjoint vectors {Xi }i= 1. By using the decomposition property, it is readily verified that 111·1111 is a norm in X(r) such that, for each x E X(r)' we have

IIIXIII ~ IIIxII11 ~ M Illxlll '

f. Uniform Convexity in Banach Lattices and Re1ated Notions 89

where M is the lower q/r-estimate constant of X(r). Furthermore, it follows from the definition of 111·1111 that Y=(X(r» 111·1111) is a Banach lattice satisfying a lower q/r-estimate with constant equal to one. Thus, the r-convexification y(r) of Y is a Banach lattice which is r-convex with M(r)(y(r» = 1 and satisfies a lower q-estimate with constant equal to one. The proof is now completed if we observe that y(r) is order isomorphie to X. 0

Proofofl f.10. Fix 1 <r<p and observe that, by l.f.7, Xis r-convex. By l.f.ll, we may assume that both M(r)(x) and the lower q-estimate constant of X are equal to one. It is clear that a Banach lattice satisfying a lower q-estimate for some q< 00 does not contain a sequence of disjoint elements which is equivalent to the unit vector basis of co. Thus, by 1.a.5 and 1.a.7, Xis u-complete and u-order continuous. Since, moreover, X can be assumed to be separable we get, by 1.b.14, that Xis a Köthe function space on a suitable probability space (D, E, J.l).

Fix e>O and let x, Y EXbe so that Ilxll=llyll= 1 and Ilx-yll~e. Put

u= Ix+ YI/2, v=(lxlr + lylr)1/r/21/r, w= Ix-Yl

and consider the sets

uo= {w E D; u(w) < v(w)-u(w)} ,

uj = {w E D; u(w)/2j < v(w)-u(w):::;u(w)/2j - 1}, j= 1,2, ... ,

By 1.f.2, there exists a constant C< 00 such that, for j = 0, 1, 2, ... , we have

Iwxo)CI2+luXaJ:::;lvXaJ,

from which we deduce that IlwXaoll:::; Cllvxaoll and for j~ 1

IWXaJ:::; C2IuXaJ«(l + 1/2j - 1)2 -1):::;4C2IUXajI2/2j-1 ,

l.e.

OC)

Observe also that outside U uj ' v(w)=u(w) and thus, w(w) =0. j=O

By the r-convexity of X and Hölder's inequaIity applied with the (conjugate) indices q/r and q/(q - r), we get that

er:::; Ilwllr = Ilj~O wXajllr:::; j~O IlwXaJ:::; C(llvxaollr + j~1 IIUXajllr/2U-3)r/2)

= C(llvxaollr + j~l (1Iuxa)lr /2U- 3)r/q)(l/2U- 3)r(q- 2)/2q))


It folIo ws that there exists a constant D < 00, depending only on q and r, so that

Let now x* E X* be a positive functional such that Ilx*11 = land X*(U) = Ilull· Since X satisfies a lower q-estimate we get, for j?: 1,

IIUXajW/q"'; Ilull- (11uW -lluxajIIQ)l/q ",; Ilull-Ilu- uXajl1

",;x*(u)-x*(u-UXa)=x*(UXa} . J J

Moreover, since Ilvll"';l (by the r-convexity of X) and O",;u"';v, we have

IlvxaoW/q",; Ilvll-llv -vXaoll",; 1-llu-uXaoll

",; 1 - x*(u - UXao) = 1 -llull + x*(UXa) .

In conclusion, we obtain

f.q ",; Dq( 1 -llull + x*(UXa) + i~l x*(uXa)/2i - 3)

"';Dq(I- llull+ i~O X*«V-U)2iXa)/2i-3)

"';Dq(l-llull +8x*(v-u» "';9Dq(1-IIUII) .

This, of course, proves that X can be renormed equivalently as to become a uniformly convex Banach lattice with modulus of convexity of power type q. By duality, we conclude that X can be also renormed equivalently as to become a uniformly smooth lattice with modulus of smoothness of power type p. In view of l.e.16, it is clear that Xis of type p and of cotype q. 0

In the assumptions required in order to be able to renorm the Banach lattiee X as to have modulus of convexity of power type q, the actual value of p > 1 does not matter (except that it affects the coefficient of f.q in the estimate from below satisfied by the modulus of convexity). In other words, what we really need is to know that X satisfies a non-trivial upper p-estimate (i.e. with some p> I). The existence of non-trivial upper or lower estimates is related to the non-existence of isomorphie copies of Ir, respective1y, I~ as is shown by the following result proved by Shimogaki [120] and W. B. Johnson [56] (Shimogaki worked with a slightly different notion but used essentially the argument presented here. Johnson proved I.f.12 for space having an unconditional basis and his proof is different.)

Theorem I.f.12. Let X be a Banach lattice. (i) There does not exist a p> I so that X satisfies an upper p-estimate if and


only if, Jor every 8> 0 and every integer n, there exists a sequence {xJ? = 1

oJ pairwise disjoint elements in X such that

Jor every choice oJ scalars {aJi= l' (ii) There does not exist a p < 00 so that X satisfies a lower p-estimate if and

only if, Jor every 8> 0 and every integer n, there exists a sequence {xJi= 1 oJ mutually disjoint elements in X such that

m~x lail::::;II.± aixill::::;(1 +8) m~x lail, l~l~n 1:;1 l~l~n

Jor every choice oJ scalars {aJi= l'

Proo! Observe first that (i) can be immediately deduced from (ii) by a simple duality argument. Suppose now that X satisfies no lower p-estimate for p< 00.

For every integer n, let IXn be the smallest constant for which every sequence {X;}i=1 ofpairwise disjoint elements in X satisfies

It is easily seen that I =1X1 ~1X2~'" ~O and we shall show now that {lXn }:=1 is a sub-multiplicative sequence, i.e. that

for all m and n. Indeed, if {Xi, J i!:\, j!: 1 is a double sequence of pairwise disjoint elements in X then

for every I ::::;i::::;m. We also have

inf II.± Xi,jll::::;lXmll.f .± Xi,jll l~l~m r=l 1=1 ]==1

which, in conclusion, implies that

i~f IlxiJI ::::;lXmlXnll.f t Xi,jll· l~l'::::;m 1=1 }=1 1 ~j~n


This proves that ocmn:s::; ocm ocn • Assurne now that ock < 1 for some integer k> 1 and put y = -log ockjIog k. Let n be an arbitrary integer and choose j so that ki:S::;n<ki+ 1. Then,

In other words, there is a constant K< 00 and a number y > 0 so that ocn:s::; K/nY,

for all n. Choosep<oo so thatpy> 1 and let {Xi}~=l be a sequence ofmutually disjoint elements in X such that IIx1 11 ~ ... ~ Ilxnll. Then, for each 1 :S::;j:S::;n, we have

and consequently,

This shows that X satisfies a lower p-estimate for some p< 00, contrary to our assumption. Hence, we must have ocn= 1, for all n. In other words, for every e>O and every integer n, there exists a sequence {X;}~=l of pairwise disjoint elements so that

1= i~f Ilxill:s::;ll.± Xill<l+e. l~l:S:;n 1=1

It follows immediately that, for any choice of {aJ~= l' we have

m~x lail:S::; II.I aixi 11:s::; m~x lai I·II.± Xi 11 < (1 + e) m~x lai I . l~l~n 1=1 l~l.~n 1=1 l::s.;l~n

This completes the proof of (ii) since the converse is trivial. 0

The following corollary illustrates weIl the use of IJ.12.

Corollary 1.f.13. A Banach fattice X, wh ich is of type p for some p > 1, is q-concave for some q< 00.

Proof By 1.f. 7, it suffices to show that X satisfies a lower r-estimate for some r< 00. If X satisfies no such lower estimate then, by I.f.12, it must contain nearly isometrie co pies of l~, for all n. However, for each k, the space l~k contains an isometrie copy of I; and this clearly contradicts the fact that X is of type p, for some p > 1. (A subspace of l~k which is isometrie to I; can be 0 btained by considering the span of the so-called Rademacher elements over the unit vector basis {ei }?~ 1


of l~k, i.e. the vectors

We study now so me questions in which properties related to the number 2, like 2-concavity, type 2 or cotype 2, etc., playa special role. We show first that, for p = 2, Proposition l.d.9 can be generalized so as to apply for general bounded operators (and not only positive ones).

Theorem l.f.14 [66]. Let X and Y be two Banach laltices and let T: X -> Y be a

bounded linear operator. Then,for every choice 0/ {X;}?=1 in X, we have

where KG is the universal Grothendieck constant (cf. I.2.b.5).

Prao! The proof will be carried out in three steps. In the first step we shall prove l.f.14 in ca se X = I::; and Y = I'{', for some m< 00. In this case, I.f.14 is a formally slightly stronger version of the assertion that every operator from Co to I1 is 2-absolutely summing (cf. 1.2.b.7). The proof of step I is based on Grothendieck's inequality (1.2.b.5) and is very similar to that of 1.2.b.7. In the second step of the proofwe apply a standard approximation argument in order to deduce from step I that l.f.14 holds if X is a C(K) space and Y an LI (p,) space. In the proof of step 3 we apply the representation theorems of Kakutani in order to reduce the general ca se to that verified in step 2.

Step 1. Let T be an operator from I: into I'{', for some integer m. Let Xi = (ai, l' ... , ai,rn)' i= 1,2, ... , n be a sequence of elements in I: and let (O(k,)~j=1 be the matrix representing Twith respect to the unit vector ba ses of I: and Ir Choose vectors Yi=(b i, l' ... , bi,rn) E I: = (I'{')*, i= 1, 2, ... , n so that

(use the duality between 1'{'(l2) and 1:(12))' Consider now the vectors uk =(a1,k>"" an,k) and vk=(b1,b"" bn,k) as

elements of l~. Then,

n m m m m

L L L O(k,jai,kbi,j= L L O(k,/Uk,V). i=1 j=1 k=1 j=l k=l


Notiee that

and

It follows from Grothendieck's inequality I.2.b.5 (applied to the matrix (ak ,j/IITII)k,j=l) that

II(tl ITxd2 Y/2111 ~KGIITlllr:::m Iluklb lr:::m Ilvklb

=KGIITIIIICtl Ixi l2Y/2t .

Step 2. Let Tbe an operator from C(K) into L 1(/J.) and let {.r.}~=l be a finite sequenee in C(K). Then, for every e > 0, there exists a partition of unity {q> j }j= 1 in C(K) and funetions.i: E [q> jJj= 1 so that 1I.r.-.i: 11< ell.r.ll, for all i (a partition of

m

unity in a C(K) spaee is a set of funetions {q> j }j= 1 of norm one so that L q> ik) = 1 j= 1

and 0 ~ q> j(k) ~ 1, for all k E K and I ~j ~ m). Let {j j }j= 1 be simp~ functions in L 1(JJ.) so that IITq>j-l/tjll~elm, I ~~m and put !q>j=l/tj' Then Textends to a linear operator from [q>j Jj=l into [l/t j Jj=l and IIT-TI[tpjlm=lll~e. Sinee [q>j Jj=l is isometrie to I: and [l/tjJj= 1 is isometrie to a subspace of It, for some k, it follows from step 1 that

Sinee this is true for every e>O we obtain, by letting e ~ 0, that l.f.l4 holds if X=C(K) and Y=L1(/J.).

Step 3. Let Tbe an operator from Xto Yand let {X;}~=l be a finite sequenee

( n )1/2 ( n )1/2

in X. Put xo= i~1 Ix;j2 and Yo= i~1 ITxi l2 • Let I(xo) be the (in general

non-closed) ideal generated by X o endowed with the norm

whose eompletion is, by 1.b.6, order isometrie to a C(K) spaee. Let Y~ be a positive funetional in Y· so that IlY~11 = 1 and Y~(Yo) = IIYoll. Put IIYlll = y~(lyl), for Y E Y, and observe that 11·111 is a semi-norm on Y sueh that the eompletion Yo, of Y


endowed with 11·111 modulo the elements Z E Y having Ilzlll = 0, is an abstract L 1

space (and, therefore, by Kakutani's Theorem I.b.2, order isometrie to a concrete L 1 (11) space). If J 1 and J2 denote the formal identity maps from I(xo) into X, respectively, from Y into Yo then J2TJ1 is a linear operator from I(xo) into Yo with IIJ2TJ111::::; IITII. Hence, by step 2,

Remark. It can be easily verified that KG is the smallest constant for which l.f.l4 holds for generallattices.

Corollary I.f.15 [66]. Let X and Y be Danach faltices and T: X ~ Ya bounded linear operator.

(i) If Xis 2-convex then T is 2-convex and

(ii) If Y is 2-concave then T is 2-concave and

(iii) If X is 2-convex and Y is 2-concave then T can be Jactorized through an L2(1l) space in two different ways:

a) There exist a probability measure 111' an operator Tl: X ~ L2(1l1) with 11 Tl 11 ::::;KG M(2)(X)II T II and a positive operator Sl: L 2(1l1) ~ Y with IIS111::::;M(2)(Y) such that T=SlTl.

b) There exist a probability measure 112' an operator T2: L2(1l2) ~ Y with IIT211 ::::;KG M(2)(Y)II111 and a positive operator S2: X ~ L 2(1l2) with IIS211 ::::;M(2)(X) such that T= T2S2.

Proof (i) Für every sequence {X;}7=l in Xwe get, by I.f.14, that

II(tl ITxiI2Y/211::::;KGIITIIIICtl Ixi l2Y/211

::::;KG IITIIM(2)(X)Ct II Xi ll 2Y/2


The proof of (ii) is similar while (iii) follows immediately from (i), (ii) and l.d.12 (i) and (ii). 0

We are now prepared to prove a theorem which eombines a result ofB. Maurey [94] «i) ~ (ii)) and one of E. Dubinski, A. Pelczynski and H. P. Rosenthai [31].

Theorem I.f.16. The Jollowing conditions are equivalent in every Banach lattice X. (i) Xis oJ co type 2.

(ii) Xis 2-concave. (iii) Every operator Jrom Co (or Jrom any C(K) space) into X is 2-absolutely

summing.

Proof The fact that (i) ~ (ii) is a direet eonsequenee of l.d.6 whieh shows that the

expressions Iletl lXi 12 ) 1/2 11 and lilit r;(u)xi 11 du are equivalent in X. Notice that,

in order to be able to use l.d.6, we have to prove first that a lattice of eotype 2 is s-coneave, for some s< 00. This follows however from 1.f.7.

Suppose now that Xis 2-eoneave and T is a bounded linear operator from a C(K) spaee into X. Then, by l.f.15 (ii), T is 2-eoneave, i.e., for every ehoice of {!;}7=1 in C(K), we have

and the proof of the implieation (ii) = (iii) ean be eompleted by reealling that

(see e.g. eondition (+) in the proof of 1.d.lO). Finally, (iii) = (ii) follows from 1.d.IO. 0

There is also a partial dual version of I.f.16 whieh follows easily from 1.d.6 and l.f.13.

Proposition I.f.17. A Banach lattice is oJ type 2 if and only if it is 2-convex and q-concave,Jor some q< 00.

Observe that, in view of 1.b.13, 1.f.16 and l.f.l7 provide in partieular a proof ofKwapien's theorem [69] for lattices: a Banach lattiee, which is oftype 2 and of co type 2, is isomorphie to a Hilbert space (even order isomorphie to some L 2(p)).

We establish now the duality between type and eotype in Banach lattiees whieh are "far" from La) (cf. B. Maurey [94]).

Theorem I.f.18. A Banach lattice X is oJ type p, Jor some p> 1, if and only if its dual X* is oJ co type q, where l/p+ l/q= 1, and satisfies an upper r-estimate Jor some r> 1.


Proof Suppose that X is of type p, for some p > 1. The fact that, in this case, X* is of cotype q=p/(p-l) is valid for every Banach space X and was proved in 1.e.17. Moreover, it folIo ws from 1.f.13 and the duality between r-convexity and r/(r-l)-concavity that X* satisfies an upper r-estimate for some r> 1.

In order to prove the converse, we first treat the case when q >. 2. Then,. by duality, X satisfies an upper p-estimate for 1 <p<2 and a lower s-estimate for s=r/(r-l). Hence, by 1.f.10, Xis of type p. If, on the other hand, q=2 then, by IJ.16, X* is 2-concave and thus, in view of l.d.4 (iii), Xis 2-convex. !<,urthermore, since X* satisfies an upper r-estimate, it follows from l.f.7 that X* is r~ -convex for every r' < r. Hence, again by 1.d.4 (iii), X is s'-concave for s' = r' /(r' ~ 1) imd, therefore, of type 2, by 1.f.3. ·0

We conclude this section with some examples.

Example l.f.19. (G. Pisier). There exists a uniformly eonvex spaee with asymmetrie basis whieh satisfies a lower 2-estimate but is not of eotype 2 (or, equivalently, 2-eoneave). Consequently, this spaee eannot be renormed equivalently as to have modulus of eonvexity of power type 2.

Fix 1 <p<2 and consider the Lorentz sequence space X=d(w,p), where w= {wn}~ 1 is defined by wn=nP/2 -(n-l)p/2, n= 1,2, ... (Recall that the norm of

x=(a1 , a2, ... ) in d(w,p) is defined to be IIx ll =C~1 a:Pwn Y/P, where a: = la1t{nll with n apermutation of the integers for which {Ia,,(nll}:'= 1 is a non-increasing sequence.) We prove first that X is not 2-concave. Let m be an integer and let {rj }j:,.1 be the maps of {I, 2, "', m} into itself defined by r/i)=(i+ j) mod m. For 1 :~,j~m put

m

Xm,j= L edrii)1/2, i=1

where {eJ~1 denote the unit vector basis of d(w,p). A simple computation shows that

IIC~1IXm,jI2 Y/211=IIC~1 it1 edrj(i))1/211

=(t1 lüY/21ljt1 ejll~m1/2(log m+ 1)1/2.

On the other hand,

Since m is arbitrary this proves that d(w, p) is not 2-concave. lt follows from


1.f.I6 that X is not of cotype 2 and bence, by 1.e.I6, for every equivalent norm in X the modulus of convexity is not of power type 2.

In order to verify that d( w, p) satisfies a lower 2-estimate it suffices to show that d(w, 1) (which is the p-concavification of d(w,p» satisfies a lower 2/p-estimate, Le. (in view of I.f.5) that d(w, 1)* satisfies an upper r-estimate, where l/r+p/2 = 1. To this end, note first that if x* =(bl , b2 , ••• ) E d(w, 1)* with bl ~b2 ~ ... ~O then

{ (al-a2)bl +(a2-a3)(bl +b2)+"'+(aj-ai_I)(bl +"'+b;}+'" .

= sup (at -a2)w1 +(a2 -a3)(wt +w2)+ ... +(ai-aj-1)(w1 + ... +w;)+ ... '

{a j ~ O} ~ I decreasing}

{ Clbl+C2(bt+b2)+···+Ci(bl+···+b;}+ .... { }} = sup , Cj~O ~l C1W1 +ciw1 +w2)+"'+Ci (WI +···+wi )+···

= sup {.± bi /.± wi }= sup {n- p/ 2 .t bi }. n J=l J=l n J-I

Let u=(u1 , U2' ... ) and v=(v1 , v2' ... ) be two disjointly supported elements of d(w, 1)*; then

Ilu+vII= sUP {(k+I)-P/2Ct u1+ itl V1} O~k,l<oo,k+l~l} ~ sup {(k+ 1)-P/2(llullkP/2 + IlvI11P/2); O~k, l< 00, k+l~ I}

~ (Ilulir + Ilv11') l/r

and this proves our assertion. Since X is the p-convexification of d(w, 1) it is p-convex and M(Pl(X) = 1.

Therefore, by the proof of l.f.10 (taking as q any number >2), Xis uniformly convex. 0

Remark. Let Yand Z be two Banach lattices and suppose that Y is linearly isomorphie to a subspace of Z. If Z satisfies a lower p-estimate, for some p > 2, then so does Y since, bya result from [94] (stated before l.f.lO), Z, and thus also Y, are of cotype p. However, contrary to the case of p-concavity described in I.d.7(ii), this assertion is, in general, false when p = 2. A eounterexample ean be, for instanee, eonstructed by taking Z = (X EB X EB "')2' where X is the Lorentz sequence spaee d(w,p) eonsidered in 1.f.19. As is easily verified, Z, too, satisfies a lower 2-estimate. Suppose now that any Banaeh lattice Y, whieh is linearly isometrie to a subspace


of Z, also satisfies a lower 2-estimate. Then a simple uniformity argument proves that there exists a constant M< r:fJ so that the lower 2-estimate constant of any Y as above is ::::; M.

Let {XJ?=l bevectors inXand, for l::::;i::::;n, put

2n - i

~

ui=2- n/2(Xi , ... , Xi'

2n - i 2n - i 2n - i

,-----'-------- ~ ,-----'--------

-Xi' ... , -Xi' ... , Xi' ... , Xi' -Xi' ... , -Xi' 0, 0, ... ),

where the number ofthe blocks of size 2n - i is 2i . A direct computation shows that, for every choice of scalars {aJ?= l'

i.e., the unconditional constant of {uJ?= 1 is equal to one. It follows that

M//.± ui// ~(.± "uill~)1/2,n=1, 2, ... , ,=1 Z ,=1

which implies that Xis of cotype 2, contrary to the assertion of 1.f.19.

Example l.f.20. For every q > 2 there exists a Banaeh spaee with a symmetrie basis whieh satisfies a lower q-estimate and has a modulus 01 eonvexity 01 power type q, but is not q-eoneave.

Proof Let I <p<2, let w be the sequence defined in 1.f.19 and let q>2. The Lorentz sequence space d(w, pq/2) is the q/2-convexification of d(w, p). Hence, d(w,pq/2) satisfies a lower q-estimate with constant one but is not q-concave. Also, d(w,pq/2) is pq/2-convex (with pq/2-convexity constant equal to one) and hence, by 1.f.1O, its norm has a modulus ofconvexity ofpower type q. 0

We present now two diagrams. The first diagram describes the various connections between modulus of convexity, concavity, cotype and lower estimates in general Banach lattices. The second diagram describes the connections between the dual notions.

100

'" " r::r

Modulus of convexity of

power type tor some equivalent

norm

'" '" A r::r r::r

r::r I!

.e .,

.!!! E

Concavity

~.,.

q>2 q>2

1. Banach Lattices

'" '" A M r::r r::r

'" " r::r

oE ., .!!! E

'" '" A '" r::r r::r

Cotype

Interdependence diagram for the power type of modulus of convexity, q-concavity, cotype and lower q-estimate for general Banach lattices.

f. Uniform Convexity in Banach Lattices and Related Notions

N .., 0. V

Modulus of smoothness of power type for

some equivalent norm

N V

N 0.

" V 0.

0.

~ ~

.E

Convexity t-------,false for r=pr------I

1<p<2

101

cuE E~ oKJ N

~ 8 VI NV 0. V o.t ~~

v

N

" 0.

.E

~ N N V VI 0. 0. V V

Interdependence diagram for the power type of modulus of smoothness. p-convexity, type and upper p-estimate for general Banach lattices.


g. The Approximation Property and Banach Lattices

In this seetion we continue the study of the approximation property (A.P.) undertaken in Sections I.I.e and I.2.d. The seetion does not however rely on the results or methods presented in Vol. land can therefore be read independently of I.I.e and I.2.d. Our main purpose here is to present the example of Szankowski [123] of a uniformly convex Banach lattice which fails to have the A.P. (and thus also fails to have a Schauder basis). This lattice can actually be chosen to be a sublattice of a lattice Yp which is isomorphie to LiO, 1), 2<p< 00. (The lattice Lp(O, 1) itself clearly does not have any sublattice which fails to have the A.P., in view of I.b.4.) The second part ofthis section is not directly connected to the theory of Banach lattices. The central result in this part is a proof (of the main part) of another result of Szankowski which states that a Banach space X has a subspace which fails to have the A.P. unless it is of type 2 -e and cotype 2 + e for every e>O.

We start by presenting a general criterion for aspace to fail to have the A.P. This criterion is a modified version of that used by Enflo [37] in his original solution ofthe approximation problem. This is actually a criterion for a Banach space to fail to have the apparently weaker property called the compact approximation property (C.A.P. in short). Recall that a Banach space X has the C.A.P. if, for every compact set K in X and every e > 0, there is a compact operator Tin L(X, X) so that IITx-xll ~e for every x E K.

Proposition l.g.1. Let X be a Banach space. Assume that there are sequences {X j };' 1

and {xj};' 1 oJvectors in X, respectively X*, a sequence {Fn}~ 1 oJfinite subsets oJ X and an increasing sequence oJ integers {kn}:= 1 so that the Jollowing hold.

(i) xj(x,) = I,jor every j, (H) xj ~ 0, sup Ilxjll < 00,

j

(Hi) Ißn(T) - ßn-l (T)I ~ sup {IITxll; XE Fn}, Jor n = 1, 2, 3, ... and every TE L(X, X), where ßo(T) = ° and, Jor n ~ 1,

k n

ßiT)=k;:l L xj(Tx) , j= 1

00

(iv) L Yn< 00, where Yn = sup {Ilxii ; x E Fn } •

n= 1

Then X JaUs to have the C.A.P.

Proof It follows from (Hi) and (iv) that

ß(T) = lim ßiT) n-+ 00

g. The Approximation Property and Banach Lattices 103

exists for every TE L(X, X) and defines a linear functional on L(X, X). Let 00

{'1,,}:'= 1 be a sequence ofpositive numbers tending to 00 so that C= L '1"Y" < 00, ,,=1

00

and put K= {O}u U ('1"y")-lF,,. Clearly, Kis a compact set and ,,= 1

/ß(T)/ ~ C sup {/lTx/l; XE K} .

It is also clear from (i) that if I is the identity operator on X then ß,,(/) = 1 for all n~1 and thus ß(/)=1. We shall show that ß(T)=O whenever TE L(X, X) is compact. This will conclude the proof since it will imply that, for every compact T,

sup {/lTx-x/l; XE K}~C-1/ß(I-T)/=C-1.

Let T thus be compact and let {) > O. Pick {Yi}i= 1 in X so that, for every integer j, there is an i0) with /lTXj-YiW/l~{). We have, for n= I, 2, "',

kn kn

ßiT )=k;1 L xt(Yiw)+k;1 L xt(TXj-Yi(j» , j= 1 j= 1

and thus

m k n

/ßiT)/ ~ L k; 1 L /xt(YJ/ + {) sup /lxt/l . i=1 j=1 j

Since xt ~ 0 it follows that /ß(T)I~{) sup /lxt/l and thus, since {) was arbitrary, j

ß(T)=O. 0

Remark. The proposition remains clearly valid ifthe ß,.(T) are defined to be ofthe form k; 1 L xt(Txj ), where 0'" is any set ofintegers of cardinality k,., n= 1,2, ....

jean For instance, in one of the applications of l.g.l below ß,.(T) will be defined to be

2"+1

r" L xt(Txj ).

j=2n +l We state now the theorem of Szankowski [123].

Theorem l.g.2. Let 1 ~r<p< 00. There is a sublattice of Ip(L,(O, 1»=(L,(O, I) E9 L,(O, 1) E9"')p whichfails 10 have lhe C.A.P.

The construction of this sublattice is based on a lemma of a combinatorial nature which deals with the existence of certain partitions of[O, 1]. F or the statement ofthis lemma we introduce first some notations. For every integer n, let fJl" be the algebra of subsets of [0, 1] generated by the 2" atoms [(i-l)/2", i/2n), i= 1, ... ,2". A subset of [0, 1] is fJl,,-measurable if it is the union of some of these atoms. For every n, let qJn be the permutation of {1, 2, ... , 2"} defined by qJ,,(2z) = 2i -1 and


cpi2i-I)=2i, i= 1,2, ... , 2n- 1 . Themap CPn induces in an obvious way apermutation between the atoms of f!Jn and therefore also a map (denoted again by CPJ from the set of aH f!Jn-measurable subsets of [0, I] onto itself. The Lebesgue measure on [0, I] is denoted by Jl.

Lemma l.g.3. For every integer n~26 there exists a partition An oJ[O, 1] into Mn disjoint f!Jn-measurable sets oJ equal measure (i.e. oJ measure Mn-i) so that

(i) M >-:2n/16 n>-:26 n7 ,r,

(ii) Jl(cpiA)(]B) ~4Jl(A)Jl(B), Jor every A E An' BE Am' n, m~26.

The restriction n ~ 26 appearing in the statement of 1.gJ is made just for convenience and has no special significance. We postpone the proof of l.g.3 and turn now to the construction of the sub1attice in l.g.2.

ProoJ oJ l.g.2. Let 1 ~r<p~ 00 and let Xbe the space of all measurable functions Jon [0,1] so that

where Mm and Am are given by l.g.3 and (l( is a number satisfying

Clearly, for every JE X, we have

where M = C~2.M~+"p-P/r }/p< 00. Hence, Xis a Köthe function space on [0, 1].

It is also obvious that Xis a sublattice of lp(Lr(O, 1». We shall now prove that X fails to have the C.A.P. byapplying l.g.l. We have first to define the quantities which enter into the statement of this proposition.

Let {wJj~ 1 be the Walsh functions on [0, 1] defined by

and, in general,

where j-l =2'" +2"2+'··+2'" with O~kl <k2 <"'<k, and {r,,}l°=l denote the Rademacher functions. Note that Ilwjll oo = 1 for aHj and that the {wj}.i= 1 form an orthonormal system in L 2(O, 1). Hence, {Wj}.i=l is a bounded sequence if we


consider it as a sequence in X or in X* and, moreover, as a sequence in X* we have wj ~ 0. We define for TE L(X, X) and n= 1,2, ... ,

2 n

ßn(T)=r n I wiTw) , j=l

1

where wiTw) means of course J w/t) (Tw) (t) dt, and claim that (iii) and (iv) of o

l.g.l hold for a suitable choice of {Fn}:= 1 .

In order to verify that this is the case we note first that {w J f: 1 is a basis of the space of all gßn-measurable functions on [0, 1]. Since {2n/2X;'};:1' where X;'= XW-l)/2n,i/2n), is a basis of the same space, wh ich is also orthonormal in L 2(0, 1), it follows that

2n

ßn(T)= I x7(Tx7) i= 1

(recall that the trace of a linear transformation on a finite dimensional inner product space does not depend on the choice ofthe orthonormal basis). Using this expression for ßnCT) we get that

2n

ßnCT)-ßn-l(T)= L x7(Tx~n(i))' n=l, 2, ... , i= 1

where q>n is the permutation of {I, 2, "', 2n} defined before the statement of l.g.3. The partition Lln of [0, 1] given by l.g.3 induces in an obvious way a partition Ll~ of {l, 2, ... , 2n}. To each A' E Ll~ there corresponds the set A = U [(i-l)/2n,

ieA'

i/2n) in Ll n . Clearly,

L X;'( T.x;;.n(i)) = Average [( I () i X;') (L T() i .x;;.n(i))] , ieA' ieA' ieA'

where the average is taken over all the 2A' choices ofthe signs (}p i E A' (A' denotes the cardinality of A' which is 2n/Mn). Each of the expressions appearing in the average is in absolute value equal to at most J ITf(t) dt withf= L: (}jX;n(i) a f!4n-

A jeA'

measurable function whose absolute value is the characteristic function of q>nCA). Hence, if

En = {J;f gßn-measurable and Ifl = Xrpn(A) for some A E Ll n}

then

Ißn(T) - ßn-l (T)I:::; A'~Ll' I i~A' xiTx~n(i) I :::;Mn sup {~ITf(t)1 dt; A E Lln,J E En} .


By the definition of the norm in X and Hölder's inequality, we have, for every gE X, that

Ilgll ~ M:M! -11' S Ig(t)1 dt, A E J n ,

A

and, henee, for every n~26

Also, by (ii) of I.g.3 we have, for every fEEn with Ifl = Xtpn(A)

Consequently, (iii) and (iv) of I.g.I hold ifwe put Fn = Mn1/,-a En for every n~ 26 • 0

Remark. If r=2<p< 00 the spaee liLz{O, 1» is isomorphie to a eomplemented subspace of liLiO, 1» whieh, in turn, is isometrie to LiO, 1) (reeall that 12 = L 2(0, 1) is isomorphie to a eomplemented subspace of LiO, 1); take e.g. [r n]:'= 1 in LiO, 1) cf. Vol. I, p. 72). Henee, in this ease, the spaee X of I.g.2 is a sublattiee of the Iattiee Y p= Ip(L2(0, 1» EB Lp(O, 1) and Yp is Iinearly isomorphie to Lp(O, 1).

Proof of Lemma l.g.3. We identify [0, 1] with D= {-I, 1 to endowed with the usual produet measure. In this identifieation ff1n eorresponds to the algebra of those subsets of D whieh depend only on the first n coordinates. The map qJn from the set of ff1n-measurable subsets onto itself beeomes under this identifieation the transformation indueed by the mapping

of D onto itself. 00

We represent Das the produet n D i , where D i = {-I, 1 y', and let 1ti be the i= 1

natural projeetion from D onto D i. F or eaeh i ~ 5 we ehoose a system of partitions {Qn}~': ;::;:} ,?f D i - 1 into disjoint sets eaeh having eardinality (D i _ 1)1 /2 = 22 <-2 so

that if a E Qn and 1] E Qm with n#m then a(l1]= 1. To see that such partitions do indeed exist put q = (D i _ 1) 1/2, eonsider D i _ 1 as a finite field and let F be a subfield of D i -1 of eardinality q (this is possible sinee q is apower of a prime, namely of 2). We let eaeh Qn be a partition of D i - 1 into Iines parallel to a fixed line of the form xF, O#x E Di - 1 • The number ofsueh Iines is (q2-I)/(q-I)=q+ 1 whieh is Iarger than the required number of partitions (namely 2i + 2).


We are now ready to define the partitions of D. For 2i+ 1~n<2i+2, i=5, 6, ... we let the elements of Ll n be the foIIowing sets: {t E D; 1T.i - 1(t) E a, tn = -I} with a E 02n and [t E D; 1ti - 1 (t) E a, tn = I} with a E 02n+ 1. The number of the sets in An is thus 2(1)i_l)1/2=21+2i-2 and since n<2i +2 we see that (i) of l.g.3 holds.

Let us verify that also (ii) of l.g.3 holds. Let

A = {t E D; 1ti - l (t) E a, tn =O}, a E 02n+(1 +0)/2' 2i + 1 ~n <2i + 2

B= {t E D; 1tj _ 1(t) E '1, tm=O'}, '1 E .Q2m+(1 +9')/2 2 j + 1 ~m < 2 j +2 .

Clearly, qJiA) = {t E D; 1tn - 1 (t) E a, tn = - O} and

(B) -2-1 ( -1 ( »_2-2j-2-t Jl - Jl 1T. j- 1 '1 - .

If i#j then 1ti-_\(a) and 1T.j-_\('1) depend on different coordinates and hence,

Jl( qJiA)nB) ~ /l(1ti-_11 (a)n1t j--\ ('1»

= /l(1ti--\ (a»· Jl(1t j--\ ('1» =4Jl(A)Jl(B) .

Ifi=jthen either n=m and 0=0' in which case qJiA) and B=A are disjoint or, by our choice ofthe On's, an'1= 1. In this case we have

We pass now to another result of Szankowski.

Theorem 1.g.4 [124]. For every I ~p<2 the space Ip has a subspace without the C.A.P.

RecaII that for 2 <p a similar result was proved in 1.2.d.6. (See also remark 2 below.)

The proof of I.g.4 is also based on a combinatorial lemma. In order to state this lemma we introduce first some notations. For n= 1,2, ... let an = {2n, 2n + 1, ... , 2n+ 1 _I}. To each integerj~8 we associate nine integers {fiMr=l defined as folIows:

A(4i+l)= 2i+k-l, i=2, 3, 4, ... , 1=0, 1,2,3, k= 1,2

A(4i+l)=4i+(/+k-2) mod 4, i = 2, 3, 4, ... , 1 = 0, 1, 2, 3, k = 3, 4, 5

A(4i+l)=8i+k-6, i=2, 3, 4, ... , [=0,1, k=6, 7, 8,9

fk(4i+/)=8i+k-2, i=2, 3, 4, ... , [=2,3, k=6, 7, 8,9.

These functions will arise from the construction of the subspace in l.g.4. An important fact about these functions is thatfk(j) # j, for every k andj. This enables us to partition the integers into relatively large subsets so that, for every 1 ~k~9,


as j runs through one set of the partition, the eorresponding integers !tU) belong to different sets of the partition. More precisely, we have the following.

Lemma l.g.5. There exist partitions .LI n and Vn of un into disjoint sets and a sequence of integers {mn}:'= 1 with mn ~ 2n/8 - 2, n = 2, 3, 4, ... , so that

(i) Each element of V n has cardinality between mn and 2mn . (ii) Every element of V n contains at most one representative from any element

of .LIn , i.e.

AnB~l, AEVn, BE.LIn, n=2,3,4, ....

(iii) For every A E V n' n ~ 3 and every 1 ~ k ~ 9 the set h(A) is contained entirely in an element of .LIn- 1 , .LIn or .LIn+ l'

Note thath(un)cUn_l for k=l, 2,h(un)cun for k=3, 4, 5 andh(un)cUn+l for k=6, 7, 8, 9. We postpone the proof ofthe lemma and pass to the

Proof of 1.g.4. Let! ~p<2andletXbethespaeeofallsequeneesx=(a4' as' a6 , ••• )

so that

where.LIn is the partition of U n given by l.g.5. The space Xis a direet sum in the lp . sense of finite dimensional inner produet spaees and is therefore isomorphie to a subspace of lp. As a matter of fact, Xis even isomorphie to lp for 1 < P < 2 (cf. Vol. I, p. 73). We denote by {ej}~=4 the unit veetor basis of Xand by {eJ}i=4 the eorresponding biorthogonal funetionals in X*. We let Z be the closed subspace of X spanned by the sequenee

We shall prove, using l.g.l, that Z fails to have the C.A.P. Put

i=2, 3, "',

and, for TE L(Z, Z),

ßn(T)=rn L zi(TzJ, n= 1, 2, 3, .... iean

In order to establish that (iii) and (iv) of l.g.l hold (see also the remark following l.g.l), we note first that, for every i~2, the restrietion of (e:i+e:i+ 1 +e:i+ 2 + et+ 3)/4 to Z is equal to that of zi (they eoincide when evaluated on Zj for every J).


Hence, for n ? 2 and TE L(Z, Z),

ßnCT)-ßn-l(T)=

2- n - 1 I (ei i - et+ 1)T(e2i - e2i+ 1 + e4i + e4i + 1 + e4i +2 +e4i + 3)

- 2- n - 1 L (e!i + et+ 1 + et+ 2 +et+ 3)T (e2i - e2i+ 1 + e4i + e4i + 1 + e4i +2 + e4i + 3) iEl1n - 1

{

e!iT(e4i-e4i+ 1 +eSi + .. , +eSi+3 -e2i +e2i +1 -e4i - .. , -e4i +3)

=2- n - 1 I +et+1T( -e4i+e4i+l -eSi -'" -eSi +3 -e2i+e2i+l -e4i -'" -e4i +3)

ieUn-l +e!i+2T(~i+2 -e4i +3 +~i+4 +.~ :.~i+7 -~i +e2i + 1 -e4i _.~ ~.e~+3) +e4i +3T( e4i+2+e4i+3 eSi +4 eSi +7 e2i+e2i+l-e4i e4i +3)

=r n - 1 I ejTYj, jEUn + 1

where

9

L Aj,kefk(j)=YjEZ, j=8,9, ... , k= 1

the h being the functions defined before l.g.5 and, for every j, IA j, kl = 1 for eight indices k and IAj,kl =2 for the ninth k.

As in the proof of 1.g.2 we write now

where the average is taken over a112A choices of signs {BjLeA' By the definition of the norm in X and by (ii) of l.g.5 we have, for every A E Vn + 1 (n? 2) and {Bj LeA'

where l/p + l/q= 1. By (iii) of l.g.5 we have, for every such A and {BJjeA and every 1~k~9,

and, consequentIy,


Hence,

where

Consequently, (iii) and (iv) of l.g.l hold ifwe put Fn =2m;;+1/PEn • 0

Proof of 1.g.5. For n~2 and 1=0, 1,2,3 we put O"~= {j E O"n J=/(mod 4)} and let cp!: O"~ ~ O"! be the map defined by cp!0)=j+I. For n~2 and r=O, 1 we let t/J~: O"~ ~ 0"~+1 be defined by t/J~0)='2J+4r.

By an easy inductive procedure, we can represent O"~ for n ~ 2 as a Cartesian product Cn x Dn , where

so that:

foreachcECn + 1 thereisanr=O, I andadEDnsothatt/J~(Cnx{d})={c}xDn+l and, for each dEDn+1, there is a CE Cn so that t/J~({c}xDn)ut/J~({c}xDn)= Cn + 1 x {d} .

We represent further each Dn , n=2, 3, ... as a Cartesian product offour factors 3

D n = n D~ so that 1=0

We can now define the desired partitions.

Vn={cp!({J} xD!);fE Cnx TI D!, 1=0, 1,2, 3}, i*'

Lln={CP!( Cn x Jl D!x {d}} dE D!, 1=0,1, 2, 3}.

It is c1ear that (i) of 1.g.5 holds with

It is also evident that (ii) of 1.g.5 holds. The verification of (iii) of 1.g.5 is straightforward but somewhat long and we omit it. . 0


Remarks. I. In the proof of l.g.4 presented above it was essential that the norm in [eJjEan ' for n=2, 3,4, ... , is given by

i.e. that [ej]jEan is a subspace of I; for a suitable m. As for the norm in the whole space X = [eJi=4 we used only the fact that X has aSchauder decomposition (cf. I.l.g) into {[ej]jEa):'= 1. Consequently, the proof of I.g.4 shows that if, for a Banach space X, there is a 1 ~p < 2 and K< 00 so that X has aSchauder decomposition into {Xn }:'= 1 with d(Xn , I;) ~ K for all n, then X has a subspace which fails to have the C.A.P. The argument used in the proof of I.l.a.5 shows that if Y is a Banach space so that, for some I ~p < 00 and K, there is for every integer n a subspace Yn of Y with d( Yn , I;) ~ K then Y has a subspace X which has aSchauder decomposition into {Xn}:'=l with d(Xn , I;)~K+ 1. Hence, if I ~p<2, every such Y has a subspace which fails to have the C.A.P.

2. The proof of l.g.4 presented above can be easily modified so as to apply to the case 2 <p ~ 00 (and thus to yield an independent proof of 1.2.d.6). We have only to arrange the partitions Vn and Lln so that every A E Vn is contained in some element of Lln while, for every A E I7n , k = I, ... ,9 and every BE Lln _ l' Lln or Lln + l'

Bnh(A)~ 1. 1fthis is the ca se we get (in the notation ofthe proof of I.g.4) that, for every A c Vn+ l' n= 2,3, ... and {Oj}jEA'

In Vol. III we shall present a deep result of Krivine [67] and Maurey and Pisier [96] which asserts that if, for a Banach space X,

p(X) = sup {p; Xis oftypep}, q(X) = inf {q; X is of cotype q}

then, for every n, X contains almost isometrie copies of I;(x) and I;cx). An immediate consequence of this result and the preceding remarks is the following theorem.

Theorem l.g.6. Let X be a Banach space. If every subspace oJ X has the C.A.P. then Xis oJ type 2 - sand co type 2 + s Jor every s > o.

In connection with I.g.6 let us recall a result of Kwapien which was already mentioned above (and actually proved in the case of lattices after I.f.17): if a Banach space Xis of type 2 and cotype 2 then X is isomorphie to a Hilbert space. Thus, l.g.6 asserts that, unless Xis "very elose" to being a Hilbert space, X has a subspace which fails to have the C.A.P. There are however Banach lattices not isomorphie to Hilbert spaces in which every subspace has the C.A.P. and even the B.A.P.


Example 1.g.7 [57]. There is a sequenee of integers {kn}:'=l and a sequenee of

numbers {Pn}:'= 1 with Pn L 2 so that X = (~ EB I;:) is not isomorphie to 12 but every n=l 2

subspaee Y of X has the bounded approximation property.

The proof of I.g. 7 is based on the following four facts:

(l) There is a constant K (independent of {kn}:'= 1 and {Pn}:'= 1 provided Pn:( 3 for all n) so that, for every X of the form appearing in I.g.7 and every finite dimensional subspace E of X with dim E = m, there is a projection Q from X onto E with

This fact is a special case of a general result from [93] to be proved in Vol. III. The assertion is actually true for every Banach space X of type 2 and the constant K depends only on the type 2 constant of X.

(2) For every integer m there is an 8> ° so that if Ip - 21 < 8 and E is an mdimensional subspace of lp then d(E, 1~)<2. In Vol. III we shall prove a more precise version of this fact, namely that, for every 1 :(p < 00 and every E c Ip with dim E=m, we have d(E, 1~):(mI1/2-1/pl (cf. [72]).

(3) For every p i= 2, lim d(lpm, I;') = 00. This fact is obvious. Actually, it is not m~oo

hard to verify that d(l;', 1;)=mll/2-1/pl. (4) Let T: X ---> Z be a quotient map and let E c Z be a subspace of dimension

m. Then there is a subspace G of X with dim G:( 5m so that TG = E and, for every Z E E, there is an x E G with Tx=z and Ilxll:(311z11.

We prove the vaJidity ofassertion (4). Let {Z;}l=l be a subset of {z;zEE, Ilzll = I} so that Ilzi - Zjll ~ 1/2 for every ii= j and which cannot be included in a larger set having this property (and thus, whenever Z E E with Ilzll = 1, there is an 1:( i:( 1 with Ilz-zill < 1/2). The balls BE(zi' 1/4), i = 1, ... ,1 have pairwise disjoint interiors and are all contained in BiO, 5/4). Hence, by considering the volumes of these balls, we get that (5/4)m~/(I/4r i.e. 1:(5m. For each 1 :(i:(/let Xi be an element of X so that Ilxill < 3/2 and TX i = Zi and put G = [xJ: = 1. Then, clearly, TBG(O, 3)::>2 conv {±ZJl=l and it remains to verify that this latter set contains BE(O, 1). For every Z E E with Ilzll:( 1 there is a 1 :(jl:(1 such that Ilz-llzllzhll:( Ilzll/2 and, by induction, we can continue and choose {AHX)= 2' all integers between 1 and I, so that, for every n ~ 1,

Hence, Z E 2 conv {± zJl= 1 and this completes the proof of assertion (4).

Proof of l.g.7. We choose inductively a sequence of numbers {Pn}:= 1 decreasing to 2 and a sequence of integers {kn}:= 1 in the following manner. We start by taking


Pl = 3. Then select k l , P2' k 2 , ... in this order so that (i) d(l;:, I';»n, n= 1,2, ... (this is possib1e by (3) above)

(ii) For every n= 1,2, ... , 2<Pn+1 <Pn a~d, whenever Ec.lp with 2~P<Pn+1

is a subspace of dimension h~2·5'~lk" then d(E,m~2. (This is possib1e by (2) above.)

Let now Y be any subspace of X, let F be a finite dimensional subspace of Y "-I

and put m = dirn F. Pick an integer n so that m ~ 5 '~I k, and let Yn be the subspace of Y consisting of all those vectors whose components in I;: are zero for i<n.

n-1 Clearly, dirn Y/Yn ~ L k i · By assertion (4) above and its proofthere is a subspace

i= 1

G of Y containing F of dimension at most

so that ifTdenotes the quotientmap from Yonto Y/Yn then TBG(O, 3):=> By/yn(O, 1). Since

where 11: i is the natural projection from X onto I;:, it follows that d( Gn Yn , I~) ~ 2, where h = dirn Gn Yn . Hence, by fact (i) above, there is a projection Q from Y (even from X) onto Gn Yn with IIQII ~2K. The restriction of Tto (I-Q)G is one to one (since kern TIG = Gn Yn) and thus S = (TI(f _ Q)G) - 1 is well defined. Since, for every Z E Y/Yn with Ilzll = 1, there is a Y E G with Ilyll ~3 and Ty=z, we get that Sz=(l-Q)y and hence IISII~31II-QII~9K. The operator P=ST+Q is a projection of norm ~ 11K from Y onto G. 0

We conc1ude this section by mentioning that, by an approach which is somewhat similar to the proofs of l.g.2 and l.g.4, it was proved in [125] that the space L(/2 , 12) with the usual operator norm fails to have the A.P. (this is the solution to apart of Problem I.I.e.1 0).

2. Rearrangement Invariant Function Spaces

a. Basic Definitions, Examples and Results

Many of the lattiees of measurable funetions whieh appear in analysis have an important symmetry property, namely they remain invariant ifwe apply a measure preserving transformation to the underlying measure spaee. Sueh lattiees are ealled rearrangement invariant funetion spaees or r.i. spaees, in short. They are the natural generalization of the notion of a symmetrie basis (or asymmetrie sequenee spaee) to the setting oflattices. The importance of r.i. funetion spaees sterns mainly from two (closely related) faets: they form the natural framework for the study of some important questions eoneerning L/Jl) spaees and they arise naturally in interpolation theory. In this seetion we present some basie faets eoneerning r.i. spaees and prove also a quite simple but general interpolation theorem.

Before giving the formal definition of an r.i. space we prefer to diseuss in some detail the notions whieh enter in its definition. The main requirement imposed on an r.i. funetion spaee X will be that it is a Köthe funetion space on some u-finite measure spaee (0, L, Jl.) (ef. l.b.17) so that, for every automorphism r of 0 into itself and every ! e X, the funetion !(r-1(m»t also belongs to X. By an automorphism r of a measure space 0 into itself we mean a one-to-one map from 0 onto a measurable subset r(O) ofitselfso that both rand r- 1 are measurable and Jl.(u)=Jl.(r(u» for every u eL. If Jl.(0) < 00 then clearly O-r(O) has measure zero and thus we ean (and shaIl) assurne that r is onto. Observe that like any Köthe funetion spaee every r.i. space is, in partieular, u-complete. Henee, C(O, I), for example, will not be eonsidered as an r.i. space on [0, I].

We shall restriet our attention to the ease in whieh (0, L, Jl.) is a separable measure spaee (i.e. L, with the metrie d( u 1 , u 2) = Jl.( u 1 Au 2)' is a separable metrie spaee, where u 1 Au 2 = (u 1 - u 2)U(U 2 - u 1)). The strueture of sueh a measure spaee is simple and weIl known (ef. [50]): it eonsists of (a perhaps empty) eontinuous part whieh is isomorphie to the usual Lebesgue measure spaee on a finite or infinite interval on the line and of an at most eountable number of atoms. (By an isomorphism of two measure spaees we mean a one-to-one eorrespondenee between the u-algebras whieh preserves the measure and the eountable Boolean operations. In general (i.e. unless we have a measure theoretie pathology whieh is of no interest in the present context), this eorrespondence is induced by a point transformation between the measure spaees.) Sinee an automorphism r of a measure spaee 0 maps the eontinuous part of 0 into itself and maps eaeh atom of

t !(r- 1(w» is defined to be zero for w not in the range of r.

a. Basic Definitions, Examples and ResuIts 115

Q to an atom with the same mass it is clear that the study of r.i. spaces over a separable measure space reduces immediately to the study of such spaces when Q is either a finite or infinite interval on the line or a finite or countably infinite discrete measure space in which each point has the same mass. Thus, up to some inessential normalization, we are reduced to the study of the following three cases:

(i) Q = integers and the mass of every point is one. (ii) Q= [0, IJ with the usual Lebesgue measure.

(iii) Q = [0, 00) with the usual Lebesgue measure. Those three cases are different as will become apparent in the sequel. Every space with a symmetric basis is, in a natural way, an r.i. space on the Q given in (i) above. However, we will include in the definition of r.i. spaces on such Q also some nonseparable sequence spaces, e.g. 100 or, more generally, the Orlicz sequence spaces IM' where M does not satisfy the Llz-condition at ° (see 1.4.a). Though some ofthe theorems proved in the sequel have a meaning and are of interest also in case (i), our main emphasis in this chapter will be on the continuous cases, i.e. on (ii) and (iii).

It is worthwhile to make so me comments on the operator U.f(w)=f(,-I(W)) induced by an automorphism, of Q on a Köthe function space X on Q which is invariant under automorphisms. We note first that Ur' being a positive operator, is bounded. The family ofthese operators is actually uniformly bounded (this fact was observed in [87J). We show this, for example, in case (ii), i.e. when Q = [0, 1]. Let k be an integer and let Pi k' 1 ::::; i::::; k, be the projection on X defined by Pi J = f X[(i _ IJ/k, i/kJ' If the U;s are ~ot uniformly bounded on X then, since '

k k

L L Pj,k UrPi, k= Ur' j= t i= t

we get that for every k there are 1::::; ik,A::::;k so that sup IIPjd UrPik,kll = 00. Let k r

be an integer and let 0' 1 and 0' z be two sub sets of [0, IJ of measure k- 1• Since there are automorphisms of [0, IJ which map 0'1 onto [(ik-l)k-1, ikk- t ), respectively, O'z onto [Uk -I)k- 1, Ak- 1) and since these automorphisms induce isomorphisms of X onto itself we deduce that sup IIP"'2 U.PoJ =00, where P.,.f=

r

fXu ' In other words, we get that sup IIP~ UrP.,.II=oo, for every choice of subsets

'1, O'c [0, IJ ofpositive measure. He~ce, we can find a sequence {!..};;"= 1 offunctions in X and automorphisms {'n};;"= 1 of [0, 1J so that II!..// ::::;n- 2 , 11 Ur"!.. 11 ~n for every n and so that the sets {O'nu'n(O'n)}:'=t, where O'n=suPPfn' are mutually disjoint. Thus if '0 is an automorphism on [0, IJ so that '01.,." = 'ni.,." we would get that U is not bounded (i.e. cannot be defined) on X. ro

Since the U;s form a semigroup it folIo ws from the preceding remark that

IIlflll= sup {IIUJII;, an automorphism ofQ into Q}

is an equivalent lattice norm on X with respect to which each Ur is a contraction i.e. /11 Ur/ll::::; 1. We claim that actually each Ur is an isometry in 111·111. This is evident

116 2. Rearrangement Invariant Function Spaces

if .. is invertible, i.e. maps Q onto Q. To prove this fact in general, let I E X, let k k

be an integer, write Q= U (1i' with (1i being mutually disjoint sets each having i=1

infinite measure, and let I; = IX(1i' I ~ i ~ k. Since both 1-11 and U,(f -/1) vanish on a set of infinite measure there is an invertible automorphism "1 which maps 1-/1 onto U,(f-/1) and thus

Similarly, for every 1 ~i~k, 111/-1;111 ~IIIUJIII. By summing over ifrom I to kwe get that

k

(k-l)III/III~ L 1I1/-I;III~kIIlUJIII i=1

and, since k was arbitrary, we deduce that 111/111 = IlIuJIII. We have thus seen that if X is a Köthe function space which is invariant under

automorphisms we can renorm it so that every U, becomes an isometry of X. In the definition of an r.i. space given below we shall require from the outset that the U;s are isometries. In other words, the norm of anl E X will be assumed to depend only on the distribution Junction

d/t)=J.l({WEQ;f(w»t}), -oo<t<oo,

of/or, in fact, on the distribution function ofi/l. More precisely, if/E Xando is a measurable function such that dlgl(t)=dltt(t), for every t~O (i.e.1/1 and Igl are J.l

equimeasurable) then also 0 EX and 11011 = 11/11. This remark needs some additional explanation if J.l(Q)= 00. In this case dl/l(t) may become infinite for some t>O and the identity dlfl(t)=dlgl(t) does not necessarily imply that there is an automorphism .. of the measure space into itself which carries 1I1 into 101 or viceversa (let e.g.l= land letg be equal to Ion some set (1 and to 1/2 on Q"'(1, where J.l«(1) = J.l(Q", (1)= (0). However, it is clear that, for every I EX, dl/l(t) is finite for large enough t and that if dl/l(t)=dlgl(t) there are, for every 6>0, automorphisms

"1 and"2 ofQ into itselfso that U'2Igl~(l+e)U,J/I. This shows that U'2gE X; consequently,g EX and Ilgll= Ilfll· (The fact that U'2g E Ximpliesg E Xis obvious ifthe complement ofthe support of 0 has infinite measure, since in this case there is an automorphism .. of Q onto itselffor which U,U'20= o. For a general 0 we deduce this fact by writing 0 = 01 + 02 with the support of each Oi having a complement of infinite measure.)

The distribution function of a non-negative function I is clearly a right continuous non-increasing function on [0, (0) (which, in case J.l(Q) = 00, mayaiso take the value + 00 ; we assume, however, as will be the case for every function in an r.i. space, that it is finite for sufficiently large t). Of special importance in the investigation of r.i. spaces is the right continuous inverse 1* of dl (for ~O) which is

a. Basic Definitions, Examples and Results 117

defined by

f*(s) = inf {t>O; d/t)~s}, O~s<f.1(Q).

The functionf*, wh ich is evidently non-increasing, right continuous and has the same distribution function asj, is called the decreasing rearrangement off Iffis a general element in an r.i. space we denote by f* the decreasing rearrangement of Ifl. Notice thatf* is, by definition, a function on [0, f.1(Q» even if Q is the space of integers. In this latter case, however, f* corresponds in a natural way to a nondecreasing function on Q which is also denoted by f*. We adopt here the symbolf* for the decreasing rearrangement of a function since it is commonly used in the literature. In contrast to the notation x*, which is used in this book to denote an element in the conjugate space X*, the decreasing rearrangement does not have, of course, any connection to conjugate spaces. As a matter of fact, if X is an r.i. space and f a function belonging to X then f* also belongs to X. The meaning of f* will always be c1ear from the context and we trust that there will not arise any confusion between these two uses of the symbol *.

It is worthwhile to note that we do not have, in general, that (/1 + f2)*(t) ~ f/(t) + f2*(t) (take e.g. f1 (t) = t and f2(t) = 1- t on [0, 1]). The following useful identity does however hold for every choice of f1 ,f2' t 1 and t 2

This identity is a consequence of the fact that

{w E Q; If1(w)+f2(w)l>f/(t1)+ft(t2)}c

{w E Q; If1(w)1 >ft(t1)} U {w E Q; If2(w)1 >ft(t2)} .

In particular, we have (/1 + f2)*(2t) ~f1*(t) + f2*(t). If a Köthe function space X on (Q, 1:, f.1) is invariant with respect to auto

morphisms of Q the same is true for X', the subspace of X* consisting of the integrals (see the end of l.b). Indeed, if r is an invertible automorphism of Q andf and g are non-negative measurable functions on Q then clearly

J j(w)g(r(w» df.1= J f(r- 1(w»g(w) df.1. [1 [1

This identity and the definition of X' show that X' is invariant under invertible automorphisms of Q. It follows however from the preceding discussion that X' is also invariant with respect to every automorphism of Q into itself. In all interesting examples of spaces invariant with respect to automorphisms, X' is a norming subspace of X* (or, equivalently, by l.b.l8, O~fn<w) i j(w) a.e. withfE Ximplies Ilfll = lim 111.11)· We shall indude also this assumption in the definition of an r.i.

n

space.

Definition 2.a.1. Let (Q, L, J1) be one of the measure spaces {I, 2, ... }, [0, 1] or


[0, 00) (with the natural measure). A Köthe function space X on (0, L, Jl.) is said to be arearrangement invariant (r.i.) space if the following conditions hold.

(i) If 'I: is an automorphism of Q into itself and I is a measurable function on Q then/e Xif and only if/('I:- 1(w» e X and ifthis is the case then 11 l(w)11 = 11/('I:- 1(w)ll·

(ii) X' is a norming subspace of X* and thus X is order isometrie to a subspaee of X". As a subspace of X", Xis either maximal (i.e. X = X") or minimal (Le. Xis the closed linear span of the simple integrable functions of X").

(iii) a. If 0 = { I, 2, ... } then, as sets,

and the inclusion maps are ofnorm one, i.e. if/e 11 then 11/11x~11/111 and if/e X then 1l/1100~ll/llx'

b. If Q = [0, I ] then, as sets,

Loo(O, l)cXcL1(0, I)

and the inclusion maps are of norm one i.e. ifl e Loo(O' I) then 11/11x~ 11/1100 and if/e Xthen 11/111 ~ll/llx'

C. If 0 = [0, 00) then, as sets,

and the inclusion maps are of norm one with respect to the natural norms in these spaces, i.e. if fE L oo nL1 then IIfllx~max(11!1I1' IIflloo) and if fEX then 1

J f*(t) dt~llfllx' o

We have already explained in detail the role of assumption (i) and of the restrietion imposed on 0 in 2.a.1. Assumptions (ii) and (iii) require, however, additional explanation. We consider first (ii). By definition, every Köthe function space contains the simple integrable functions and thus the terminology ofminimal (and clearly maximal) used in (ii) is justified. If X is separable then, since it is a-order complete, it is also order continuous (cf. the remark preceding 1.a.8) and therefore, in this case, assumption (ii) is always satisfied with X being the minimal subspace of X". By the remark following 1.b.l8, Xis a maximal subspace of X" (i.e. X = X") if and only if it has the Fatou property. The non-separable space Loo(O, I) is a minimal r.i. function space (it is clearly also a maximal one). Every minimal non-separable r.L function space X on [0, I] is equal to Loo(O, I) (up to an equivalent norm). Indeed, if lim Ilx[o,tlllx=O then Xis separable (the charac-

t .... O

teristic functions of dyadic intervals span a dense set) while if lim Ilx[o, tllix > ° t .... O

then X=Loo[O, I]. Similarly, every non-separable minimal r.i. function spaceX on [0,00) has the property that its restrietion to [0, I] is equal to Loo(O, I) (with an equivalent norm); however, X itselfneed not be isomorphie to Loo(O, 00).


A separable r.i. funetion spaee X is maximal if and only if X does not have a subspaee isomorphie to eo (use 1.e.4). In partieular, every reflexive r.i. spaee is both minimal and maximal.

The reason for assuming (ii) (besides beip.g satisfied for separable spaees X and the eommon non-separable examples) is that the basie results on r.i. spaees, to be proved below, may fail to hold without it (see example 2.a.11 below).

In order to avoid possible eonfusion we point out that there is no eonneetion between the terms minimal and maximal r.i. spaees defined here and the notion of a minimal Orliez funetion (or of a minimal symmetrie basis) introdueed in l.4.b.7. Minimality in the sense of 1.4. b will not be used in this volume.

Assumption (iii) in 2.a.1 is just a normalization eondition. The inclusion relations in (iii) are already a eonsequenee of (i) and the normalization is imposed just to ensure that the inclusion maps have all norm one. Condition (iii)a means that the unit veetor (1,0, ° ... ) in Xhas norm one. Similarly, eondition (iii)b means that Ilx[o. dlx= 1. Indeed, for every I E Loo(O, 1), 1/1:::; 11/1100x[o, 1) and, by a simple averaging argument it follows that, for every simple funetion of the form

n n

1= L aiX[(i-1)/n,i/nP we have 1I/IIx~n-1 L lai IIIX[o,1)llx=11/1111Ix[o,dlx' Con-i= 1 i= 1

dition (iii)e requires a litde more explanation eoneerning the spaees appearing in its formulation.

Proposition 2.a.2 [48]. The spaee Y = L 1 (0, (0) + Loo(O, IX)) eonsisting 01 all the lunetionslon [0, (0) whieh eanbe written asg +h withg E L 1 (0, (0) andh E Loo(O, (0), beeomes a Banaeh spaee if we define the norm in it by

This norm on Y ean also be eomputed by

11I11 = i I*(t) dt= sup Ü I/(t)1 dt; /1«(J) = I} . The spaee Y has the Fatou property and is an r.i.lunetion spaee on [0, (0). The spaee Y' isorder isometrie to L 1 (0,00 )nLoo(O, oo)endowedwith thenormmax (11/111' 1111100)'

Proof It is trivial to verify that Y is eomplete. Let us prove that the two ex-1

press ions for the norm eoineide, denoting for the moment J I*(t) dt by 111/111· o

If I=g+h then, for every (J C [0, (0), J I/(t)1 dt:::;llgl11 + Ilhlloo/1«(J), and henee "

111/111::::; 11/11· Conversely, fix I E Yand put A= 11/* - I*x[o, 1]1100' Then

11I11 = 11/*11:::; 11/* - min (A,f*)lll + Ilmin (Je,f*)lloo 1

= 11(1* - A)x[O, dl1 +Je= J I*(t) dt= 111/111· o


The fact that Yhas the Fatou property is evident from the form of 111/111. Let now 00

k E Y' be an element of norm one; then, in particular, J Ik(t)g(t)1 dt:::; 1 for every o

gE Lt(O, (0) with IIgllt:::;1 and also for every g ELoo(O, (0) with Ilglloo:::;1. Hence, Ilklloo:::; 1 and IIkllt :::; 1. It is just as trivial to verify that, conversely, Ilklloo:::; 1 and Ilklit :::; 1 imply that k E Y' with Ilklly.:::; 1. 0

Proposition 2.a.2 explains the notions appearing in (iii)c of 2.a.1. Condition iii( c) is equivalent to the normalization condition IlxiO. d Ix = 1 (provided, of course, that we assume (i) of 2.a.l). Indeed, since 1I/IIx~II/*I[o.dlx it follows by the

1

discussion preceding 2.a.2 that IlxIO. lllix = 1 implies 11/11x~ J I*(t) dt. Conseo

quently, we get that also IlxiO. dlx.= 1 and hence, by duality, it follows that

1l/llx:::;max (11/111' 1111100)' The most commonly used r.i. function spaces on [0, 1] and [0, (0), besides the

L p spaces, l:::;p:::; 00, are the Orlicz function spaces. Let M be an Orlicz function on [0, (0) (i.e. a continuous convex increasing function satisfying M(O) = ° and M(t) ~ 00 as t ~ (0) and let a be either 1 or 00. The Orlicz space LM(O, a) is the space of all (equivalence dasses of) measurable functions/on [0, a) so that

a

J M(I/(t)l/p)dt<oo, o

for some p>O. The norm in LM(O, a) is defined by

11/11= inf {p>o; I M(I/(t)l/p) dt:::; I}. It is easily checked that LM(O, a) has the Fatou property and thus if M is normalized so that M(1)= 1 then LM(O, a) is a maximal r.i. function space according to 2.a.1. In the study of Orlicz function spaces, the subspace HM(O, a) of LM(O, a), which

00 consists of all I E LM(O, a) so that J M(lj{t) 1/ p) dt< 00 for every p > 0, is of particular

o interest.1t is easily verified that HM(O, a) is the dosure in LM(O, a) ofthe integrable simple functions and thus it is also an r.i. function space according to 2.a.l (namely a minimal r.i. space). It is also quite easily checked (in a manner similar to the proof ofI.4.a.4) that LM(O, 1) =H M(O, 1) if and only if M satisfies the L1 2-condition at 00 i.e. lim sup M(2t)/ M(t) < 00 and that LM(O, (0) = H M(O, (0) if and only if M

t-+ 00 satisfies the .1 2 -condition both at ° and at 00.

Another dass of r.i. function spaces which has received attention in the literature, especiaIly in connection with interpolation theory, is that ofLorentz function spaces. Let I :::;p < 00 and let W be a positive non-increasing continuous function

1 00 on (0, (0) so that lim W(t) = 00, lim W(t)=O, J W(t)dt=1 and J W(t)dt=oo.

t-O t-oo 0 0


The Lorentz funetion space LwjO, 00) is the spaee of aU measurable funetions f on [0, 00) for whieh

(CO )l/P Ilfll = t f*(t)PW(t) dt < 00 .

If we impose on W only those eonditions whieh involve the interval [0, I] and define the norm by integrating over (0, 1) we obtain the Lorentz funetion spaee LwjO, 1). Obviously, the Lorentz funetion spaees have the Fatou property. The

1

eondition J W(t) dt = 1 is imposed to ensure that the Lorentz spaees satisfy the o

normalization eondition (iii) of 2.a.l. The other three eonditions imposed on W are meant to exclude trivial eases.

If Xis an r.i. funetion spaee the same is true for X'. We have already noted that (i) of2.a.1 holds in X'. Sinee X' always has the Fatou property, (ii) holds too. It is evident that the normaIization eondition (iii) of 2.a.l is self dual. This remark shows, inpartieular, that every maximal r.i. spaeeXis ofthe form X=Z', for so me r.i. spaee Z (take Z = X'). We shaU show next that, with one exeeption, such an X is of the form X = Y* for some r.i. spaee Y.

Proposition 2.a.3. Let X be a maximal r.i. funetion spaee on [0, 1] whieh is not order isomorphie to L l (0, 1). Then X = Y*, where Y is the closed linear span of the simple funetions in X'.

Proof Sinee clearly X = Y ' it suffiees to show that Y ' = Y* or, equivalently, that Y is a-ordereontinuous (see the diseussion on general Köthe spaees preeeding l.b.18).

By its definition, Y is a minimal r.i. funetion spaee on [0, 1] and, thus, if it is not a-order eontinuous then, by 1.a.7, it is not separable. Henee, as remarked above, Y is, up to an equivalent renorming, equal to Lco(O, 1) from wh ich it is immediately dedueed that X is order isomorphie to L l (0, 1). 0

Remarks. 1. The unit ball of L 1 (0, 1) does not have any extreme point. Henee, by the Krein-Milman theorem, L 1 (0, 1) is not isometrie to a eonjugate spaee. This same observation eoneerning extreme points ean be used to show that L l (0, 1) is not even isomorphie to a subspace of a separable eonjugate spaee. (We shall diseuss this matter in Vol. IV.)

2. There are maximal r.i. funetion spaees on [0, !Xl) whieh are not isomorphie to either L 1 (0, 00) or to a eonjugate spaee. Consider, for example, L l (0, !Xl) + LiO, 00) with I <p < 00 (the spaee is separable and has a subspace isomorphie to L 1(0, 1)). The proof of 2.a.3 shows that if X is an r.i. funetion spaee on [0, 00) having the Fatou property then X= Y* (Ybeing the minimal r.i. subspace of X') provided that the restrietion of X to [0, 1], i.e. the subspace of aUf EX, whieh are supported on [0, I], is not isomorphie to L l (0, I).

The special role of block bases with eonstant eoeffieients in the study of spaees


with a symmetrie basis is played in the theory of r.i. funetion spaees by a-algebras Pl ofLebesgue measurable subsets of[O, (0) (or of[O, 1]). By the Radon-Nikodym Theorem, for every fELl (0, (0) + Loo(O, (0) and every a-algebra Pl of measurable subsets of [0, (0) so that the Lebesgue measure restrieted to ffI is a-finite (i.e. so that Pl does not have atoms of infinite measure), there exists a unique, up to equality a.e., Pl-measurable loeally integrable funetion E fJ8f so that

00 00

S gE fJ8fdt= S gfdt, o 0

for every bounded, integrable and ffI-measurable funetiong on [0,(0). (Reeall that a function g is said to be Pl-measurable if g-l(G) E Pl, whenever G is an open sub set ofthe line). In partieular, we have that

for every Pl-measurable set a with J1(a) < 00. The funetion E fJ8j, defined above, is ealled the conditional expectation off with respect to ffI. (The term eonditional expeetation and the notation E fJ8f are taken from probability theory where the term expeetation, denoted by EI, means the integral of f with respeet to the underlying probability spaee.) The map E fJ8 , which is obviously linear, is also called sometimes an averaging operator. This term originates in examples of the following type: take as measure spaee the unit square [0, 1] x [0, 1] endowed with the usual Lebesgue measure (which is measure theoretieally equivalent to [0, 1]) and eonsider the a-algebra Pl of all subsets of [0, 1] x [0, 1] having the form a x [0, 1] with a ranging over the measurable sub sets of [0, 1]. In this ease, for every fELl ([0, 1] x [0, 1]),

1

E fJ8f(s, t) = S fes, u) du . o

The linear map E fJ8 is clearly positive and aets as a projeetion of norm one in Loo(O, (0) and in L 1 (0, (0). Therefore, E fJ8 is a projeetion of norm one also in L 1(0, (0) + Loo(O, (0) and in L 1(0, oo)nLoo(O, (0). The same is true for every r.i. funetion spaee X on [0, (0) or [0, 1].

Theorem 2.a.4. Let X be an r.i. function space on the interval I, where I is either [0, 1] or [0, (0). Then, for every a-algebra Pl of measurable subsets of I so that the Lebesgue measure restricted to Pl is a-finite, the conditional expectation E fJ8 is a projection of norm one from X onto the subspace XfJ8 of X consisting of all the Plmeasurable functions in it.

Proof Assurne first that X is maximal. We start by proving that if fEX and {a;}?= 1 are disjoint sets of finite measure in I then the eonditional expeetation g of I, with respeet to the algebra generated by {ai }?= 1 and the points of the

a. Basic Definitions, Examples and Results 123 n

complement of U (Ji' belongs to X and satisfies Ilgllx~ Ilfllx. It is c1early enough i= 1

to consider the case where n = 1 and (J 1 = [0, a]. Then 9 is given by

a

g(t)=a- 1 Jf(s)ds, O~t~a, g(t)=f(t), t>a. o

For every O~s~a let fs(t)=f«t+s) mod a) if O~t~a and fs(t)=f(t) for t>a. Then, since Xis r.i·,fs EX and Ilfsllx= Ilfllx for every O~s~a. Let h be a simple integrable function on I. Then

a

J g(t)h(t) dt=a- 1 J ds Jfs(t)h(t) dt~llfllxllhllx', 101

and since X" =X (as a consequence of the maximality of X which is equivalent to the Fatou property), our assertion on g. is proved.

Let now f!B be a (J-algebra of sub sets of I. Clearly, X ~ is a Köthe function n

space on (I, f!B; /1) having the Fatou property. Let k= L biXa , be a simple f!Bi= 1

measurable integrable function. Since, for every fE X, there is an JE X~ so that IIJllx~ Ilfllx and J fk dt = J ]k dt (namely the restrietion of the 9 appearing in the

I I n

beginning of the proof to U (Ji) it follows that IlkllX' = Ilkllx&i' i= 1

Hence, if PO in X and k is as above then

J (E~f)k dt= J fk dt~ Ilfllxllkllx· = IlfllxllkIIX';f' I I

Since X~ has the Fatou property i.e. X';'=X~ this proves that E~fE X~ and IIE~fllx~ Ilfllx·

Assurne now that Xis a minimal subspace of X'. By what we have already shown, E~ is an operator of norm one from X into X". Since E~ maps L l (I)nLoo(l) into itself, it maps also its c10sure in X", namely X, into itself. 0

Theorem 2.a.4 is actually a consequence of a general interpolation theorem (cf. 2.a.1O below). In the proof of this interpolation theorem, as well as in other investigations of r.i. spaces, a certain order-like relation in L 1 + L 00' which was introduced by Hardy, Littlewood and Polya [51], plays an important role. Before defining this relation for functions we consider briefly the simpler case of vectors in Rn which illustrates very well the general case.

Let x=(a1 , az' ... , an) and y=(b1 , bz ... , bn) be two elements in Rn. We write X -<y if a! +a! + ... +at~ b! +b! + ... +bt for every k~n, where (a!, a!, ... , a:), respectively (b!, b!, "', b:), are the decreasing rearrangements of(la11, lazl, "', lan!) and (Ib 11, Ibzl, ... , Ibn!)' As we shall presently see, this order-like relation is c10sely related to a certain set of matrices. We let ~ n be the set of all n x n matrices


" " (lXi) such that L IcXi.jl~ 1, for every j, and L IIXi.jl~ 1, for every i. It is easily

i= 1 j= 1

seen that a matrix belongs to qp" if and only if the operator which it defines on R"(with respeet to the unit veetor basis {ek}:=l) is ofnorm at most one in both I~ and I:'. (We shall often identify a matrix in qp" with the corresponding operator T.) We denote by 8" the subset of qp" consisting of operators of the form Tek = Oke"(k)' 1 ~ k ~ n, where lOk I = 1, for every k, and 1t is a permutation of {I, 2, ... , n}.

Proposition 2.a.5 [51]. For vectors x, y E R" we have x -<y if and only if x= Ty for some TE f!)".

" Proof To prove the "if" part assume that a i = L IXi,)./)j' i= 1, ... , n. Then, for

j= 1

every subset (J of {I, ... , n} of cardinality k, we have

since .L IIX;J ~ 1 for every j and .~ (.L IIXiJ)~ k . • ea J= 1 .ea

Toprovethe "onlyif"part, assumethatx=(a1, a2 , ••• , a,,) -<y=(b1, b2 , ... , b,,) and x lf eonv {Ty; TE tS',,}, Then, by the separation theorem, there are {Äk}: = 1

" " so that L Ä~k> 1 and L IÄ~"(k)l~ 1 for every permutation 1t. We may clearly

k=1 k=1 assume that IÄ11~IÄ21~IÄ31~' . '. The following computation leads then to a contradiction.

" I~ L IÄklbt=(IÄ11-1A.21)b!+(1A.21-1A.31)(b!+b!)+ . .. k= 1

+ IÄ"I(b! +b! + ... +b:)~ (IÄ11-IÄ21)la11 +(IÄ21-IÄ11)(la11 + la21) + "

... +IÄ"I(la11+la21+' .. + la"l) = L IÄkllakl> 1. 0 k=1

Remarks. 1. The proof shows that, for every y and every SE f!)", we have Sy E conv {Ty; TE 8,,}. If 1\·11 is any nOrm in R", with respect to whieh, the unit veetors have symmetrie eonstant one then, since every TE 8" is an isometry of (R",II·II), it follows that every SE f!)" is a contraction in this space. This is the essential content of the interpolation theorem 2.a.lO below (in the ease 01' linear operators). We also get that Ilxll ~ Ilyll whenever x -<y (again, this is the main point in 2.a.8 below).

2. From the remark above it follows also that, for every y ER", the extreme points of the convex set {Sy; S E f!) ,,} are of the form Ty with TE 8". A weIl known result, which essentially goes back to Birkhoff [lI], states that 8" is precisely the set of extreme points of f!)". Since we shall not need this somewhat stronger result we omit its proof.


3. If X and Y are positive (i.e. ak~O, bk~O for all k) and x <y then there is a positive operator TE !!J n so that x = Ty. In order to see this, we have just to replace in the proof above the set C n by that of all the operators of the form Tek = ()ke"(k) with ()k=O, 1 for all k.

We pass now to function spaces.

Definition 2.a.6. Let/, gE L 1(0, oo)+Loo(O, (0) (respectively L 1(0, I». We write I <g if, for every O<s<oo (respectively O<s:::;; I),

• • J I*(t) dt:::;; J g*(t) dt; o 0

Whenever we use in the sequel the relation < between functions we shall assume implicitly that they belong to the function spaces appearing in 2.a.6. Clearly, I< 9 is equivalent to 1/1< Igl, to 1* < g* and to AI< Ag, for every real A:F O. Also I< 9 and 9 < h imply I< h. The relations I< 9 and 9 -<Ihold if and only if/*=g*. For every two functions/1 and/2 we have ifl+12)*-</l*+12*' The relation -< has the following decomposition property (cf. [83]).

Proposition 2.a.7. Assume that 9 -</1 +12 with g,/l andl2 non-negative. Then there exist non-negative 9 1 and 9 2 with 9 1 + 9 2 = 9 and gi -<h, i = I, 2.

Proof Consider first the case of vectors in K'. If x -<Yl + Y2 with non-negative vectors then, by remark 3 following 2.a.5, there is a positive TE!!J n so that x = T(Yl + Y2)' Put xi= TYi' i= 1,2. Then x=x 1 +x2, Xi -<Yi and the Xi are nonnegative, i= 1,2.

We consider now the case offunctions. There is no loss of generality to assume that J; = J;*, i = 1, 2. Let n and k = k(n) be integers and let g<n) be a positive function

n

so that g(n) = L aj,nXoj, n :::;;g, where {(Tj, n}j= 1 are disjoint sets with J1«(Tj, n)=k- 1, j= 1

for every j, and so that g(n)(t) i g(t) a.e. as n ~ 00. Leth(n), i= 1,2, be defined by j/k

h(n)(t)=k J h(s) ds if tE [U-I)jk, jjk),j= I, 2, ... , n andJ;(n)(t) =0 for t~njk. (j-l)/k

Then gIn) -<11") + Ir and it follows readily from the case of vectors in K' that gIn) = g~n) + g~) with gIn) ~ 0 and gIn) -<hIn) -<h, i = 1, 2. Since the functions gIn) are bounded by 9 there is a subsequence {nj}.i= 1 of the integers so that, for i= 1,2, {glnj )}.i=l converge in the w topology of L 1(r!> for every finite interval '1 of[O, (0) to limits gi> i= 1,2. It is easily verified that these gi have the desired property. 0

We exhibit next the connection between the relation -< and r.i. function spaces.

Proposition 2.a.8. Let X be an r.i. lunction space on I which is either [0, 1] or [0, 00 ).

Assume that g-<Iand/E X. Then gEX and IIgll:::;;II/II·


Proof Suppose first that Xis maximal and let h be a simple integrable function. Then n

h* = L aiX[O,t;l for suitable ai ~O, i= 1, 2, ... , n and 0< t1 < t2 < ... < tn • We have i= 1

n ~ n ti

If gh dsl~ f g*h* ds= I ai f g*(s) ds~ L ai f f*(s) ds I I i= 1 0 i= 1 0

= f f*h* ds~ Ilf*llxllh*llx' = Ilfllxllhllx' . I

This proves that g E XI/=X and Ilgllx~ Ilfllx. Assume now that Xis a minimal subspace of X' and let 1:>0. We may clearly

assume thatfand g are non-negative and since LlnLoo is dense in X, we can write fasfl +f2 withfl ~0,f2~0,fl E LlnLoo and IIf21Ix~l:· By 2.a.7, g=gl +g2 with gi -<fJor i = I, 2. It follows from the first part ofthe proofthat gl' g2 E X', IlgIIIX" ~ Ilflllx~llfllx and Ilg211X"~I:· We also get from the first part of the proof that gl E L lnLoo and thus gl EX i.e. d(g, X) ~I:. Since I: was arbitrary this concludes the proof. 0

Before stating the interpolation theorem we need one more concept.

Definition 2.a.9. A mapping T from a Banach space X into a Banach lattice Y is said to be quasilinear if

(i) lT(etx)1 = letllTxl, x E X, et scalar. (ii) There is a constant C< 00 so that

IT(xl +x2)1~C(ITxII+ITx21), Xl' x2 EX.

A quasilinear operator is said to be bounded if IITII = sup {lITxll; Ilxll ~ I} < 00.

It is clear that every linear operator is quasilinear. There are several important examples of non-linear quasilinear operators. The most commonly used example ofthis type is the so-called"square function", which is introduced in the following situation. Let {x;} be an unconditional basic sequence offinite or infinite length in a q-concave Banach lattice X for some q< 00. Then, for every X= L aixi E [Xi]'

the square function

is weH defined (since X is q-concave it is a Köthe function space and thus

( )1/2

71aixil2 is a weH defined function even if the summation is infinite).

Proposition l.d.6 ensures that this function belongs to X and its norm is actuaHy equivalent to Ilxll. It is obvious that S is quasilinear (with C= 1) but not linear.


The following interpolation theorem is due to Calderon [22J (cf. also Mitjagin [101J).

Theorem 2.a.l0. Let X be an r.i.function space on I which is either [0, IJ or [0, (0). Let T be a quasilinear operator defined on L oo (/) + LI (/) which is bounded on both L oo(/) and L 1 (/). Then T maps X into X and

where C is the constant appearing in 2.a.9(ii).

Proof Let fEX, let sEI, put

{f(t) - f*(s) if f(t) > f*(s)

g.(t) = f(t) + f*(s) if f(t) < - f*(s)

° if If(t)1 ~f*(s)

and h.(t) = f(t) - g.(t). Clearly, Ilh.lloo =f*(s) and

• Ilg.111 = J f*(t) dt-sf*(s) .

o

Sinee ITfl ~ C(ITg.1 + I Th.l) it follows that

I (Tf)*(t) dt ~ c(I (Tg.)*(t) dt + I IITh.lloo dt) o 0 0

• ~ C(IITg.III +sIITh.lloo)~ C max <lITIII' IITlloo) J f*(t) dt .

o

Consequently, Tf -<C max (11Tlito IITlloo)fand the desired result follows byapplying 2.a.8. 0

Remarks. 1. The assumption in 2.a.1O that X is an r.i. funetion spaee is also neeessary as far as the main requirement in 2.a.l (i.e. (i) there) is eoneerned. Indeed, if-r is an automorphism of Iinto itselfthen VJ(t)=f(-r-I(t» is a linear operator of norm one in L 1(/) and L oo (/)' The eonclusion of 2.a.lO thus asserts, in partieular, that V, has norm one on X.

2. Theorem 2.a.1O holds also, and with the same proof, if X is an r.i. spaee on the integers. For spaees X with a symmetrie basis, 2.a.1O takes the following form. Let T be a quasilinear operator on Co whieh is bounded in Co and 11 , Let X be a Banach sequenee space in which the unit vectors form a basis whose symmetrie constant is M. Then T maps X into itself with IITllx~ CM max (1ITIIIt , IITllco)'

3. It is instructive to note that, for a linear T and a separable r.i. space X, 2.a.lO can be easily deduced from 2.a.5. To fix ideas we assume that X is an Li. function space on [0, 1]. Let Tbe an operator on L 1(0, 1) so that I\TIL., and I\T\\1

128 2. Rearrangement Invariant Function Spaces n

are ~ 1. Let h be a simple function on [0, I] ofthe form L aiXai with /l(ui)=n- l , i=l

for every i. Then Th E Loo(O, I) and as such can be approximated in the Loo norm (and therefore in X) by simple functions. Since Xis separable Ilx[o,tlllx -+ ° as t -+ ° which implies that there is, for every 8>0, a finite algebra PA of subsets of [0, I] whose atoms have all the same measure so that h E E"'Xand IITh-gll~8 for some gE E'" X. The space E'" Xhas a basis with symmetrie constant one and hence, . by remark 1 following 2.a.5, IIE"'TE"'llx~ 1. Therefore,

Since 8 is arbitrary and the simple functions are dense in X it follows that IITllx ~ 1.

We conc1ude this section by presenting an example, due to Russu [117], of a Köthe sequence space (i.e. a Köthe space on {I, 2, ... }) Xl so that X~ is a norming subspace of Xi and any permutation of the integers induces an isometry on Xl' but on which the conditional expectation operator fails to be defined for a suitable u-finite algebra of sets. We present this example on {I, 2, ... } since in this form it is most transparent. With only trivial notational changes this example can be presented on [0, (0) and with some more changes also on [0, I]. This example shows, of course, the role played by the requirement 2.a.l(ii) that X be either a maximal or a minimal subspace of X". Without this assumption, 2.a.4, and thus also 2.a.8 and 2.a.1O, may fail to hold.

Example 2.a.11. Let X be the Danach space oJ all sequences x = (al' a2, ... ) so that

There is a closed ideal Xl in X wh ich is invariant under permutations (Le. (al' a2, ... ) E Xl if and only if (a"(l)' a,,(2)' ... ) E Xl' Jor every permutation 1t oJ the integers) but on which the conditional expectation operator cannot be definedJor a suitable u-finite algebra oJ sets. More precisely, there is an Xo == (b l , b2, ... ) E Xl and a partition oJ the integers into a sequence oJ pairwise dis joint finite sets {u n}:= 1

sothatifcj = L bJo-n,jEun,n=I,2, ... then(cl ,c2, ",)~Xl' iean

Proof It is evident that X has the Fatou property and is an r.i. space on the integers. We take x o=(l,2-t, ... ,rt, ... ) and let Xl be the smallest (c1osed) ideal in X which is invariant under permutations and contains xo. It is evident that Xl is the norm c10sure of the linear space of sequences (al' a2, ... ) for which sup jaj < 00. Let nk = 2k2 and put

j

nk+l

cj = L i -l/(nk+l -nk), nk<j~nk+l' k= 1, 2, .... i=nk+ 1

b. The Boyd Indices 129

We claim that Y=(c 1, c2 , .•• ) ~ Xl (clearly, Y E Xsince Xis an r.i. space). We note first that, for nk<I~nk+ l' Cj is ofthe order ofmagnitude of(log nUl -log nk)/nU 1

i.e. of k/nU1 . In other words, there is an rx>O so that cj >rxk/nU 1' nk<j~nu1' k= 1, 2, .... Let x=(a1, a2 , ••• ) be such thatjaj ~K, for some Kand allj= 1, 2 •.... We have to show that Ily-xll is bounded from below by a constant independent of x (and thus also of K). Let n be a permutation of the integers so that la j 1 = a:(j)' j=I,2, .... Let

'lk = {j; nk<j ~nk+ l' nU) ~2Knu l/rxk}, k= I, 2, ....

It is evident that, for k ~ ko = ko(K), the cardinality ~k of 'ik is larger than nk + d2. Forj E '1k' k= 1,2, ...

Hence,

~sup (rx(m(m+ I)-ko(ko+ 1))/8(1 +(m+ 1)2 log 2)) m

~rx/8 log 2. 0

Remarks. 1. Note that the space X of 2.a.l1 is the dual of the Lorentz sequence space d(l, w), where w = (l, 1/2, ... , l/j, ... ).

2. Calderon [22] proved a function space analogue of2.a.5 and showed thereby that Theorem 2.a.1O holds for a Köthe function space X on I if and only if X satisfies 2.a.l (i) and has the property that, whenever g <./ with / EX, then also gEX.

b. The Boyd Indices

In the previous section we proved that every operator, wh ich is bounded on LI (l) and Loo(l), acts also as a bounded operator on every r.i. function space on I. However, many of the interesting operators in analysis are not bounded simultaneously in both ofthese spaces, but only on suitable Lp(l) spaces with 1 <p< 00.

In this section we study r.i. function spaces X on I which are "between" LpJI) and L p2(I) in the sense that every operator, which is defined and bounded on these two spaces,' is defined and bounded also on X. This is done by assigning to each


r.i. function space two indices, called the Boyd indices. The definition of these indices resembles formally the notions of upper and lower p-estimates which were studied in section l.f. However, in spite of the formal resemblance, the Boyd indices do not coincide with the notions studied in 1.f. After investigating some simple properties of the indices and considering some examples we show that these indices really enable us to prove an interpolation theorem in the setting of r.i. function spaces for operators bounded in LpJI) and L p2(I). Actually, we prove an interpolation theorem for operators which are only ofweak type (P1,P1) and weak type (Pz, pz) (for the definition ofweak type, see 2.b.lO). We thus obtain a version of the classical Marcinkiewicz interpolation theorem for r.i. function spaces. Several applications of this theorem will be presented in the following sections.

We start by defining, for every 0< s < 00, a linear operator D s' If 1= [0, 00 )

we put, for a measurable functionf on I,

(Dsf)(t)=f(tjs), O<s<oo, O~t<oo .

If 1=[0, 1] we put, for a measurable f on land 0< s < 00,

(D f)(t) = {f(tjs)' t~min (1, s) sO, s < t ~ 1 (in case s < 1) .

Geometrically, in the case of [0, (0) the operatorDs dilates the graph off(t) by the ratio s: 1 in the direction of the taxis. In the case 1=[0, 1] we have the additional effect of restricting everything to I. It is obvious that D s acts as a linear operator ofnorm one on Loo(I) and ofnorms on L 1(!); hence, by 2.a.10, Ds is bounded on every r.i. function space X and IIDsllx ~max (1, s). Clearly, (DJ)* ~Dsf* for every fand sand hence IIDsl1 on an r.i. function space X can be computed by considering only non-increasing functionsJ Since, for every non-increasingpO and every O<r<s< 00, we have DJ~DJit is clear that IIDsl1 is a non-decreasing function of s. Also note that, for every rand s, D,Ds=D,s' with the only exception being the case r< 1 <s and 1= [0,1] in which we have D,DJ= X[o. ,)D,sf In any case we have

IID,s II ~ liD, II liDs II

(if r< 1 <s and 1= [0,1] simply use DsD, instead of D,Ds in order to verify this inequality). We are now ready to define the indices (cf. [16]) of an r.i. function space.

Definition 2.b.1. Let X be an r.i. function space on an interval I which is either [0,1] or [0, (0). The Boyd indices Px and qx are defined by

1. log s Px= 1m II II s .... oo log Ds

log s sup :----ii----.. s>1 10g11Dsll

1· log s . flog s qx= 1m = In

s .... o+ log IIDsl1 O<s<l log IIDsl1


The expression liDs II appearing above is, of course, the norm of D s acting as an operator in X. IfIIDs11 = 1, for some (and hence aIl) s> 1, we putpx= 00. Similarly, if IIDsll= 1, for all s< 1, we put qx= 00. We have to verify that the limits in 2.b.l exist and are equal to the re~pective supremum or infimum. Let q;(s) = log s/log IIDsl1 and let s, r~ 1 with sn :(r<sn+ 1, for some n. Then, since IIDsn + dl:( IIDslln + 1,

This easily implies our assertion concerning Px' The proof of the assertion concerning qx is the same.

The indices Px and qx can be computed explicitly for many examples of concrete r.i. function spaces. We shall carry out this computation in the case ofOrlicz function spaces in 2.b.5 below. Here we mention only the trivial, but important fact that if X = Lp(I), 1 :(p:( 00, then Px = qx = p. This fact influenced our decision to define the indices as in 2.b.1. In Boyd's paper and in several other pi aces in the literature the indices of X are taken to be the reciprocals of the ones we use here (i.e. r1.x = l/px and ßx = l/qx)'

Proposition 2.b.2. Let X be an r.i. function space. Then

(ii)

Proo! That Px ~ 1 follows from liDs II:(s while the assertion that Px :(qx can be easily deduced from IIDsIIIIDs-"I~IIDss-"I=1. This proves (i). To prove (ii), let fEX and g E X', pick s< 1 and assume that 1=[0, 1]. Then

s I

g(DJ)= S f(t/s)g(t) dt=s Sf(u)g(su) du=s(Ds-,g)(f)· o 0

By taking suprema over all fand g in the respective unit balls, we get that liDs Ilx = sllDs - ,Ilx" This proves that I/qx + I/pX' = 1. The proof of the other assertion in (ii) as weIl as the proof in case 1=[0, 00) are the same. 0

We considered so far only r.i. function spaces. The Boyd indices can be defined also for r.i. spaces on the integers. In this case the operators Ds are defined only if s is an integer or the reciprocal of an integer. If f=(a p a2 , a3 , ... ) and n = 1,2, ... we put

n n

~~ Dnf=(a l , a l , ... , a l , a2 , ... , a2 , a3 , ... ),

The indices Px and qx are defined as in 2.b.I by taking the limits onIy over s=n


(respectively, s= I/n), n= I, 2, .... The results proved in this section are all valid also for r.i. space on the integers. We shall however not give the proofs in this case since, while they are essentially the same as those for function spaces, they do often require a somewhat different notation.

It is worthwhile to note that if 1= [0, 00) and I is any measurable function then DJ can be written as 11 +12 + ... + In, where the 1; are mutually disjoint and each 1; has the same distribution function as f The same is true if 1=[0, I] and I is supported on [0, I In]. Hence, Px is the supremum of all the numbers p which have the following property: there exists a number K so that, for every choice of an integer n and of a function I having norm one (supported on [0, I/n] if 1 = [0, I]), we have

11/1 +12 + ... +/nll :;:;Kn1/p ,

where the {1;}~=1 are disjointly supported and have the same distribution function as f Similarly, qx is the infimum of all the numbers q for which there is a K so that, for every n and {1;}7= 1 as above,

To justify the first assertion for 1=[0, I] we have to note that, since DJ= Dn (X[O,l/n l /), the norm of Dn can be computed by considering only functions supported on [0, I/n).

It follows from this observation that if an r.i. function space X satisfies an upper p-estimate (cf. I.f.4) then P:;:;Px and if it satisfies a lower q-estimate then qx:;:;q. In general, Px (respectively, qx) is strict1y larger (respectively, smaller) than sup {p; X satisfies an upper p-estimate} (respectively, inf {q; X satisfies a lower q-estimate}). For example, consider the Lorentz function space X =

Lw, 1(0, 00), where W(t)=1/2jt. For every non-increasing/in Lw, 1(0, 00) and every 0< s < 00 we clearly have

J (f(tls)/2jt) dt=Js J (f(t)/2jt) dt ° 0

and thus IIDsl1 =Js, i.e. Px=qx=2. On the other hand, it is easily seen that, for any sequence {/n}:'= 1 of elements of norm one in X such that either sup {1f,,(t)l; 0< t< oo} or Jl(support l/nl) tend to zero, there is a subsequence which is equivalent to the unit vector basis of '1' Hence, X does not satisfy an upper p-estimate for any p > 1. Note that X* is non-separable. Hence, there are non-reflexive and even non-separable r.i. function spaces with I <Px:;:;qx< 00.

Any r.i. function space X satisfies L 1(/)nL oo(/)cXcL1(/)+L oo(/)' If we have information on the indices of X then a stronger assertion is valid.

Proposition 2.b.3. Let X be an r.i. lunction space on an interval 1 which is either [0, I] or [0,00). Then,for every I :;:;P<Px and qx<q:;:; 00, we have

L/I)nLq(/) cX.:: L p(/) + Li/) ,

with the inclusion maps being continuous.


The spacesL/I)nLq(I) and Lp(I) + Lq(I) aredefined in analogyto thecasep= I, q=oo treated in 2.a.2. Clearly, (Lp(I)nLq(I))*=Lp,(I)+Lq,(I), where I/p+ l/p'=I, l/q+l/q'=I, and if 1=[0, I] then LiI)nLiI)=Lq(I) and LiI)+ Lil)=L/I). We also note that in case Px= I we can take p= 1 and, similarly, if qx=oo we can take q= 00 in 2.b.3.

Proof It suffices to prove that LiI)nLq(I) eX with a continuous inclusion map; the second assertion will then follow by duality. Letp<Po<Px and qx<qo<q. Then there is a constant K so that IIDsll~Ksl/PO for s):1 and IIDsll~Ksl/qO for O~s~ 1. Since DsX[o, 1]= X[O,s] and Ilx[o, 1]llx= 1 we deduce that for (JeI

(The first of these inequalities makes sense only if 1=[0, 00).) Let now 9 be a non-negative simple function on I so that Ilgllp, Ilgllq ~ 1. Choose

n a simple function g on I with g/2~g~g so that g= L 2kXak for some integer

n and mutually disjoint sets {(Jd~= -n' Since

we have that

Ilgllx~KC~n 2krkp/po+ k~O 2krkq/qo) ,

and therefore

00

k= -n

Ilgllx~21Igllx~2K L (2-k(l-p/po)+rk(q/QO-l») . . 0 k=O

Remarks 1. Unless Px= 1 (or qx= 00), we cannot take, in general, in 2.b.3 P=Px (or q= qx)' For example, if X is the Lorentz space Lw, 1 (0, 00) with W(t) = 1/2y't then, clearly, 2.b.3 does not hold withp=q=2.

2. In connection with 2. b.3 and remark 1 above we should note that the following is true. Assume that X is an r.i.function space on [0, I] which is p-convex and q-concavefor some 1 ~p~q~ 00. Thenfor every fEX

where M(p)(X), respectively, ~q)(X) is the p-convexity, respectively, the q-concavity constant of X. We shall prove the assertion concerning q (the assertion concerningp will follow by duality). If q= 00 there is nothing to prove. If q< 00 then X must be a minimal r.i. function space and thus it suffices to consider simple functions.

n

Assume that f= Laiei' where ei= XW- 1)/n,i/n)' 1 ~ i~n, and let n be the set of j= 1


cyclic permutations of {I, 2, ... , n}. Then

Ilfll = C~u 11 it an(ijei Ilq/n) l/q ~ M(qj(X) IIC~u ttl an(ijei Iq/n) l/qll

=M(iX)IIC~u itl lanmlqe;/n)l/qll=M(qj(X)Ctl lailqy,qllCtl e;/n y/qll

=M(qj(X)Ct la;jqy,qn-1/q=M(qj(X)lIfllq .

3. The converse of 2.b.3 is not true in general. If X is an r.i. function space so that

with p<q then it need not be true that P~Px or that qx~q. Notice that in the proof of 2.b.3 we have just used estimates for IIX[o,alllx which follow from but are not equivalent to our assumptions on IID.II (viapx and qx)·

Proposition 2.b.3 can be used to describe the behaviour of the Rademacher functions in an r.i. function space X on [0, I]. For instance, if qx<oo and q>qx then, by 2.b.3, there exists a constant K< 00 such that Ilfllx ~ Kllfllq for all fE Lq(O, 1). Hence, by Khintchine's inequality I.2.b.3, we have that

for every choice of {aJ i n = o' This proves that in any r.i. function space X on [0, I] with qx< 00 the

Rademacher functions are equivalent to the unit vector basis in 12 , If, in addition, 1 <Px or, equivalently, qx' < 00 then the same is valid in X'. This implies that the span of the Rademacher functions in such an r.i. space is also complemented.

The conditions imposed above on the Boyd indices are however far from being necessary. For example, the fact that in a given r.i. function space X on [0, I] the Rademacher functions are equivalent to the unit vector basis of 12 does not even imply that X-:::JLq (0, I) for some q< 00. The following is a sharp result in this direction containing the above assertions as a particular case. Part (i) of 2.b.4 was proved by V. A. Rodin and E. M. Semyonov [137].

Theorem 2.b.4. Let X be an r.i. function space on [0, I] and let 11· 11M and 11· liMO denote the norms in the Orlicz function spaces LM(O, I), respectively LMo(O, I), where M(t) = (et2 -I)/(e-I) and M* is the function complementary to M (which at 00 is equivalent to t(log t)1/2).

(i) The Rademacher functions {ri}~O in X are equivalent to the uni! vector basis in 12 if and only if there exists a constant K 1 < 00 so that

IIfllx~KlllfllM ' for al/fE Loo(O, I).


(ii) The subspace [rJ~ 0 is complemented in X if and only if there is a constant K2< 00 such that

Jor al/JE Loo(O, 1), in which case it is, oJ course, isomorphie to 12,

In the proof we shall use the well known centrallimit theorem from probability theory (cf. for example [135]). This theorem states that if {gJ~ I is a sequence of independent and identically distributed random variables on a probability measure space (Q, 1:, j1) so that gl E L 2(j1) and J gl(W) dj1=O then the sequence

g

where (J2 = J gi (w) dj1, tends in distribution to the normal distribution Le. g

lim j1({w E Q; hn(w»r})= ~ 7 e- u2 / 2 du, n->oo y'2n,

for every real r. Here we shall use only the most simple and classical special case of "coin tossing" where gl (and thus every gi) takes only the values ± 1, each with probability 1/2.

Proo! (i) Suppose that there exists a K< 00 so that I/JI/x::::; KI/JIIM for every JE Loo (0, 1) and observe that a computation identical to that presented in the remark following 1.e.15 (based on the expansion of the function el2 into apower

00

series) proves that L: airi E LM(O, 1) whenever {aJ~o E 12, It follows from the i=O

closed graph theorem that there is an A < 00 so that

KA(.~ a~)1/2 ~KII.~ aJil1 ~II.~ airill ~II.~ airill ~AI(.~ a~)1/2, .-0 .-0 1 M .-0 X .-0 I .-0

for every choice of {aJ~o, i.e. that {rJ~o in Xis equivalent to the unit vector basis of 12 ,

Suppose now that there exists a constant B< 00 so that

for any choice of {ai} ~ o. In order to show that the norm in X is dominated by that in LM(O, 1) it suffices to prove that the function


belongs to X" (the maximal r.i. function space on [0,"1] containing X). Indeed, letJbe a non increasing function in Loo(O, 1) and notice that in any Li. function space Yon [0, 1] we have, by 2.a.4, that

t=llx[o,tlllyllx[o.tllly*, O::::;t::::;l.

Hence, in the particular case where Y=LM(O, 1) we find that

t

J(t)::::; r 1 J J(u) du::::; t-llIJIIM Ilxro, tlIIM'::::; IIJIIMCP(t), o

for aB O<t::::;1. Therefore, if CPEX" then IIJllx::::;KIIJIIM' where K=llcpllx" and this obviously completes the argument.

To prove that cp EX" or, equivalently, that the function (log l/t)1/2 belongs to X", we use the centrallimit theorem which, as pointed out above, asserts that if

tP' n(t) = (tl ri(t») I Jn then, for each ° <! < 00,

n .... oo n .... oo

2 00 -t 2

J 2 e =-- e- uI2 du::::;--·

J2n t J2n Thus, by passing to the inverse functions, we get that the pointwise limit tP' of { tP'n~)= 1 satisfies

On the other hand, since IItP'nllx::::;B for aB n (by our hypothesis) it foBows that tP', and thus also (log l/t)1/2, belong to X" (use Remark 3 following l.b.18).

(ii) If there is a constant K 2 < 00 as in the statement of (ii), then in addition to having

we obtain, by duality, that also

Hence, by part (i) of the theorem, the Rademacher functions in both X and X' are equivalent to the unit vector basis in 12 i.e.

for some constant K< 00 and for every choice of {ai} ~ 0' This implies that the

b. The Boyd Indices

orthogonal projection P defined by

1

is bounded in X. Indeed, if ai(f)=J j(u)ri(u) du then o

i.e.

It follows that

for every jE X, and this completes the proof of the assertion.

137

In order to prove the converse assertion it will suffice to show that if [ri]~ 0 is complemented in X then the orthogonal projection P defined above is bounded. Indeed, once this is shown, the fact that

for every choice of {aJ~o, implies, bya standard duality argument, that {ri}~O is equivalent to the unit vector basis in /2 both in X and X' (or X*) and the desired result follows from part (i).

Actually, what we need for the duality argument in the previous paragraph is just to know that the projections Pn=P1Xn ' n= 1,2, ... , have uniformly bounded norms, where

Fix an integer n. Let Q be a projection from X onto [r;]r;o. Then Qn=RnQlxn is a projection of norm ~ IIQ 11 from Xn onto [rJ~= 0' where Rn is the projection of

00 n

norm one on [rJ~o defined by Rn L air j = L airi· Let {Wj }7:1 be the first 2n

i=O i=O


Walsh functions on [0, I] introduced in the proof of l.g.2. These functions form an (orthonormal) basis of Xn. For I ~j, k~2n let (Jj,k be the value taken by wj on the interval [(k-l)r n, k2- nWj ,k= ± 1) and let T j be the linear map on Xn

defined by

The proof of the theorem will be concluded once we prove the following two facts:

2n

2. Pn=2- n L TjQnTj' j= 1

To prove statement I, notice that, for every 1 ~h~2n, we have that

Hence,

2 n

XHh- 1)2 -n, h2 -n) = 2- n L (Jk,h Wk . k= 1

2n

T j X{(h_1)2-n,h2- n)=2-n L (Jk,h(Jj,kWk' k= 1

Since, as is easily verified, (Jj,k=(Jk,j for all k andj and si~ce the product oftwo Walsh functions is again a Walsh function, we deduce that there is an index 1 ~i~2n so that (Jk,h (Jj,k= (Jk,i for 1 ~k~2n. Consequently,

T j X{(h-1)2 -n,h2 -n) = Xw- 1)2 -n, i2 -n)

and this means that Tj is a map induced by an automorphism of [0, I]. Since Xis an r.i. function space we deduce that Tj is an isometry on Xn thus establishing I.

To verify 2, denote by An C {I, 2, ... , 2n } the subset of those indices j for which wj is a Rademacher function (and not a product of two or more distinct Rademacher functions). Let Ci,k' i E An' 1 ~k~2n be such that

Then

QnWk= L Ci,kWi' ieAn

Statement 2 folIows now from the orthogonality of the matrix «(Jj, k);,~= l' 0

Remark. The proof of part (ii) of 2.b.4 actually shows that if [ra~o is complemented in an r.i. function space X on [0, 1] then the orthogonal projection P from X onto [rJ~o is automatically bounded.


Before continuing the study of general r.i. function spaces we evaluate the Boyd indices in an important special case.

Proposition 2.b.5 [120], [17]. Let X=LM(O, 1) be an OrliczJunction space. Then

Px=sup {p; inf M(At)/M(A)tP>O} -<,t::> 1

= sup {p; X satisfies an upper p-estimate} ,

qx=inf {q; sup M(At)/M(A)tq < oo} -<, t::> 1

= inf {q; X satisfies a lower q-estimate } .

Proof Put

aM,oo=sup {p; inf M(At)/M(A)tP>O} -<, t::> 1

and

ßM,oo=inf{p; sup M(At)/M(A)tP<oo}. A, t~ 1

We shall prove now the assertion concerning Px' The assertion on qx will then follow by duality. Let p<aM,oo' Then there is a y>O so that M(At)~yM(A)tP, whenever t, A~ 1. By replacing y by a possibly smaller constant we assurne, as we clearly may, that this inequality holds for every t ~ 1 and A ~ 1/2. Let {J;}7= 1 be disjointly supported non-negative functions in LM(O, 1) and put I 1/;/1 =ai, 1 ~ i~n,

and 11 it1

/;11 = b. Clearly, b ~ ai for all i and thus,

M(/;(u)/a j ) ~yM(/;(u)/b)(b/aY ,

for every u E [0, 1] for which/;(u)/b~ 1/2. Put

O'={UE [0, 1]; itl.t;(U)~b/2}. Then

The integral of MC~l/;/b) over [0, 1]~0' is clearlY less or equal to M(l/2).

Hence,


and therefore, ifwe note that M(1/2)<M(l)= 1, we get that

whieh shows that X satisfies an upper p-estimate. Consequently,

IXM, 00 ~sup {p; X satisfies an upper p-estimate} ~Px .

In order to eonclude the proof it is therefore enough to show that, whenever P<Px, then p ~IXM, 00' If P<Px then there is a eonstant K so that IIDsll ~Ksl/p for s> 1. Since DsX[o,u/Sj= X[O,uj we deduee that

Observe that 1/lIx[o,uJII=M-1(1/u) and thus we get that

By putting A=M- 1(1/u) and t=Ks1/P, we deduce thatp~IXM,oo sinee

M{At)~S/u=M(A)tP/KP, A> 1, t~K. 0

Remarks. 1. An Orliez spaee LM(O, 1) is contained in Lp(O, 1), for some p, if and only if M{t)~KtP, for some constant K and every t> 1. From the inequality M{t)~KtP we cannot however deduce information concerning the behavior of M{).t)/ M(A). It is easy to construct Orlicz funetions M so that, say X = LM(O, 1) c LiO, 1), but Px= 1. This justifies remark 2 following 2.b.3:

2. Analogous results hold for function spaees LM(O, 00) and for sequenee spaces [M. In partieular, for every Orlicz space X, we have that

p x = sup {p; X satisfies an upper p-estima te }

and

q x = inf {q; X satisfies a lower q-estimate} .

In the ease of an Orlicz sequence space X = IM' the indices Px and qx turn out to be the numbers IXM and ßM' respectively, which were introduced in 1.4.a.9. These numbers were characterized there by the fact that Ir is isomorphie to a subspace of IM if and only if rE [IXM' PM]. An inspection of the proof of this fact, as given in I.4.a, yields the following additional information. If rE [IXM , PM] then, for every e>O and integer n, there exist disjointly supported vectors {X;}7=1 in IM' all having the same distribution (i.e. they form a finite block basic sequenee ofthe unit vector basis in IM which is generated by one vector in the terminology


of l.a.3.8), so that

( n )1/' 11 n 11 ( n )1/' (1-e) i~1 laJ ::::; i~1 aixi ::::;(1 +e) i~1 la;/' ,

for every choice of scalars {a;}?= 1· Of course, if r ~ [aM' PM] = [Px, qx] then we cannot find such {Xi}? = 1 for every e and n. Exactly the same result holds also for X =LM(O, 1). For every rE [Px, qx], e>O and integer n, there exist n disjointly supported functions {xJt)}?=1 in LM(O, 1), all having the same distribution function, so that the preceding inequalities hold for every choice of scalars {a;}? = 1. The proof is very similar to that of 1.4.a.9 and thus we do not reproduce it here.

It turns out that a similar result holds for a general r.i. space X. There exist in X nice copies of I; for so me (but in general not all) r in [Px, qx]. More precisely, we have the following result.

Theorem 2.b.6. Let X be an r.i. spaee. Then Px' respeetively qx' is the minimum, respeetively the maximum, oJ all the numbers p whieh have the Jollowing property. For every e > ° and every integer n, X eontains n disjointly supported Junetions {J;}? = l' having all the same distribution Junetion, so that

Jor every ehoiee oJ sealars {a;}? = 1 .

This theorem is an easy consequence of an important result of Krivine [67] (cf. also Rosenthai [116]) wh ich will be presented (and proved) in Vol. III. We shall prove here only the following weaker version of 2.b.6 whose proof is much simpler.

Proposition 2.b.7. Let X be an r.i. spaee. Then (i) qx< 00 if and only if X does not eontain, Jor all integers n, almost isometrie

eopies oJ I;;, spanned by disjoint Junetions having the same distribution Junetion.

(ii) 1 <Px if and only if X does not eontain,jor.all integers n, almost isometrie eopies oJ I~ spanned by disjoint Junetions having the same distribution funetion.

Proof. (i) If qx < 00 then, as we have already no ted above, for each q > qx there is an integer K so that, for every choice of disjointly supported {J;}?= l' all having

norm one and the same distribution function, we have lIit/;II~K-1n1/q. Thus,

X does not contain uniformly isomorphie copies of 1':., spanned by disjointly


supported functions with the same distribution function. The proof of the converse assertion is identical to the proof of l.f.12(ii) if we require throughout that proof that the functions {Xi }7= 1 have the same distribution function. Assertion (ii) follows from (i) by duality. 0

We turn now to the interpolation theorem which motivated the definition of the Boyd indices and which will play an important role in the sequel. We have first to define the notion of an operator of weak type (p, q). The perhaps most natural way to introduce this notion is by using the Lp,q spaces.

Definition 2.b.S. Let (Q, 1:, v) be a measure space. For 1 ~p< 00 and 1 ~q< 00, Lp , q(Q, 1:, v) is the space of alliocally integrable real valued functions J on Q for which

00

IIJllp,q= [(q/p) J (t1/PJ*(t))q dt/tr 1q < 00 . o

For 1 ~p ~ 00, L p , oo(Q, 1:, v) is the space of all functionsJ as above so that

Note that, for p=q, Lp,q coincides with Lp (with the same norm). If 1 ~qp, it is easily seen that 1IIIp,q does not satisfy the triangle inequality and thus, it is not really a norm (the function W(t) written above is increasing rather than decreasing when q > p). Nevertheless, Lp,q is a linear space also for q>p. It can be shown that it can be made into a Banach space if p> I by introducing an actual norm 111 Illp,q which satisfies IIJIIp,q~ IIIJIIIp,q~ C(p, q)IIJllp,q' We do not give these details since our interest in Lp,q spaces lies not in their structure but only in the quantities 11 IIp,q which arise naturally in interpolation theory. Note also that we have not defined L

00, q' 00

for q< 00, since J J*(t)q dt/t< 00 impliesJ=O. o

We shall be interested in the sequel in inclusion relations between the spaces Lp,q'

Proposition 2.b.9 [82], [54]. Let I~p<oo and I~ql <q2~00. Then

and moreover,Jor every J E Lp,q/Q, 1:, v),


Proof If q2 = 00 the result follows from

t I/Pf*(t) =f * (t{(q t!p) I u'l,/p-I duy/q,

~ ((qdP) I (u 1/Pf*(u»q, du/u y/q" 1>0.

Assume now that q2 < 00. It clearly suffices to prove that IlfllM2 ~ II filM' for n

simple and decreasing functions. Assume thatf= L akX[tk_"tk)' with a1 >a2 ••• > k= 1

an>O and 10 =0<11 <'''<In, and put y=q1/qz{< 1), bk = aZ2 and Sk=tZ2IP,

k= 1, ... , n. Then the inequality, we want to establish, gets the form

This inequality is proved by induction on n. Indeed, the function

n-l qJ(x)= L bk(Sk- Sk_l)+X(Sn- Sn_l)

k=1

(n-l )~

- "b1(S1_ S1 )+X1(S1_ S1 ) ~ k k k-l n n-l

k= 1

is convex (Le. qJ"(X)~O) on [0,00) and thus, by assuming that qJ(O)~O and qJ(bn-l)~ 0, we get that qJ(bn)~ O. 0

There are also inclusion relations between the Lp,q spaces as P varies. These inclusion relations are proved by simply applying Hölder's inequality. If (Q, l.:, v)

is a probability space then, for every r<p<s and every q,

while, for a general measure space (0, l.:, v), we have

In other words the inclusion relation between the spaces {Lfi,qf(O, E, V)}~=1 with PI <P2<P3 are the same as those between {Lpi(O,E, V)}i=1 regardless of qi' i = 1, 2, 3. The index P is thus the "main" index in 11·llp, q while the index q is used for a finer estimate of the size of a function once P is given.

Definition 2.b.l0. Let (Oi' Ei' v), i= 1,2, be two measure spaces. Let 1 ~Pl ~oo and let Tbe a map defined on a subset of Lp ,(OI) which takes values in the space of al1 measurable functions on °2 ,


(i) The map Tis said to be of strong type (Pl,P2)' for some 1 ~P2 ~ 00, ifthere is a constant M so that

for every f in the domain of definition of T. (ii) The map Tis said to be ofweak type (Pl,P2)' for some 1 ~P2 ~ 00, ifthere

is a constant M so that

for every f in the domain of definition of T with the convention that if P 1 = 00 we have to replace Ilflloo.l above, which is not defined, by Ilflloo. 00 = Ilflloo·

It follows immediately from the definitions involved that a map T is of weak type (Pl' P2) if and only if there is a constant M so that

00

sup tv2({w E Q2; ITf(w)l~t})1/P2~MP11 J tl/PI-lf·(t) dt 1>0 0

(if either P2 or Pl are 00 the expression on the right-hand side, respectively lefthand side, has to be replaced by IITflloo' respectively Ilfll oo)' Note that, by 2.b.9, every map of strong type (Pl,P2) is also of weak type (Pl,P2)' with the same constant M. The d~finition of weak type, as presented in 2.b.1O, turns out to enter naturally in the interpolation theorem proved below and is also of importance in some applications in harmonie analysis. It differs however from the c1assical notion of weak type which goes back to Marcinkiewicz. The original definition was: T is of Marcinkiewicz weak type (p l' P2) if

for some constant M and every fin the domain of Tor, alternatively, if

sup tV2({WE Q2; ITf(w)l~t})1/P2~Mllfllp, 1>0

(where the left-hand term is replaced by IITflloo in case P2 = (0). It is c1ear from 2.b.9 that Ilfllp, =llfll p, .p, ~llfllp'.l' for every fE L pl • l . Hence, every map, which is of Marcinkiewicz weak type (Pl,P2)' is also of weak type (Pl,P2) and thus the interpolation theorem below applies, in particular, to operators which are ofthe suitable Marcinkiewicz weak types. There are many important operators appearing in various parts of analysis which are of weak type but not of strong type. Here we just mention a trivial example. If qJ:F 0 is a continuous linear functional on Lp(Q) , 1 ~P < 00 then the operator Tf(t) = r l/PqJ(f) from Lp(Q) into the space of all measurable functions on [0, (0) is of Marcinkiewicz weak type (P,p) but not of strong type (P,p). In the proof of 2.b.13 below, we shall


use simple examples of operators which are of weak type (P,p) but not of Marcinkiewicz weak type (P,p).

We are now ready to state the interpolation theorem ofBoyd (cf. [16]).

Theorem 2.b.H. Let 1 be either [0, 1] or [1, (0), let 1 ~p < q~ 00 and let T be a linear operator mapping L p, 1 (I) + Lq, 1 (I) into the space oJ measurable Junctions on l. Assume that T is oJweak types (P,p) and (q, q) (with respect to the Lebesgue measure on I). Then,Jor every r.i.Junction space X on I, so that P<Px and qx<q, T maps X into itself and is bounded on X.

Note that, by 2.b.3 and the observation preceding 2.b.lO, X cLp, 1 (/)+ Lq , 1 (I) i.e. T is already defined on all of X. The main step in the proof of 2. b.ll is the following lemma due to Calderon [22].

Lemma 2.b.12. With the the same assumptions on T as in 2.b.II there is a constant M< 00 so that

(TJ)*(2t)~MGJ*(tU)U<1-P)/P du+ I J*(tU),j.l- q)/q dU),

Jor every O<t<oo if 1=[0, (0) (respectively, 0<t~I/2 if 1=[0,1]) and every JE Lp,l(/)+Lq,l(/).

Proof Supposethat Tisofweaktypes(p,p)and(q, q) with theconstantsappearing in 2.b.1O being M p' respectively, M q. LetJE Lp,l(l)+Lq,l(l) and, foru, tEl, set

and

{

J(U) - J*(t) ifJ(u) > J*(t)

gl(u)= J(u)+J*(t) ifJ(u) < -J*(t)

° if IJ(u)1 ~*(t)

The function hl is the "flat" part of J (relatively to the level J*(t» while gl is the "peaked" part off We apply the fact that T is of weak type (p,p) to gl and of weak type (q, q) to hl • Note that g:(u) vanishes outside the interval [0, t] while, for ° < u < t, we have g:(u) ~J*(u). Hence, for tEl,

00 1

tl/P(T91)*(t)~Mpp-l J g:(S)S<l- p)/p ds~Mpp-l J J*(S)S<l- p)jp ds o 0

1

=M pp-l t l/P J J*(tU),j.l- p)/p du. o


Observe also that, for every u E I, we have Ihr(u)l=min (lf(u)l, f*(t)). Hence, for tEl,

00

tl/q(Thr)*(t)~Mqq-l f h:(S)s(l-ql/q ds o

~ Mqq- {I f*(t)S(l- ql/q ds + I h:(s)s(1-ql/q ds )

=Mqq-l (qt 1/1lj*(t) + t 1/q f h:(tU)U(l- Ql/q dU)

~Mqq-ltl/Q(qp-l f f*(tU)U(l- Pl/P du+ j f*(tU)U(l- Ql/Q dU) o 1 .

Since ITfl ~ ITgt I + ITht I it follows that

(Tf)*(2t) ~(Tgr)*(t) + (Thr)*(t)

1 ~(Mpp-l +MQp-l) f f*(tU)U<l- Pl/P du

o 00

+MQq-l f f*(tU)U(l- QlQ du. 1

This proves our assertion with M=p-l(Mp+MQ). 0

Proofof2.b.ll. Choosepo and qo so thatp<po<Px and qx<qo<q. Then there is a constant K so that

Let g E X' with Ilgllx' = 1. We have

00 1 1 (00 ) f f f*(tu/2)g(t)u(1- pl/p du dt= f U<1- pl/p f (D2/J*)(t)g(t) dt du o 0 0 0

and similarly,

00 00

1 ~ Ilfllx21/POK f Ul/p-l/po-l du

o

f f f*(tu/2)g(t)U<1- Ql/Q du dt~ Ilfllx21/QOK(I/qo _1/q)-l . o 1

(lf 1=[0, IJ we take in the formula above g(t)=O for t> 1.)


Hence, by 2. b.I2, we get that

00

J (TJ)*(t)g(t) dt~MollJllx , o

for every g E X' with Ilgllx' = 1, where M o=KM(21/p°(1/p-I/Po)-1 + 21/q°(1/qo -l/q) -1). If Xis a maximal r.i. function space this already proves that TJ E X and that IITJllx~MoIIJllx' as desired.

If X is a minimal r.i. function space we get, by what has already been proved, that Tmaps X" into itself and is bounded there and also that Tmaps Lpo(I)nLqo(I) into itself. Since Xis the closure of Lpo(I)nLqo(I) in X" it follows that T also maps X into itself. 0

Remarks. 1. If X is a maximal r.i. space then the proof above works also when T is only a quasilinear operator. The only change needed in the proof is to replace the M of 2.b.12 by the constant CM, where Cis the constant appearing in the definition of quasilinearity. In the case of a minimal r.i. space the proof given above does not work for a quasilinear operator. A bounded quasilinear operator need not be continuous. Thus, the fact that it maps Lpo(I)nLqo(I) into itself does not ensure immediately that the same is true for its closure in X".

2. Suppose that T is a linear operator defined on (a non-closed) linear subspace Yof Lp(I) + LiI). Assume that Tis ofweak types (p,p) and (q, q) and that JE Yimplies max (I,j) E Y. Then the proof of2.b.ll shows that, whenever Xis a maximal r.i. function space on I, T maps X nY into X and is bounded on X n Y. If, in addition, LiI)nLiI)nYis dense in XnY(in the norm induced by X) the same is true when X is a minimal r.i. function space on I. The operator T can then, of course, be extended in a unique way to a bounded linear operator from the closure of X n Y in X into X. A typical example where this remark is used (and in which the two assumptions made on Y are satisfied) is the case where Y is the space of all simple integrable functions or, altematively, the space of all finite linear combinations of characteristic functions of dyadic intervals.

We shall prove now a converse to 2.b.II.

Proposition 2.b.13 [I5J, [16]. Let X be an r.i.function space on I (where 1=[0, IJ or [0, (0». Assume that, Jor some 1 ~p<q< 00, every linear operator defined on L p,l (I) + L q, 1 (I) which is oJ weak types (p, p) and (q, q) maps X into itself. Then P<Px and qx<q·

The main point in 2.b.l3 is the assertion that we have strict inequalities (not only P~Px and qx~q)·

Proof We shall prove only the assertion concemingpx' The assertion conceming qx can be proved in an entirely similar way, by using the formal adjoint of the operators Ty appearing below.

Consider the operators

t 1

(Ty/)(t)=r Y J sY- 1J(s)ds= J sY- 1J(st)ds, O<y~l, tEl. o 0


We claim first that Ty is ofweak type (r, r) for every r~y-1. Indeed, forfE Lr• 1 (I) and r~y-l, we have

We used here the fact (proved e.g. by differentiation) that (TJ*)* = TJ* and Tylfl~ Tyf*. Note further that, for y >e >0 and for every simple integrablefon I,

1 1

TyTy_J(u)=J ty- 1 J sy-e- 1f(stu)dsdt o 0

1 1 = J vy- 1 J s-e- 1f(vu) ds dv

o v

=e- 1(1 vy- e- 1f(vu) dv- t vY - 1f(vu) dV)

= e- 1((Ty_J)(u) - (Tyf)(u))

and hence, TJ=Ty_J-eTyTy_J. It follows from our assumption that T1/p is a bounded operator on X. If

e< IIT1/p ll- 1 then (Ix- eT1/p)-1 is also a bounded operator on X. Fix now an e with O<e<min (p-l, IIT1/p ll- 1) and put 1J=I/p-e. Iffis a simple integrable function such that T~f E X then we get from what we have shown above that Tl/pf=(Ix-eTl/p)T~fand thus

T~f=(Ix-eTl/p)-lTl/pf .

In particular,

It should be noted that in the preceding step it is essential to know a-priori that T~f E X. In case 1=[0, 1] it is clear that if fis a simple function then T~f is a bounded function and hence T"fE X. Thus the estimate above für IIT"fllx applies für every simple f Since, for a nün-increasing simple positivef and O~s, t~ I,

• 1

s~(Dl/.f)(t)/f/ = fest) J U,,-l du~ S U~-l f(ut) du= (TJ)(t) o 0

it follows that IID1/.11 ~f/C(l/s)~ and hence l/Px~f/= l/p-e i.e. P<Px. In case 1= [0, (0) it is nüt a-priori clear that T~f E X. We üvercüme this

difficulty by considering the operators (T;f)(t) = XIO. alTJ)(t) for O<a< 00,

O<y~1. Note that for simple integrablej, T;,T;J=T;,TyJand that T;fEX für every a and y. Hence, with the same e and f/ as above,


and thus, for a eonstant C independent of a and J, IIT:fllx~ Cllfllx. By Ietting a - 00 wededueethat T"fE X"forevery simple integrablefand IIT"fllx"~ CllfHx· The proof is now eoncluded as in the ease 1= [0, 1]. 0 . ,

Remark. The assumption made in 2.b.l3 that X is an r.i. funetion spaee on I is neeessary to the extent that the main assumption in 2.b.13 already implies that, for every automorphism r or I, the operator UJ(t)=f(r-l(t)) maps Xinto itself. Moreover, it follows from the main assumption on X in 2. b.l3 that fEX and 9 -<fimply that 9 E X (see remark 2 following 2.a.Il).

Theorem 2.b.ll reduees for X=L r to a special ease of the two classical interpolations theorems of Riesz-Thorin and Marcinkiewiez. These two fundamental theorems inspired all the subsequent development of interpolation theory. The Riesz-Thorin theorem, and often also the Marcinkiewiez theorem, are proved in most textbooks on funetional analysis and harmonie analysis. We shallapply them in the following sections only in the special ease eontained in 2.b.ll and therefore we shall not reproduee here their proofs. We find it however appropriate to state these theorems here in their general form. A very detailed diseussion of these theorems and other material on interpolation in Lp spaees and related spaees ean be found in the book [10]. We refer to this book also for a diseussion of the history of the various variants of these theorems. (In Seetion g below we outline the general Lions-Peetre interpolation theory from whieh, in partieular, 2.b.I5 follows.)

Theorem 2.b.14 (The Riesz-Thorin interpolation theorem). Let (Oi' 1:i, J1.;), i = 1, 2, be two measure spaces, let PI#- q land p 2 #- q 2 be numbers in [1, 00] (00 is included). Let T be a linear operator mapping L p1(J1.I) + L q1 (J1.I) into L p2(J1.2) + L q2(J1.2)' which is of strong types (PI,P2) and (ql' q2). Then,for every 0<0< 1, T is of strong type (r I' r 2)' where

10(1-0) -=-+--, i=I,2. ri Pi qi

Moreover,

The symbolllTllrh r2 means ofeourse sup {IITft; Ilfllrl~I}. The preceding inequality eoneeming the norms holds ifwe use eomplex scalars. For real scalars the factor 2 has to be added to the right-hand side of the inequality.

Theorem 2.b.15 (The Marcinkiewicz interpolation theorem). Let (Oi' 1:i , J1.i)' i= 1, 2, be two measure spaces and let PI #-ql andp2 #-q2 be numbers in [1,00] (00 is included). Let T be a quasilinear operator mapping L p1 (J1.I) + L ql (J1.I) into the space of measurable functions on 02' which is of weak types (PI,P2) and (ql' q2). Then,


Jor every 0< (J < 1, T is oJ strong type (r l' r 2) where

1 (J (1 - (J) -=-+--, i=I,2, r i Pi Pi

provided that r 1 ~ r 2.

c. The Haar and the Trigonometric Systems

The present seetion is devoted mostly to the study of the uneonditionality of the Haar system in r.i. funetion spaees on [0, 1]. We treat this matter in the more general setting ofmartingales and present a eomplete eharaeterization ofthose r.i. funetion spaees on [0, 1] for which the Haar basis is uneonditional. Next, we introduee the notion of reproducibility for bases and show that the Haar basis has this property in every separable r.i. funetion spaee. We eonclude by presenting some applieations of reproducibility and also by diseussing briefly the trigonometrie system.

The Haar system {Xn}:': 1 was introdueed in I.l.a.4. For eonvenienee, we recall here that Xl(t) = 1 and, for 1= 1,2, ... , 2k and k=O, 1, ... ,

{I if tE [(2/-2)2- k - 1 , (2/-1)2- k - 1)

X2 k +/(t)= -1 if tE[(2/-1)r k -1,2/·2- k - 1)

° otherwise.

The vectors {Xn}:'= 1> as defined above, are normalized in Loo(O, 1) and, unless stated otherwise, we shall always use this normalization.

The Haar system ean be associated in a natural way with an inereasing sequenee of Ci-algebras {.91 n}:': 1 of measurable subsets of [0, 1]. The Ci-algebra .911 eonsists only of the sets 121 and [0, 1] and if n = 2k + I, for some 1 ~ 1 ~ 2k and k ~ 0, then .91 n is defined to be the Ci-algebra generated by .9In- 1 and the two intervals [(2/- 2)2- k - 1, (2/-I)r k - 1) and [(2/-1)2 -k-l, 2/· 2 -k-l). It is evident that .91 n is, for every n, the smallest Ci-algebra .91 for whieh the funetions {Xl' ... , Xn} are .9I-measurable.

We have seen in I.l.a that the Haar system forms a monotone basis of every LiO, I) spaee, 1 <P< 00. This faet is true in every separable r.i. function space on [0, I].

Proposition 2.c.l [36]. The Haar system is a monotone basis oJ every separable r.i. Junction space on [0, I].

Proof Let X be a separable r.i. funetion spaee on [0, I]. Sinee X is not isomorphie to Loo(O, 1), we have that lim Ilx[o"111=0 and thus, every simple funetion on [0, I]

1-+0

ean be approximated, in the norm of X, by step funetions over the dyadie intervals

e. The Haar and the Trigonometrie Systems 151

[I. 2- k , (I + 1)2 -k), 0::;;; I ::;;;2k -1, k= 0, 1, .... It follows that the linear span of the step functions over the dyadic intervals or, equivalently, of the Haar functions is dense in X and, therefore, it remains to show that {xn}:: 1 forms a monotone basic sequence in X. This fact can be proved e.g. by using the observation that, for every n < m and every choice of scalars {ai}7'= l' we have

and by the assertion of 2.a.4 that the conditional expectations E""n act as norm one operators in X. Actually, it is easier to check directly that {Xn}::'= 1 is a basic sequence.

n n+1 Put 1= L aiXi and g = L aiXi and notice that land g coincide in [0, I J with the

i=1 i=1

exception of some dyadic interval1] on which/is a constant, say it takes the value b there, and 9 is equal to b + an + 1 on the first half of 1] and to b - an + 1 on the second half of 1]. Let"C be an automorphism of [0, I] which permutes the first half of 1] with its second half and leaves invariant every point outside 1]. Then it is easily seen that

I(t) = (g(t) + g("C(t»)/2,

for every t E [0, IJ, and thus 11/11::;;; Ilgll. 0

In order to characterize those r.i. function spaces on [0, IJ for which the Haar basis is unconditional we work in the framework of martingale theory. We first recall the definition of a martingale. Let (0, E, v) be a probability space and let

be an increasing sequence of u-subalgebras of E. A sequence {/n}:'= 1 ofintegrable random variables over (0, E, v) is said to be a martingale with respect to {gBn}:'= 1 if

gM"/,. + 1 = /" ,

for all n. Also a finite sequence {/,,}!= 1 satisfying gMn/,,+ 1 = /" for I::;;; n < k will be called a martingale. (This finite sequence can be identified with an infinite martingale which is constant for n~k.)

It follows immediately from the definition of a martingale {/,,}::,= 1 that /" is &iJn-measurable, for every n, and that

for I ::;;;j::;;;n. In particular, we get that any subsequence {/"J[';1 of {/"}::'=1 is a martingale with respect to {gBnJ~1'

The connection between the study of the Haar system and the theory of martingales is quite obvious. If X is a separable r.i. function space on [0, 1] and


{sln }:'= 1 is, as above, the (increasing) sequence of a-algebras associated to the 00

Haar basis {Xn} ~ 1 of X then, for every J= L aiXi E X, the sequence i= 1

n

!,.=E"',,/= LaiXi' n= 1,2, ... i= 1

defines dearly a martingale with respect to {sln} ~ 1 .

We begin the study of martingales by proving the following inequality (cf. [29]):

Proposition 2.c.2. Let {f..}~ = 1 be a (finite) martingale with respect to a sequence {~n}~= 1 oJ a-subalgebras oJ some probability space (Q, 1:, v). Then,Jor every t> 0, we have

a,

whereat={wEQ; max 1f..(w)l>t}. l~n~k

Proof Fix t > 0 and define a function T on Q with values in the set of integers {I, 2, ... , k, k + I} by putting, for w E Q,

_ {min {j; 1~(w)1 > t} if w E a t

T(w)- k+ I if w ~ at .

Let 1 O::;;jO::;; k and observe that

is precisely the set where IHw)1 > t and max I Jn(w) I 0::;; t. This implies that Q. E ~. l~n<j J ]

for 1 O::;;jO::;;k. Furthermore, since~=F!BJjk and since the conditional expectation is a positive operator, it follows that 1~lo::;;F!BjIJkl, 100000jO::;;k. Using these facts we get that

k

tv(at)=tv({WEQ; T(w)O::;;k})=t L v(Q) j= 1

k k

0::;; L J IHw)1 dvO::;; L J (EBIlthl)(w) dv j=lQj j=lQj

k

= L J Ih(w)1 dv= J IJiw)1 dv. 0 a,

Remarks. 1. The function T introduced in the proof above is a special instance of a fundamental notion used in probability theory, namely that of a stopping time.

c. The Haar and the Trigonometric Systems 153

Specifically, if {,qßj}~ = 1 (with k finite or 00) is an increasing sequence of u-subalgebras of 1: then a function T: Q - {I, 2, ... , k} is called a stopping time with respect to {,qßj}~=l if {w; T(w)=j} E ,qßj' for everyj.

2. The proof of 2.c.2 shows also that if {gBn} ~ 1 is an increasing sequence of u-subalgebras of 1: then, for every fE L1(Q, 1:, v),

i.e. the quasilinear operator Uf= sup IEat'il is ofweak type (1,1). It is evident that n

U is of strong type ( 00, 00). Hence, by the Marcinkiewicz interpolation theorem, U is of strong type (p, p) for every p > 1. This is essentially the assertion of 2.c.3 below which is called the maximal inequality of Doob (cf. [29]). We prefer to prove it direcdy since the proofis simple and gives also information on Iluilp'

We recall that, for every 1 <p< 00 and every gE LiQ , 1:, v), we have

00

(*) S Ig(w)jP dv= S ptp-1v({w E Q; Ig(t)1 > t}) dt. Q 0

This identity can be easily checked for simple functions while for a general g is deduced by approximation with suitable simple functions.

Proposition 2.c.3. Let {J,.}:'=l be a martingale with respect to a sequence {,qßn}:'=l of u-subalgebras of some probability space (Q, 1:, v). Then, for every 1 <p < 00,

where ".//p denotes the norm in Lp(Q, 1:, v) and q is the conjugate index of p i.e. l/p+ l/q= 1.

Proof Observe first that it suffices to prove the maximal inequality of Doob only for finite martingales since the general case follows from it by Fatou's lemma. Fix k and put

f= max Ifnl and ut={wEQ;f(w»t}, t>O. l~n~k

By (*) and 2.c.2, we get that

00 00

IIfll~= S ptp-1v(ut) dt~ S ptp- 2 S Ifk(W)1 dv(w) dt o 0 ~

00 f(w)

= S I f,,(w) I S ptP- 2 X",(w) dt dv(w) = SIf,,(w)1 S ptp- 2 dt dv(w) Q 0 Q 0

=q S Ifk(W)lfiw)p-l dv(w). Q


Hence, by Hölder's inequality, it follows that

We are prepared now to state a lemma which is crucial in the proof of the unconditionality ofthe Haar basis in LiO, 1), 1 <p< 00.

Lemma 2.c.4. For every p of the form 2m , m = 1, 2, '" there exists a constant Cp < 00 such that if C[,,}!= 1 is afinite martingale with respect to a sequence {Bln}:= i of u-subalgebras of some probability space (D, E, v) then

where S is the square function of the differences of {!,,}!= 1 i.e.

Proof Putfo=OandLl.fj=.fj-.fj_l> I~j~k,sothat

For p = 2 (i.e. m = 1) the assertion is true with C2 = 1. Indeed, if {!,,}:= 1 is a martingale we get, for any 1 ~j < n ~ k, that

J LI.fj LI!" dv = J Jtl8j (LI.fj LI!,,) dv = J LI.fj Jtl8j (LI!,,) dv = ° , u u u

and the orthogonality in L 2 (D, E, v) of {LI!,,}! = 1 clearly implies that

Fora generalp ofthe form 2m , m= 1,2, ... the proofis done byinduction on m. To this end, suppose that the assertion of 2.cA is valid for some p and for every finite martingale. Let {!,,}!= 1 be a finite martingale and observe that

m<n k n-1 k

=S2+2 L LI!" L Llfm=S2+2 L !,,-1 LI!" 11=2 m=1 n=2

C. The Haar and the Trigonometrie Systems 155


Notice now that the sequence {g)~=1 defined by g1 =0 and

j

gj= L fn-1 iJf., j=2, ... , k, n=2

is also a martingale with respect to the same sequence of a-subalgebras as that corresponding to {fn}~= l' Thus, by applying the induction hypothesis to this martingale, we get that

Put f= max Ifn!- Then, by using the Cauchy-Schwarz inequality and Doob's l~n~k

maximal inequality 2.c.3, it follows that

IIJ2 fn-1 iJf.llp ~ CpllfSllp~ CpllsII2pllfll2p

~ Cp(2p)(2p-1)-1I1 SlbllfkIl2p'

By combining this inequality with a preceding one, we obtain

IISII~p-2Ci2p)(2p-1)-1I1SI12pllfkIl2p~ 1IJ,.II~p~

IISII~p+2Ci2p)(2p-1)-1I1SI12pllfkIl2P .

Set a.=2Ci2p)(2p- 1)-1 and A=lIsII2plllfkIl2p' Then the above inequalities become

which easily yields that

This proves that the assertion of 2.cA is true for 2p with

Theorem 2.c.5. The Haar system {Xn}:'= 1 is an unconditional basis of LiO, 1),for every 1 <p< 00.


Proof First, we ass urne that pis ofthe form 2m , m = 1, 2, .... Then, for every choice of scalars {an}~= 1 and every choice of signs ()n = ± 1, 1 ~ n ~ k, the functions

j j

~= L anXn, 1 ~.i~k and gj= L an()nXn' 1 ~.i~k form martingales with respect n=1 n=1

to {si n}~= l' Since both these martingales have clearly the same square function

( k )1/2 n~1 lanXnl 2 it follows from 2.c.4 that

This, of course, proves the unconditionality of {xn}::'= 1 . Since the unconditionality of {Xn}::'= 1 is equivalent to the uniform boundedness of the natural projections associated to {xn}::'= 1 it follows from the Riesz-Thorin interpolation theorem 2. b.l4 that the Haar system is an unconditional basis of LiO, 1), for every p?; 2. For 1 <p< 2 the proof is achieved by duality since the Haar system, normalized in L p (O,1), 1 <p<2, is clearly biorthogonal to the Haar system normalized in L/O, 1), where l/p+ l/q= 1. 0

The remarkable result 2.c.5 was originally proved by Paley [108]. Later on, proofs in the more general setting of martingales were given by Burkholder and Gundy [20], Burkholder [19], Garsia [46] and others. It follows from these proofs that the unconditional constant of the Haar basis in Lp(O, 1), 1 <p < 00 is ~Kmax (p, l/(p-l)), for some universal constant K. It can be shown that this is the right order of magnitude. The idea of the proof presented here is taken from Cotlar [23]. It is easily seen that this proof gives (for p?; 2) C p ~ Kp but for the unconditional constant of the Haar basis we get only the estimate ~ K 2 p2.

In some of the proofs of 2.c.5 it is shown first that, for every choice of signs

{();};':1' the operator TeC~1 aixi) = i~1 a;BiXi is ofweak type (1,1). The proofis

then completed by using the Marcinkiewicz interpolation theorem and duality since obviously Te is of strong type (2,2). The operator Te is not of strong type (1, 1) for every choice of signs {();}?: 1> i.e. the Haar basis is not unconditional in L 1(0, 1). This follows from I.1.d.l but can also be easily verified directly. Indeed, for every integer k,

while

Theorem 2.c.5 can be used in conjunction with the interpolation theorem 2. b.ll in order to give a characterization of the separable r.i. function spaces on [0, 1] for which the Haar system is unconditional.

Theorem 2.c.6. The Haar system {Xn}~ 1 is an unconditional basis in a separable r.i. funetion spaee X on [0, 1] if and only if X does not eontain uniformly isomorphie


copies 0/ l~ or 0/ l~ ,for all n, on dis joint vectors with the same distribution /unction or, equivalently, if and only if 1 <Px and qx < 00.

Proo! Suppose that X does not contain l~'s and l';,'s as in the statement. Then, by 2.b.7, its Boyd indicespx and qx satisfy 1 <Px and qx< 00. Hence, by using 2.b.1I, for any 1 <P<Px and qx<q<oo, and 2.c.5 (in Lp(O, 1) and in LiO, 1», it follows that the Haar basis is unconditional in X.

Conversely, suppose e.g. that X contains uniformly isomorphie copies of l~ , for all n, on mutually disjoint vectors having the same distribution function. In other words, there exists a constant M< 00 with the property that, for every n, there exist 2" mutually disjoint positive functions {Ui}t~ 1 all having the same distribution function such that Iluill = 1 for 1 ~ i~ 2" and

Consider now the "Haar system {hk}f:1 over the vectors {Ui};~ 1'" which is defined by

h1 =(u1 +"'+u2")/2"

h2 = (u l + ... + u2"-1 -u2"-I+ 1 -'" - u2")/2"

We may assume without loss of generality that each ui is a finite linear combination of characteristic functions of intervals of the form [(lj-l)2- k, IjT k) for some fixed kindependent of i. Hence, by applying a suitable automorphism to [0, IJ, we may assume that on the first 2" dyadic intervals oflength 2- k each Ui is non-zero on exactly one of these intervals and takes there a value independent of i (say ßI)' that the same is true on the next 2" dyadic intervals oflength T k (with ß1 replaced by ß2) and so on. In other words, we may assume that, for some integer m and scalars {ßj}j= I , we have

m

ui = L ßj X[(i-I+(j-1)2")2- k ,(i+(j-l)2")2- k l' l~i~2". j= 1

Note that m

2"h2 =u I + u2 + "'u2"-1 -U2"-1 + I -'" -u2" = L ßj X2k -"+ 1+ j j= 1

m

2"h3=U1 +U2+"'+U2"-2- U2"-2+1-"'- U2"-I= L ßj X2 k -"+2+2j-1 j= 1

2"h4 =U2"-'+1 +"'+U2"-1+2"-2- U2"-1+2"-2+1-"'- U2" m

= L ßjX2k -"+2+2j j= 1


and so on. In other words, {hl}?~ 2 forms a block basis of a permutation ofthe Haar basis {Xs } ~ 1 of X (observe that necessarily m ~ 2k - n). Hence, the unconditional constant Kn of {hl}?~ 2 does not exceed that of the Haar basis of X. On the other hand, the definition of the {hl}?~l in terms of the {UJ?~l and the fact that the {uJ?~ 1 are M-equivalent to the unit vector basis of Ir shows that, up to a factor M, the unconditionality constant of {h l }f~ 1 is the same as that of the first 2n elements of the Haar basis in LI (0, 1). Consequently, Kn ~ 00 as n ~ 00 and we conclude that the Haar basis of X is not unconditional.

The case when X contains uniformly isomorphie copies of /':.0, for all n, on disjoint vectors having the same distribution function is treated similarly. 0

We introduce next a notion of reproducibility for general bases and study it mainly in connection with the Haar basis. It is easily checked (see e.g. I.l.a.12) that when a Banach space X, with a normalized basis {Xn},~'= 1 which tends weakly to 0, is isomorphie to a subspace of some space Y, with a basis {Yk}k'= l' then there exists a subsequence {XnJj=1 of {xn}~1 which is equivalent to a block basis of {Yk}kX'= 1. We describe this situation by saying that a subsequence of {Xn}~ 1 can be "reproduced" as a block basis of {Yk}f= l' Of particular interest is the case when {Xn}~ 1 itself can be reproduced as a block basis ofany basis {Yk}k'= 1 ofan arbitrary space Y containing an isomorphie copy of X.

More precisely, we have the following definition from [76].

Definition 2.c.7. A Schauder basis {Xn}:= 1 of a Banach space X is said to be K-reproducib/e, for some K~ I, if, for every isometrie embedding of X into aspace Y with a basis {Yk}k'= 1 and every 8>0, there exists a block basis {Zn}:= 1 of {Yk}f= 1

which is K + 8-equivalent to {Xn}:= 1. A basis is said to be reproducible if it is K-reproducible for some K< 00. When K = 1 the basis is said to be precisely reproducible.

We recall that two bases {Xn}:=l and {zn}~l are said to be K-equivalent for some constant K< 00, whenever there exist constants K 1 and K2 whose product is equal to K, such that

for every choiee of scalars {an }:= 1 •

It is clear that all subsymmetrie ba ses with subsymmetrie constant equal to one and, in particular, the unit vector bases in Co and in Ip ' 1 ~p ~ 00, are precisely reproducible. While this fact is entirely trivial, the reproducibility of the Haar basis is less obvious (cf. [107] and [76]).

Theorem 2.c.8. The Haar basis 0/ every separab/e r.i. function space X on [0, 1] is precisely reproducible.

The proof of 2.c.8 will be based on a theorem of Liapounoff [73] which has many other applications in functional analysis.


Theorem 2.c.9. Let {JL;}? = 1 be a sequence of finite (not necessarily positive) nonatomic measures on a measure space (Q, 1:'). Then the set

is a compact convex subset of Rn.

Proof[75]. By decomposing each JLi into its positive and negative part we reduce the ca se of general measures to that ofpositive measures (with n replaced by 2n). It suffices therefore to prove 2.c.9 for positive measures. The proof will be done by induction on n. The proof for n = I is the same as that of the induction step so we present only the induction step.

n

LetJL= L JLi' Wbe the sub set {g; O~g~l} of Loo(Q, 1:, JL) and let Tbe the i= 1

map from Loo(JL) to Rn defined by

The set W is w* compact and convex and T is linear and w* continuous (since the {JLi}?= 1 are absolutely continuous with respect to JL). Hence, T( W) is a convex and compact subset of Rn.

It is clear that T( W)::J r( {JL;}?= 1)' In order to complete the proof, it suffices to show that these two sets coincide, i.e. that, for every (al' a2, "', an) E T(W) Wo = T- 1(a 1 , a2, ... , an)n W contains a characteristic function. The set Wo is w* compact and convex; thus it has extreme points, by the Krein-Milman theorem, and it is enough to prove that any extreme point g of Wo must be a characteristic function.

Assurne that gE ext Wo is not a characteristic function. Then there is an 8> ° and a sub set 0"0 in 1: so that JL(O"o) > ° and 8~g(W)~ 1- 8 for W E 0"0' Since JL is non-atomic there exists a 0" 1 C 0"0 so that JL(O" 1»0 and JL(0"2) >0, where 0"2 = 0"0 ~ 0" l'

By the induction hypothesis, there are ''11 cO" 1 and '72 cO" 2 so that

Pick real s, t so that Isl, Itl < 8, S2 + t 2 > ° and

Let h=s(X.u1 -2XI1I )+ t(X.u2 -2XI1J Then J h df.1i=O, for I ~ i~n, and Ihl ~g~ l-Ihl n

on Q. Hence, g±h E Wo and, since h#O, this contradicts the assumption that gEext Wo. 0

Proof of2 .c.8. Assurne that Xis a subspace of a Banach space Y having a basis PI

{Yk}k'= 1 whose basis constant is K. Fix 8> ° and let z 1 = L akYk be a vector in Y so k=l


that Ilzl -hIli <sllh111/23 K, where h 1 is the function identically equalto one on [0,1] (h 1 is an element of X and thus also of Y). Observe next that, for every Y* E Y*, the function v(a} = Y*(xa} defines a finite non-atomic measure on [0, 1] (since Ilx,,11 ~ ° as Jl(a) ~ 0, where Jl denotes the Lebesgue measure}.Hence, by 2.c.9, there exists a a 1 c [0, 1] so that if we put a 2 = [0, 1}- a1 we have

where {ynk: 1 denote the functionals bi orthogonal to {YkHx'= l' Hence, if we put h2 = X"I - X"2 then y:(h2} = 0, 1 ~ k ~Pl . It follows that there exists a P2 > PI and a

P2 vector Z2 E Y so that Z2 = L akYk and IIz2 -h211 <sllh211/24K. By applying

k=PI+l 2.c.9 again, we find disjoint sets a 1, 1 and a 1,2 with a 1, 1 ua 1,2 = a 1 and

Jl(a 1, 1} = J1(a 1, 2} = 1/4,

Hence, y:(h 3 } =0, 1 ~k~P2' where h3 =X"I,I-X"I,2' We continue in an obvious inductive procedure and find a sequence {hn}:,= 1 in X and a block basis {zn} ~ 1 of {Yk}k'= 1 SO that Ilzn -hnll < sllhnll/2n+2 K. Thus, {Zn}~ 1 is (1 +s)(l-S}-1 equiva1ent to {hn}:'= l' The proof is completed by 0 bserving that the {hn}:'= 1 are, by their construction, isometrically equivalent to the Haar basis of X. 0

Note that, in the proof above, it was not essential that e.g. y:(h 1}=0 for I ~k ~Pl' It would have been enough to construct a1 , and thus h1 , so that ly*(h 1 }1 becomes as small as we wish (say < sllh111/23 KPl)' This can be done by using, instead of2.c.9, the following proposition which is also of so me independent interest.

Proposition 2.c.lO. [137] Let X be an r.i.function space on [0, 1]. The Rademacher functions form a weakly null sequence in X if and only if X is not equal to Loo(O, I}, up to an equivalent norm.

Proo! We have already studied in 2.b.4 the case when the Rademacher functions form a sequence which is equiva1ent to the unit vector basis of 12 , For a general r.i. function space X, the Rademacher functions {rn}~ 1 form asymmetrie basic sequence with symmetrie constant equal to one since they clearly are symmetrie and independent random variables with the same distribution.

Suppose now that {r n} ~ 1 does not converge weakly to zero in X. Then, by the unconditionality and symmetricity of {rn }:,= l' we easily conclude that the Rademacher functions in X form a sequence which is equivalent to the unit vector basis

of 11, On the other hand, since I/n~1 rnl/2 =k1/2, k= 1,2, ... we get that, for every

8>0, lim J1(a(k, s»=o, where k-+ 00


But

from which it folIo ws that if X is not equal to L 00(0, 1), up to an equivalent norm,

then IIXa(k,e)llx' and thus also IIC~1 r n)/ kll x' tend to zero as k-+ 00. This con

tradicts the fact that {rn};'; 1 is equivalent to the unit vector basis of 11 •

The converse assertion is trivial since in Loo(O, 1) the sequence {rn}'%'1 is c1early isometrically equivalent to the unit vector basis of 11 , 0

A simple consequence of 2.c.8 is the following.

Corollary 2.c.ll. Let Y be a Banaeh spaee whieh has a basis whose uneonditional eonstant K is finite. Then, for every r.i. funetion spaee X whieh is isomorphie to a subspaee of Y, the Haar systemforms an uneonditional basis. Moreover, Kx~Kd, where Kx is the uneonditional eonstant ofthe Haar basis of X and d= inf {d(X, Z); Z is a subspaee of Y}.

In partieular, we get that an r.i. function space X on [0, 1] has an unconditional basis if and only if Px > 1 and qx< 00. Note also that, since the Haar basis is not unconditional in LI (0, 1) or in L oo (0, 1), the unconditionality constant of the Haar basis in LiO, 1) tends to 00 if p tends to either 1 or 00. Hence, if either

inf {Pn};'; 1 = 1 or sup {Pn};'; 1 = 00 the space (~ EB LpJO, 1)) (which is reflexive n n; 1 2

if 1 <Pn < 00, for every n) is not isomorphie to a subspace of aspace with an unconditional basis.

It turns out that not only the Haar basis of an r.i. function space on [0, 1] is reproducible. In fact, the same is true for every unconditional basis of such a space. This result is due to G. Schechtman (cf. [119] for L p spaces).

Theorem 2.c.12. Let {Xn };'; 1 be an uneonditional basis of an r.i.funetion spaee X on [0, 1]. Then {Xn };'; 1 is reproducible. In partieular, {Xn };'; 1 is equivalent to a bloek basis of the Haar basis in X.

Proof We first observe that, since the interval 1=[0,1] and the square lxI are measure theoretieally equivalent, X is order isometrie to the r.i. function space X(I x I) on the unit square in which the norm IlfllX(! x I) of a measurable function f(s, t) on Ixlis taken to be equal to Ilf*llx'

Suppose now that X(I x I) is a subspace of a Banach space Y which has a basis {yJ ~ 1 with basis constant K. The basis {Xn };'; 1 of Xis, of course, isometrieally

q,

equivalent to the sequence {xn(S)}'%'1 in X(IxI). Fix s>O and let VI = L biYi i; 1

be a vector in Yso that Ilv l -x1(s)ll<sllx111/23K. Then notice that, by 2.c.l0, the


sequence {x2(sh(t)}k"=1' where {rit)}~l are the Rademacher functions on I, tends weakly to zero in X(I x I). Hence, there is an integer k 2 and a vector

q2

v2 = L biYi E Y such that IIv2 - x 2(s)rk2(t)11 < ellx211/24 K. By repeating the argu-i=qj + 1

ment for the sequence {x3(s)rk2 +k(t)}k"=1 and by continuing in this manner, we are able to construct a block basis {Vn}~l of {yJt~'l and an increasing sequence {kn}~l of integers so that k1 =0 and Ilvn-xn(s)rkn(t)II<ellxnll/2n+2K, for all n (where ro(t) = 1). This implies that the sequence {Xn(s)rkn(t)}~ 1 is (1 +e)(1 +e)-1_ equivalent to the block basis {Vn}:= 1 . Since {Xn} ~ 1 is unconditional it is equivalent to {x/s)rkn(t)}:=l. This fact is evident if X=LiO, 1). For a general Xthis is a consequence of l.d.6(ii) and of 2.d.! below. 0

It should be mentioned that if 2.c.I2 is applied to the Haar basis {Xn }:=l of an r.i. function space in which this basis is unconditional we still do not obtain the precise reproducibility of {Xn}:= l' as given by 2.c.8. Note also that the proof of 2.c.12 shows that the last assertion in its statement is valid also if {Xn}:= 1 is an unconditional basic sequence, provided that Xis q-concave for some q< 00. However, an unconditional basic sequence in X need not be reproducible even when X is q-concave for some q< 00. For instance, it follows from the discussion preceding 1.2. b.1 ° that, for each p ~ 1, the space Lp(O, 1) contains an unconditional basic

sequence which is equivalent to the unit vector basis of C~1 EB I;) p. Obviously,

this basic sequence is not reproducible. While unconditional bases in an r.i. function space as above are reproducible

this is not the case for conditional bases. We conclude the study ofthe reproducibility with the following proposition, whose proof is evident.

Proposition 2.c.13. Any eonditional basie sequenee in a Banaeh spaee with an uneonditional basis is not reproducible.

A result ofPelczynski and Singer [112] (which we partially proved at the end of section 1.2.b) states that every infinite dimensional Banach space with a basis has a conditional basis and, actually, even uncountably many mutually non-equivalent conditional bases. Hence, by the observation 2.c.13, every space with an unconditional basis has uncountably many mutually non-equivalent non-reproducible bases. In Vol. IV we shall present a result from [76] which asserts that C(O, I) has the remarkable property that all its Schauder ba ses are reproducible.

If X is a separable r.i. function space on [0, 1] with 1 <Px and qx< 00 it follows from 2.c.6 and 1.3. b.I that Xis isomorphic to a complemented subspace of aspace Y with a symmetrie basis. Moreover, if X is uniformly eonvex then it follows from 1.3.b.2 that Y may be chosen to be, in addition, also uniformly convex. It will be of interest for us in the sequel to know that Yeannot be taken to be a separable Orliez sequenee spaee, unless X=L 2(0, 1).

Theorem 2.c.14 [78]. A separable r.i.Junetion spaee X on [0, 1] is isomorphie to a

c. The Haar and the Trigonometric Systems 163

subspace of some separable Orlicz sequence space hM if and only if X = LiO, 1) (up to an equivalent norm).

Proo! The "if" part is trivial. To prove the "only if" assertion, we notice first that, by 2.c.ll, we have l<px and qx<oo. Let m be an integer and let Xm be the r.i. function space on [0, I], whose norm is defined by

where D 1 m is the dilation operator introduced in the previous section. Since IIDsllxm ~ I(Dsllx for every ° <s< I it follows, by 2.b.3 and the discussion following it, that there is a constant K, independent of m, so that, for every choice of scalars

{an}~=l'

( I )1/2 11' 11 (' )1/2 K- 1 ~ a; ~ ~ anrn ~ K ~ a; n-1 n-1 Xm n-1

In view of the definition of 11·llxm , it follows that if we put

m( ) _ {sign sin 2nmnt, I ~ t~ I/m _ rn t - ° h . , n - 1, 2, ... , ot erWlse

and

then

The same inequalities remain· c1early valid if we replace the {r::'}:'.. 1 by their "translations" {1f,n}:'..1 to the interval [(i-l)/m, i/m], for 1 ~i~m. More precisely,

{sign sin 2nmn(t-(i-l)/m) if tE [(i-l)/m, i/m]

r':' (t) = I, n ° otherwise .

Assume now that T is an isomorphism from X into a separable Orlicz sequence space hM • Since, for every 1 ~ i ~ m, w lim r~ n = ° it follows that, for every 6> 0,

n-+ ao

there are, by I.l.a.12, increasing sequences of integers {nk,i}f= 1 and vectors Xi k E hM so that

(*) IIxi,k-Tr7:nk)A.mll ~t:2-k, Xi,k= L bjej , I ~i~m, k= 1,2, ... , jeai. k


where {ej}f= 1 denotes the unit vector basis of hM and {(Ti,k}:"= 1> k= 1 are mutually disjoint subsets ofthe integers (i.e. (Ti, , k/'I(Ti2' k2 = 0, unless;1 =;2 and k 1 =k2)· We shall use this fact for a fixed but sufficiently small e (depending only on m). The required order of magnitude of e will be pointed out as the proof proceeds.

Let Fi , k be the Orlicz functions defined by

Fi,k(t)= L M(lbjlt), l~i~m,k=I,2, .... jeaitk

Recall that, for a finite set '1 ofpairs (i, k) and scalars {a(i,k)}(i,k)E~' we have

It follows from (*) that if e is small enough then, for every 1 ~ i ~ m and every choice of {ak}~=1> we have

where C=2KiITIIIIT- 111. ForeveryO~ t< C- 1 andevery I ~ i~m thereare integers k 1 (i, t) and k 2(i, t) so that

Let us verify e.g. the existence of k 1 (i, t). Given t< C- 1 we pick an integer n so that

1/2n~t2C2<1/n. Then 11 i:. tXi'kll~Ctn1/2<1 and thus i:. Fi,k(t) < 1; in parti-k=1 k=1

cular, Fi k(t)< 1/n~2C2t2, for some 1 ~k~n. m m

Let now {t;}:"=1 be reals so that L tf ~ 1/2C2. Then L Fi,kl(i,t;)(ti) ~ land thus i=1 i=1

also II.~ tiXi'k1(i,ti)11 ~ 1. Similarly, if .~ tf ~2C2 then II.~ tiXi'k2(i,t il ll ~ 1. Note .-1 1-1 1-1

that since X is, in particular, a Köthe function space we have, for every choice of

{ti}:"=1 and {ki}~1' that

Hence, by combining the preceding observations with (*), we get that if eis small enough (e< 1/4Cm1/2 , to be precise) then

.t tf=I/2C2 = II.t tiX[(i-1)/m'i/mlll~Ä.mIlT-111(I+e.f ti) 1-1 1-1 1=1

~2Ä.mIlT-111


and

In other words, there exists an absolute eonstant K o (which, in partieular, is independent of m) so that, for every ehoiee of {ti}~ l' we have

By taking, in partieular, t i = I, 1 ~ i ~ m, we deduee that

for every m. Henee, for every simple dyadie funetionJon [0, I], we have

and this proves that X=L 2(0, I), up to an equivalent norm. 0

We shall see later on in this ehapter that, in the general theory of r.i. funetion spaees on [0, 1], a eertain exeeptional role is played by those spaees X for whieh the Haar basis {Xn}:'= 1 is equivalent to a sequenee of disjointly supported elements {fn}:= 1 in X. From 2.e.l4 we ean deduee, in partieular, that this eannot happen in an Orliez funetion spaee LM(O, I) (unless, it is L 2(0, 1), up to an equivalent norm). Indeed, if {/"}:'=1 are disjointly supported elements in LM(O, I) with (Jn= supp/"

then IIn~l aJnll=1 if and only if n~l FnClanl) = I, where Fn(t) = L M(tl/,,(s)l)ds'

n = I, 2, .... In other words, the span of {/,,}:,= 1 in LM(O, I) is a modular sequenee spaee (cf. I.4.d.I). The proof of2.e.14 aetually shows that an r.i. function spaee on [0, 1] whieh embeds in a modular sequenee spaee is L 2(0, I). (This is not really a stronger statement than that appearing in 2.e.14 in view of I.4.d.5.)

The Haar system is by far the most eonvenient basis for studying the strueture of r.i. funetion spaees on [0, I]. Therefore, this system has a eentral role in Banach spaee theory. However, from the point ofview ofanalysis in general and, of course, from that of harmonie analysis, the most important basis or system of funetions is the trigonometrie system. We shall diseuss here very briefly this system. For a more detailed study of the trigonometrie system we refer to books on harmonie analysis (espeeially, Zygmund [132]).

It is somewhat more eonvenient to treat the trigonometrie system in the setting of eomplex sealars. Thus, in the rest of this seetion we shall eonsider eomplex r.i. funetion spaees on [0, I]. The trigonometrie system on [0, I] eonsists of the funetions 1, e21tit , e- 21tit, e41tit , .... The order in whieh the trigonometrie functions


are enumerated is important. They form an uneonditional basis of an r.i. funetion space X on [0, 1] only if X=LiO, 1) (up to an equivalent norm). In order to verify this, we have just to remark that the square funetion eorresponding to the expansion

( )1/2

~ ake21tkit is simply ~ lakl2 and to apply l.d.6(ii).

A major tool in the study of the trigonometrie system is the following classical result of M. Riesz.

Theorem 2.c.15. Let R be the linear projeetion defined by

on the linear spaee 01 the trigonometrie polynomials. Then, lor any 1 <p< 00, R extends uniquely to a bounded projeetion in LiO, 1).

Proof We shall present a proof whieh is due essentially to Boehner. It is very similar to, but simpler than, the proof given for 2.e.4 (obviously, it motivated Cotlar's proof of 2.e.4).

We note first that, by the Riesz-Thorin theorem and duality, it suffiees to prove the theorem for p beinga power of2 and that for p=2 the assertion is obvious. We observe next that it suffiees to prove that there exists a eonstant Cp so that, when-

n

ever lis of the form L IXke21tkit, then k= 1

where u and v are the real, respective1y, the purely imaginary parts of I (i.e. 1= u + iv). Indeed, onee this is shown we put, given any funetion h of the form

n

L ake21tkit with ao = 0, k= -n

Then

and

n

L (ak+a- k) e21tkit=l=u+iv, k=1

u+is=h, n

r + iv = L ake21tkit k=1

n

L (ak-a- k) e21tkit=g=r+is. k=1

-1

L ake21tkit k= -n

This shows that IIRllp~2Cp when restrieted to those funetions with ao=O and therefore R is bounded also without this restrietion.

c. The Haar and the Trigonometrie Systems 167

We prove the existence of Cp for p=2m by induction on m. Assume that a suitable Cp exists and letf=u+iv. Thenf2=u2-v2+2iuv. Since

it follows from the induction hypothesis, applied to f2, that

Hence, by the Cauchy-Schwarz inequality,

which clearly implies the existence of C2p ' 0

An immediate consequence of 2.c.15 and 2. b.11 is

Theorem 2.c.16. Let X be a separable r.i. funetion spaee on [0, I] with I <Px and qx< 00. Then the trigonometrie system is a Schauder basis of X.

Proo! By 2. b.II and 2.c.l5, R extends uniquely to a bounded operator on X. Since multiplication by e21tmit acts as an isometry on X for every integer m we get that, for every choice of scalars {ak}~=O and I <m <n, we have

Ilktm ak e27tkitll = Il:t: aHm e27tkitll~ IIRIII[~mm aHm e27tkitll

= IIRllllkt ak e21tkitll·

Consequently, the trigonometric system is a basic sequence in X. Since X is separable C(O, I) is dense in X and this proves that the trigonometrie system spans X i.e. is a basis of X. . 0

The operator R defined in 2.c.15 is not bounded in L 1(0, I) and thus, the trigonometric system is not a basis of L 1 (0, 1). It can e.g. be verified that

00 00

f(t)= L cos2nkt/logk= L (e27tkit+e-27tkit)/210gk k=2 k=2

00

belongs to L 1(0, 1) while L e21tkit/log k does not belong to L 1(0, 1). The operator k=2

R is, however, ofweak type (1, 1). Let us also mention that, actually, the converse of 2.c.I6 is valid. It was proved in [39] that the operator R is bounded in an r.i. funetion spaee X on [0, I] (or, equivalently, that the trigonometrie system is abasie sequenee in X) only if 1 <Px and qx< 00.


We eonclude this seetion by mentioning another simple but interesting eonsequenee of2.e.15. Let Xbe an r.i. funetion spaee on [0,1]. We denote by Xa the closed linear span of {e21tkit}~o in X. The spaee Xa ean be viewed as the analytie part of X. It ean be identified with the spaee of all funetionsf{z), whieh are analytie in the open unit dise Izl < 1 and for whieh,

IlfllXa = sup 11f..llx < 00 r< 1

wheref..(t)=f(re21tit), O~t~l. In ease X=L/O, 1), l~p~oo, the spaee Xa is eommonly denoted by Hp, 1 ~p~ 00.

Proposition 2.c.17 [12]. Let X be a separable r.i. fune/ion spaee on [0, 1] with 1 <px and qx < 00. Then X is isomorphie to Xa·

Proo! Let R be the projeetion on X defined in 2.e.15. The map (Sg)(t)= e- 21titg(l_t) is an isometry from Xa onto (Ix-R)X=[e-21tkit]r'= l' Henee X:::::: Xa E9 Xa. We have also that Xa::::::Xa E9 Xa. Indeed, the mapf(t) -+ f(2t(mod 1» defines an isomorphism between Xa and its subspace [e41tkit]~ o. This subspace is eomplemented by the projeetion Q defined by

Qf(t)=(f(t)+f«t+ 1/2)(mod 1))/2

and ker Q = [e21t(2k + l)it]k"= 0 is also isomorphie to Xa. 0

It follows, in partieular, that, for l<p<oo, the spaees Lp(O, 1) and Hp are isomorphie. It is known that this is not the ease if p = 1 or p = 00. For a detailed study of the isomorphie properties of H 1 and H 00 we refer to [111]. F or the classieal theory of Hp spaees the reader is referred to [34] and [52].

d. Some Results on Complemented Subspaces

In this seetion we present several results on the strueture of uneonditional basic sequenees and eomplemented subspaces of r.i. funetion spaees X on [0, 1]. Nearly all the proofs rely on the uneonditionality of the Haar basis and therefore we shall assurne in most plaees that X is separable and its Boyd indices satisfy 1 <px and q x< 00. F or instanee, we prove that under those assumptions Xis a primary spaee. The results presented here were originally proved for L p spaees. In many eases the extension to more general r.i. funetion spaees is easily aehieved by a suitable applieation of the interpolation theorem 2. b.l1.

We begin with some preliminary material involving the spaee X(l2) associated to a Banach lattiee X. Reeall that X(l2) is the eompletion of the spaee of all sequenees

d. Some Results on Complemented Subspaces 169

(x l' x 2 ' ••• ) of elements of X which are eventually zero, with respect to the norm,

In other words, X(l2) consists of all the sequences (Xl' X2' ... ) for which

lim sup II(.I IXi12)1/211 =0. m-+oo n>m l=m X

In the case when X is an r.i. function space on the interval 1= [0, 1] and qx< 00,

the space X(l2) can be identified (up to an equivalent norm) with the subspace ofthe r.i. space X(Ix I) spanned by the functions of the form x(s)rit), where x(s) EX and n= 1,2, ... (rit) denotes as usual the nth Rademacher function). This subspace of X(lx I) is denoted by Rad X. More precise1y, we have the following result.

Proposition 2.d." Let X be an r.i. Junction space on 1="[0, 1] with qx< 00. Then there exists a constant M< 00 so that, Jor every x = (Xl' x 2, ... ) E X(/2)' we have

M- lllxllx(l2) ~ lIi~l Xi(S)ri(t)IIx(l x I) ~ MllxII X (l2) •

Proof Let {XJ7=1 be a finite sequence of elements of X. Since {xi(s)ri(t)}7=1 has unconditional constant equal to one in X(lx I) it follows from Khintchine's inequality that

11 ± x;(s)ri(t)11 = J 11 ± Xi(S)ri(t)r;(u)11 du i = 1 X(l x I) 0 i = 1 X(l x I)

~ Ili I ± xi(s)ri(t)ri(u)I dull o i= 1 X(l x I)

~A111(f IX;(Sw)1/2

11 i=l X(lxI)

=A111Ct1 Ixi l2 Yl211x .

The other estimate can be obtained in a similar manner if Xis q-concave, for so me q< 00. Our assumption thatqx< 00 is, however, weaker and thus theproofrequires additional work.

( n )1/2

Let {x;}7= 1 cXbechosen so thatthefunctionJ= i~l Ix;j2 satisfiesllJllx~l.

It folIo ws easily from Khintchine's inequality in LiO, I) that


for everyO<v< 00, SE land forevery 1 ~q~ 00. On the other hand, since t- 1jq is a decreasing function, we have that

Hence, for every v> 0 and q ~ 1,

dl" I (v)~B: d!(s)t-'/.(V) L X,(S)',(t) i=l

from which we deduce (since the dilation operator DB: has a norm ~ 0:) that

11 ± Xi(S)ri(t)11 ~ 0:11/(s)t- 1jqll x(I x I) i=1 X(IxI)

~o:ll/(S) i 2(n+1)jQX(2-n-I,2-nj(t)11 n=O xpxD

00

~o: L 2(n+1)jqll/(S)X(2-n-I,2-nj(t)IIX(lXI)' n=O

Fix now q> qo > qx and recall that the definition of qx impIies the existence of a constant Cqo < 00 so that

liD II ~ C u1jqo u x~ qo '

for all O~u~ 1. Therefore, by using the fact that/(s)X(2-n-'.2-n)(t) in X(lx I) has the same distribution function as D 2 -n - 1I in X, we get that

11 f xi(s)ri(t)11 ~ 0: i 2(n+ l)jQ IID2 -n - dllx

i = 1 XII x I) n = 0 00

~ l1!.C "2(n+ 1)jQ2 -(n+ l)jQo ~ q qo L.. .

n=O

Since q>qo the series above converges. 0

Remark. In case X is q-concave for some q< 00 then X(l2) admits another representation which we encountered already in Section l.e (hut which will not be used in this section). The space X(12) is, for every 1 ~p< 00, isomorphie to the subspace of LiX) spanned by the functions of the form xrit) with x E X and n= 1,2, ... (use I.d.6 and l.e.13).

An important fact concerning Rad Xis the following.

Proposition 2.d.2. Let X be a separable r.i. function space on 1=[0, 1] such that 1 <Px and qx< 00. Then Rad Xis a complemented subspace of X(I x I).


Proof For fes, t) E U Lp(I X I) put p>l

We have already observed in the proof of 2.b.4(ii) that, for 1 <p< 00, there is a constant C p so that for sEI

1 1

J Ipf(s, tW dt ~Cp J Ifts, tW dt . o 0

By integrating this inequality with respect to s over [0, 1], we deduce that P is bounded on LiIxI) for every I<p<oo. Hence, by 2.b.II, Pis a bounded operator on XCI xI). It is clear that the range of P contains Rad X and that every

00

function in the range of Pis of the form g(s, t) = L xn(s)rn(t). It remains to show n=l

that if this infinite sum (taken in the sense of convergence in measure) belongs to XCIx I) then the series converges in the norm of XCIx I) and thus its sum g(s, t) must belong to Rad X. Indeed, assume that there is an e> ° and a sequence of

mf+l

00

for every i. Since the sum L gi(S, t) = g(s, t) belongs to XCI x I) and since, for every i=l

00

choice of signs (J = {(Ji};x'= 1 ,g9(S, t) = L (Jigi(S, t) has the same distribution function i= 1

as g(s, t) we would deduce that g9(S, t) E XCIx I), too. But, for any two distinct sequences of signs (J and (J', we have that IIg9 - g9' Ilx(l x I) ~ 2e and this contradicts the separability of X. 0

Remark. If X is a non-separable r.i. function space on [0, 1] with 1 <Px and qx<oo then the projection P above is still bounded. However, its range is not

r:,~:;t;:t=(~~ I::~~: )~~: ;on,istmg of all the fuoctioos of the form

From 2.d.l and 2.d.2 and the interpolation theorem 2.b.1I it is easy to deduce an interpolation theorem for operators defined on X(l2)' For simplicity ofnotation we shall write LP2) instead of LiO, 1)(/2),

Proposition 2.d.3. Let X be a separable r.i.function space on [0, 1] with 1 <Px and qx< 00. Let 1 <p <Px' qx < q < 00 and let T be a bounded linear operator from L p(l2) into itselfwhich also acts as a bounded operatorfrom Lq(l2) into itseIJ. Then T maps X(/2) into itself and is bounded on this space.

Observe that, since Lq(O, I)cXcLiO, 1) (cf. 2.b.3), we have also, in a natural


way, the inc1usions

and thus it makes sense to talk of the restrietion to Li!2) or to X(l2) of an operator defined on L p(l2)'

Proo! We first identify the three spaces appearing in the statement of 2.d.3 with the respective spaces Rad Lp(I), Rad LiI) and Rad X, of functions on I x I, via the correspondence established in 2.d.1. Let P be the projection defined in 2.d.2. Since TP is a bounded operator from Li1x I) into itself and from Li1x I) into itself it follows, by 2.b.11, that TP is also a bounded operator from X(Ix I) into itself. Consequently, T is a bounded operator from Rad X into itself. D

Remark. A similar result holds, with the same proof, for operators from X into X(l2) or from X(l2) into X. Thus, e.g. (with X, p and q as in 2.d.3) if T is a bounded linear operator from Lp(O, 1) into Li12) which also acts as a bounded operator from LiO, 1) to Li/2) then T is a bounded linear operator from X into X(l2)'

Another consequence of 2.d.1 and 2.d.2 is the following result (proved by B. S. Mitjagin [102] under somewhat different assumptions).

Proposition 2.d.4. Let X be a separable r.i.Junetion spaee on [0, 1] with I <Px and qx< 00. Then X is isomorphie to X(l2)'

Proo! By 2.d.1 and 2.d.2, X(l2) is isomorphie to a complemented subspace of X. On the other hand, Xis c1early isometrie to a complemented subspace of X(l2)' Since X and X(12) are obviously isomorphie to their respective squares the desired result follows by using Pelczynski's decomposition method .. D

Remark. The isomorphism between X and X(l2)' established in 2.d.4, is 'canonical' in the terminology of category theory. In other words, the isomorphism from X onto X(l2) is induced by an operator whose formal form is independent of X. Proposition 2.d.3 is a direct consequence of this fact and 2. b.Il. The proof of 2.d.3, presented above, is, however, conceptually simpler.

From 2.d.4 we can get some information on general complemented subspaces of r.i. function spaces.

Proposition 2.d.5. Let X be an r.i. Junetion spaee on [0, 1] with I <Px and qx< 00.

Let Y be a eomplemented subspaee oJ X whieh, in turn, eontains a eomplemented subspaee isomorphie to X. Then Y itself is isomorphie to X.

Proo!. We assurne first that Xis separable. We let Y(l2) denote the subspace of X(l2) consisting of all x = (x l' x 2' ... , X n , ... ) E X(l2)' with Xn E Y for all n. (Note

( k )1/2 that the vectors n~1 JXn J2 need not belong to Yand that Y(l2) is not aspace

intrinsieally associated to Y in the sense that it depends on the embedding of Y in


X.) Let Q be a projection from X onto Y. By l.f.14, the operator

defines a projection from X(l2) onto Y(l2) whose norm is :::S;Kd/Q// (as we have already mentioned, I.f.14 was proved by J. L. Krivine [66J; prior to this, B. S. Mitjagin [102J had proved this fact for X being an r.i. function space on [0, IJ). Hence, there exist Banach spaces Z and W so that

It follows that

Y~XEB Z~XEB XEB Z~XEB Y

and, by 2.d.4,

This conc1udes the proof if X ls separable. If X is non-separable we can apply the same argument provided we replace X(12) by the space x(f,,) of all sequences

( 00 )1/2

x=(x1, x 2 ' ... , X n, ... ) such that n~1 /xn/2 EX and Y(/2 ) by YU;). For non-

separable Xwith I <Px andqx< 00 wehave, inanalogy to 2.d.4, thatX ~x7h). The proof is similar to 2.d.4; this can be also deduced from 2.d.4 by duality (apply 2.a.3 and the discussion preceding l.d.4). . 0

We pas's now to a discussion of some facts concerning unconditional bases in X and X(/2). To this end, we need the two dimensional version of the c1assical Khintchine inequality.

Proposition 2.d.6. For every 1 :::S;p< 00 there exist constants A~) and dp2) so that, for every matrix of scalars (am,n):,n= 1 with only finitely many non-zero entries,

Proof Since the functions {rm(t)rn(S)}:=1,~1 form an orthogonal system in L 2([0, IJ x [0, IJ) we have that


Therefore, for p > 2, it suftices to prove the right-hand side inequality of 2.d.6. Using the one-dimensional inequality ofKhintchine and the triangle inequality in L p12 , we get

It remains to prove the left-hand side inequality for 1 ~p < 2. It suffices to consider p = 1. The desired inequality follows from what we have already proved for p=4 and the fact that Ilflh~lIfIW31Ifll~/3 for every functionf(by Hölder's inequality). 0

The two dimensional Khintchine inequality 2.d.6 and its proof extend in an obvious way to any finite dimension.

Our first result on bases in X(l2) is valid for generallattices.

Proposition 2.d.7. Let X bea q-concave Banach latticefor someq<oo. Let {Ym}:;=1 be an unconditional basic sequence in X. Then the double sequence

n

Ym,n=(O, ... ,0, Ym' 0, ... ), m, n= 1,2, ...

forms an unconditional basic sequence in X(l2)' In particular, if {Ym}:;=l is an unconditional basis of X then {Ym,n}::,n= 1 is an unconditional basis of X(/2)'

Proof Let K be the unconditional constant of {Ym}:;=1 and let M denote the q-concavity constant of X. Then, by Khintchine's inequality in LI (1) and Lq(/ x I), we get, for every choice of scalars {am, n}::, n = 1 with only finitely many different from zero, that

11 00 00 11 II( 00 1 00 12)

1/211 L L am,nYm,n = L L am,nYm

m=l n=1 X(il) n=1 m=l X

d, Some Results on Complemented Subspaces

~AI1KMIIO II m~ J1 rm(v)rn(u)am,nYmlq dUdvy,qlix

~Al1~2)KMIIC~1 n~1 lam,nl2lYml2Y'2l1x,

In a similar manner, it can be shown that

00 00

175

Thesetwoestimatesforthenormof L L am,nYm,ninX(/2)proveourassertion, 0 m= 1 n= 1

Note that if we put Y = [y"J:'= 1 then, with the notation used in the proof of 2.d.5, Y(l2) = [Ym,n]:',n= 1 .

In the ca se of the Haar basis in an r.i. function space we can prove the same result with the assumption of q-concavity replaced by the weaker one that qx< 00.

Proposition 2.d.8. Let X be a separable r.i. /unction space on 1=[0, I] with 1 <Px and qx < 00. Let {Xm}:= 1 be the Haar basis 0/ X. Then the vectors

n

Xm,n=(O, ... , 0, Xm, 0, ... ), m, n= 1,2, ...

form an unconditional basis of X(/2 ).

Proof Let 1 <P<Px and qx<q<oo. By 2.d.7, {Xm,n}:',n=1 is an unconditiona'l basis of Lp(/2) and Lq(l2)' Hence, by 2.d.3, this double sequence is also an unconditional basis of X(/2). 0

Before we proceed, let us recall that, by l.d.6 and the remark thereafter, for every unconditional basis {un}:'= 1 of a complemented subspace of a Banach lattice X (and thus, in particular, for the Haar basis of a separable r.i. function space on

[0, 1] with 1 <Px~ qx < 00), the expression 11 i:. anun ~ is equivalent to the norm of . n=1 Ilx

the square function !lCt lanUnl2 ) 1/211x .

In the previous section we have seen that every unconditional basic sequence {Ym }:'= 1 in an r.i. function space X on [0, 1] which is q-concave for some q< 00 is equivalent to a block basis {Zm}:'=1 ofthe Haar system of X. We shall prove now, under the usual assumptions made on X in this section, that if [Ym]:'= 1 is complemented in X then {zm}:' = 1 can be chosen so that, in addition, [zm]: = 1 is complemented in X. For the case of X=Lp(O, I), I <p<oo, this was proved by G. Schechtman [119].

176 2. Rearrangernent Invariant Function Spaces

Theorem 2.d.9. Let X be a separable r.i. function space on [0, 1] with 1 <px and qx < 00. Let Y be a complemented subspace of X having an unconditional basis {Ym}~= l' Then {Ym}~= 1 is equivalentto a block basis {Zm}~= 1 ofthe Haar basis in X with [Zm]~= 1 complemented in X.

Proof Since Xis separable the step functions on the dyadic intervals are dense in X. Hence, by a standard perturbation argument (cf. I.l.a.9(ii)), there is no loss of generality in assuming that

2km Ym= L C2k m +1 IX2km + ,I, m= 1,2, ... ,

1= 1

for a suitable sequence of integers k 1 <k2 < ... and for suitable scalars {c;}. Put

z"m Zm= L C2km+IX2km +l' m=l, 2, ...

1= 1

and notice that {zm} ~ = 1 is a block basis of the Haar basis of X so that

k 2 m

IZml = L IC2k m +/llx2km +/1 = IYml ' 1= 1

for every m. Hence, for every choice of scalars {am};:.'= l'

By applying the remark following 2.d.8 to the unconditional basis {Ym};:.'= l' of a complemented subspace of X, and to the Haar basis of X, we deduce that {Ym}~= 1

is equivalent to {Zm};:.'= l' It remains to prove that [Zm];:.'=l is complemented in X. For every m, n= 1,2, ... , put

n

Ym,n=(O, ... ,0, Ym, 0, ... ) .

These vectors belong to Y(l2)' As noted in the proof of 2.d.5, Y(12) is a complemented subspace of X(l2)' We shall show next that [Ym,m];:.'= 1 is a complemented subspace of Y(l2) and thus of X(l2)' This would be dear if {Ym,n}~n=l were an unconditional basis. However, this fact is ensured by 2.d.7 only under the assumption that Xis q-concave for some q< 00 which is stronger than our assumption that qx< 00. We shall show that there is a constant M so that, for every choice of scalars {am. J:.n=1 with only finitely many non-zero entries and ofsigns {8m }:= 1

and {'1n}~ l' we have

(*) 11 f f 8m'1nam,nYm,nll ~MII f f am,nYm,nll . m=l n=l X(l2) m=l n=l X(!,)


From (*) it follows easily that {Ym,n}~,n= 1 is a Schauder basis of Y(l2) if ordered as

Yl,I'Yl,2'Y2,2' Y2,1' Yl,3' Y2,3' Y3,3' Y3,2' Y3,1' Yl,4' ...

and that the canonical projection from Y(l2) onto [Ym m]~= 1 is bounded (the argument is the same as the one used in I.l.c.8 to prove'the boundedness of the diagonal operator). In order to prove (*), note first that the operator Te on Y defined by

00 00

Te L amYm = I ()mamYm m= 1 m= 1

satisfies 11 Tell ~K, where K is the unconditional constant of {Ym}~= l' Hence, by 1.f.14,

This proves that [Ym, m]:= 1 is complemented in X(/2)' Consider now the vectors

k m

h2k~+I=(0, ... , 0, IX2k~+II, 0, ... ), / = 1, 2, ... , 2 ~, m=l, 2, ...

and notice that {h2k~+I}?~~,~=1 is equivalent to {Xl~+I}?~~,~=I' Indeed, this folIo ws from

k

and the remarkJolIowing 2.d.8. The canonical isomorphism from [X2k~+a?=~,~= 1

onto [h2k~+af=~,~=1 maps Zm to Ym,m' for every m. Sil1ce [Ym,m]~=1 is complemented in X(/2) and thus, in I!articular, also in [h2k~+af=~,~=1 the space [ZmJ~=1 is complemented in [Xl~+I]f=~,~=1 and thus in X, by the unconditionality ofthe Haar basis in X. 0

Theorem 2.d.9 should, in principle, be of he1p in c1assifying the complemented subspaces of an r.i. function space having an unconditional basis. However, in practice it is usuaUy hard to determine which block bases of the Haar basis span a complemented subspace and to c1assify the subspaces they span. A subsequence of the Haar basis obviously spans a complemented subspace if the Haar basis is


uneonditional. The spaees spanned by subsequenees of the Haar basis have been eharaeterized by J. L. B. Gamlen and R. J. Gaudet [45J in the ease of Lp(O, 1), 1 <p < 00. The extension of their result to r.i. funetion spaees is straightforward.

Theorem 2.d.l0. Let {qJn}:."= 1 be a subsequenee of the Haar system and let

er = {t E [0, 1]; t E sUPP qJJor infinitely many n} .

Let X be a separable r.i. funetion spaee on [0, IJ so that 1 <Px and qx< 00. If f1( er) = ° then there exists a sequenee of pairwise disjoint eharaeteristie funetions in X whose closed linear span Yeontains [qJnJ:."= 1 (obviously, both [qJnJ:."= 1 and Yare eomplemented in X) while lff1(er) >0 then [qJnJ':= 1 is isomorphie to X. In partieular, every subsequenee of the Haar basis in Lp(O, 1), 1 0).

Proo! Case f1(er)=O. For every positive integer m, put

er m = {t E [0, 1 J; t E supp qJn for exaetly m distinet values of n} .

For t E erm, m = 1, 2, ... , let n(t, m) be the largest integer n for whieh tE supp qJn' Sinee two distinet Haar funetions have either disjoint supports or the support of one of them is entirely contained in that of the other it folIo ws that, for any SE ermllsupp qJn(t,mj> we have qJn(s,ml = qJn(t,ml' Henee, for every m, the distinet functions among {qJn(t,ml}tE<1~ have pairwise disjoint supports. Consequently,

forms a sequenee of pairwise disjoint eharaeteristie funetions whose c10sed linear span Y is, by 2.a.4, eomplemented in X. Moreover, if tE O"mllSUPP qJk' for some m and k, then supp qJn(t,mlCsupp qJk which implies that qJkX"JqJn(t,mll E Y.

00

Sinee U erm eoineides with [0, lJ, exeept the set er which is assumed to have m=l

measure zero and exeept the set ero on which all the {qJn},~)=l vanish, it follows that [qJnJ:."= 1 C Y. This completes the proof in ease f1( er) = ° for a general X. When X = LiO, 1), for some I 0. We begin by defining the notion ofa tree

of subsets of er. By this we mean a eolleetion of measurable sets so that er = 1]0 UI] 1 ,

f1(l] i 1,i2, ... , iJ= 2- nf1(er) and


for every n and {iJj = l' For every such tree the functions {hm}:= l' defined by h 1 = X"' hz = Xqo - Xq1 ' h3 = Xqo • 0 - Xqo• l' h4 = Xq1 • 0 - Xq1 • 1 and SO on, form clearly a monotone basis equivalent to the Haar basis of X. Moreover, [hm]:= 1 is complemented in X since it is the range of the projection Pf= x"EPAf of norm one, where EIß denotes the conditional expectation operator with respect to the (J

algebra f!J generated by the tree. Hence, by the perturbation result I.l.a.9(ii) and 2.d.5, it suffices to find a tree of subsets of (J and a sequence {Ym}:=z of vectors in [CPn]:'=l so that IIYm-hm/l<cm for m=2, 3, ... , where {hm}:'=z are the functions associated to the tree and {cm}:'=z are small enough positive numbers given in advance (to be specific, Cm = Jl«(J)4- m - Z , m =2, 3, ... ).

To construct '10 (and thus '11) we proceed as folIows. Given J > ° we find a relatively open sub set G of [0,1] so that G~(J and Jl(G"'(J)<J. For every tE (J, let n(t) be the smallest integer n for which tE supp ({Jnc G. Clearly, the distinct functions among {CPn(t)}tE" have mutually disjoint supports and thus their sum Yz belongs to [({Jn]:'= 1 and satisfies X" ~ IYzl ~ XG' The function Yz takes the value 1 on half of its support and -Ion the second half. It is clear, therefore, that if J is small enough we can find '10 so that the corresponding function hz satisfies IIYz - hzll < Cz (we used here the fact that Xis separable and thus IlxG-,,11 =0(1». By replacing in the above construction (J with '10 we can find in exactly the same manner a subset '10,0 of '10 and a vector Y3 E [CPn]:'= 1 so that /lY3 - h3 /1 < c3' The construction of the entire tree and of the vectors {Yn}:'=4 is continued by an obvious inductive argument. 0

Remark. The assumption that 1 <Px and qx< 00 was used only in the proof ofthe case Jl«(J) > 0.

We conclude this section by proving that separable r.i. function spaces on [0, 1] with non-trivial Boyd indices are primary. Recall that a Banach space Xis said to be primary if, for every deeomposition of X into a direet sum of two subspaces, at least one of the faetors is isomorphie to X itself.

Theorem 2.d.1l. Every separab/e r.i. function space X on [0, 1] with 1 <Px and qx< 00 is primary.

OriginaIly, 2.d.ll was proved by P. Enflo for Lp spaees, 1 ~p < 00. A proof in the ca se p= 1, whieh is similar in spirit to Enflo's original argument, is given in P. Enflo and T. Starbird [38] (cf. also B. Maurey [92]). The present proofas weIl as the extension to Li. funetion spaees (under slightly more restrietive eonditions) is due to D. Alspach, P. Enflo and E. Odell [2].

Proof By 2.d.4, X is isomorphie to X(lz) and, therefore, it is equivalent to prove the primariness of X(/z). Suppose that

for some subspaces Yand Z, and let P be the projection from X(lz) onto Y which


vanishes on Z. By 2.d.8, the vectors

n

Xm. n = (0, ... , 0, Xm' 0, ... ), m, n = 1, 2, ...

form an unconditional basisof X(l2)' Let {X!.n}~.m=l be the sequence of the biorthogonal functionals associated to {Xm. n}:' n = l' Then, for each m and n, at least one of the numbers X!. nPXm, n and X!, n(I - P)Xm, n is ~ 1/2. Put

lT y= {m E N; X!.nPXm,n~ 1/2 for infinitely many values of n}

and

Notice that, by 2.d.lO, either [XmJmEay or [XmJmEaz is isomorphie to X. We may assume without loss ofgenerality that lT y= {m 1 <m2 < ... } and [Xm)Y; 1 ~X. Since, for eachj, {xmj,n/IlXmj,nll}~ 1 is equivalent to the unit vector basis of 12 and thus weakly null, it follows that there exists an increasing sequence {nj }Y;l ofintegers so that {PXmj,n)f=l is equivalent to a block basis {Zj}f=l of {Xm,n}:'n=l and, in addition,

for every j. For any sequence {a j }}=l ofscalars we have

00 00

'" aJ·Xm· n . converges => '" aJ,PXm. n. converges ~ JtJ ~ J'J j= 1 j= 1 00 00

=> L ajzj converges => L ajXmjonj converges. j= 1 j= 1

Hence, {Zj}f=l is equivalent to {Xmj,n)f=l and, moreover,

00

Qx = '" X· (x)z ·Ix· (z.) L.. mj, nj J mj, nj J j= 1

defines a bounded projection from X(l2) onto [zjJf= l' Thus, if the {zJ~ 1 were chosen sufficiently elose to {PXmj, nJ~ 1 we could deduce, by I.l.a.9(ii), that [PXmj,n)f= 1 is complemented in X(12)' Therefore, in order to prove that X ~ Y, it suffices, in view of 2.d.5, to show that {Xmj, n)}= 1 is equivalent to {XmJ~ l' Note however that

and thus the desired result is an immediate consequence of the remark following 2.d.8. 0

e. Isomorphisms Between r.i. Function Spaces

e. Isomorphisms Between r.i. Function Spaces; Uniqueness of the r.i. Structure-

181

The first topic considered in this section is that of isomorphic embeddings of an r.i. function space X on [0, IJ into another r.i. function space Y. The simplest example of such an embedding is that of X = L 2(0, 1) into an arbitrary r.i. function space Y with qy< 00. Another well-known example of an isomorphic embedding, which will be discussed in detail in Section f below and especially in Vol. IV, is that of X=Lp(O, 1) into Y=Lr(O, 1), for 1 ~r<p<2. A characteristic ofthis example is that the identity mapping is a continuous operator (but in general not an isomorphism) from X into Yi.e. there exists a constant c>O such that Ilflix ~cllflly, for every fEX. There is also a third possibility which was already mentioned briefly in 2.c. This is the case where the Haar basis of X is equivalent to a sequence of pairwise disjoint functions in Y. Concrete examples ofthis type will be constructed in Section g below.

The three cases described above essentially exhaust all the possibilities of embedding isomorphically X into Y provided some natural conditions are imposed on X and Y(e.g. one c1early has to exc1ude the case Y=Loo(O, 1) when no information on X can be obtained). This result from [58J Section 6 will be stated below in a precise way but without giving a proof since its proof is too complicated to be reproduced here. Instead, we shall prove aversion of it which, though weaker, still has several interesting applications. This theorem as weIl as its applications will deal only with r.i. function spaces on [0, IJ which are, in a sense, on one side ofthe Hilbert space L 2(O, 1) (for example, r.i. function spaces of type 2 or of cotype 2). In many cases, we have actually to ensure that Xis even "far" from L 2(0,1) (for instance, in the sense that it is q-concave for some q< 2 or r-convex for some r> 2).

The results proved on isomorphisms between r.i. function spaces can be used to study the uniqueness ofthe rj. structure ofa given r.i. function space X on [0, IJ, i.e. the quest ion when any two representations of X as an rj. function space on [0, IJ must coincide (except, perhaps, for an equivalent renorming). Generally speaking, the uniqueness of the r.i. structure on [0, IJ can be proved for those spaces X for which the Haar system of any representation of X as an r.i. function space on [0, IJ is not equivalent to a sequence of pairwise disjoint (relative to the representation under consideration) functions in X. For example, by 2.c.l4 and the remark thereafter, this is the case for reflexive Orlicz function spaces LM(O, 1) and, indeed, these spaces have a unique r.i. structure.

Before stating the first result we discuss some facts concerning the notion of a p-stable randorn variable which will be treated in detail only in Vol. IV. Let (Q, 1:, v) be a probability space having no atoms. Areal valued randorn variable g on Q is called p-stable, for sorne O<p~ 2, if

J eitg(ro) dv(w)=e-cltIP , Q

für süme constant c> 0 and for every - 00 < t< 00. That such randorn variables do


indeed exist and that any p-stable variable, I <p~2 belongs to the space Lr(Q, E, v), for any I ~r<p, will be proved in Vol. IV. (See also [135J and the remark following 2.f.5.) The value of the constant e determines the distribution function of 9 and thus also the norm of 9 in Lr(Q, E, v) i.e. two p-stable random variables with the same e have the same norm in Lr(Q, E, v). This statement is a consequence of the fact that the Fourier transform on Li ( - 00, (0) is one to one. Unlessp=2, ap-stable random variable does not belong to Lp(Q, E, v) itself.

The importance of p-stable random variables in Banach space theory sterns from the fact that they can be used to embed isometrically lp into Lr(O, 1), for I ~ r <p ~ 2. If {g,,} ~ 1 is a sequence of identically distributed independent p-stable random variables with I/g 11/r = 1 then, for every choice of scalars {a,,},~"= 1 ,

00

Indeed, if we suppose that L la,,1 p = 1 then, by the independence of the sequence ,,=1

00

it L a.g.(ro) 00 00

Je .=1 dv(w)= TI J eita"gn(CJ) dv(w) = TI e-clan/lP

U ,,=lU ,,=1

i.e. L a"g" is again a p-stable random variable with the same constant e as all of ,,= 1

the g,,'s. Hence, 11"~1 ang"llr = 1.

The above property of the p-stable random variables will be used in the proof of the following result (cf. [58J Section 5).

Theorem 2.e.1. Let X be a separable r.i.funetion spaee on [0, I] having the property that,for some 1 <q<2, C< 00 andJor every JE Lq(O, I),

IIJllx~CIIJllq . Let Y be an r .i.funetion spaee on [0, I] or on [0, (0) whieh does not eontain uniformly isomorphie eopies oJ I:, Jor all n. If X is isomorphie to a subspaee oJ Y then either

(i) there exists a eonstant D< 00 so that

Jor every J E X, or (ii) the Haar basis oJ Xis equivalent to a sequenee oJ pairwise dis joint Junetions

in Y.

The proof of 2.e.1 is based mainly on the following lemma.

e. Isomorphisms Between r.i. Function Spaces 183

Lemma 2.e.2. Let T be a bounded linear operator Jrom a separable r.i. Junction space X on [0,1] into an r.i. Junction space Y on [0,1] or on [0,00) so that lim IlxIo. t)11 y = ° (i.e. the restriction oJ Y to [0, 1] is not equal to Loo(O, 1), even up to t~O

an equivalent norm). Suppose that there exist numbers s ~ 1 and R< 00 such that if, Jor n = 0, 1, 2, "', we put

2 n

Yn= V ITX[(i_1)2-n.i2- nJI and ()n={tE[O,s];Yn(t)~R} i= 1

then

a = inf IIYnXdnll y > ° . n

Then there exists a constant D< 00 so that

Jor every J E X.

Proo! Fix n and put

Observe that IIYnX~nlly ~ a12. Let {1Jn, ;}?~ 1 be a partition of 1Jn into mutually disjoint measurable subsets such that

for t E 1J n, i' i = 1, 2, ... , 2n• We assume, as we clearly may without loss of generality, that {,u(1Jn,i)}?;l is a non-increasing sequence. For any I <m~2n, we have

IIYn ~t: X~n'illy ~lliYl ITX[(i-1 J2- n,i2-nJ llly

~II! [tl1 r;(U)TX[(i-1 J2-n,i2- nJ lduL

~IITII } II~Il ri(U)XW-1J2-n.i2-nJII du ° ,= 1 X

= 11 TII IIX[O,(m-1)2 -nJllx


aI2~IIYnX~nlly~IITIIIIX[o,(m-1)2-nJllx+lkn i~m X~n'illr ~ 11 TII IIx[o. (m -1)2 - n)lIx + Rllx[o, 2nl'(~n. m»11 y •


Since Xis assumed to be separable there exists aß> ° so that

Hence, by letting m> 1 be the smallest integer for wh ich mTn > ß, we get that

Since Ilx[o, tJll y -4 ° as t -4 ° it follows that there exists a y > 0, independent of n and m, so that

This means that we have found positive reals ß and y, independent of n, such that a fixed proportion ß< mTn of the 2n sets {l'/n.J?: 1 have Lebesgue measure ~ y. Tn.

Let now N be an integer for which Ny ~ I. Then, by taking into account the definition ofthe sets {I'/n, i}?: l' it folIo ws that, for any choice of scalars {a;}7'= l' the function

satisfies

m

g = L aiX[(i-ljN- 12 -n, iN-12 -nj i= 1

Ilglly~ IIJl aiXqn,;lly ~ 2a- 11Ix[o,Slllyllitl aiXqn,;TX[(i-lJ2-n,i2-nj/ly

~ 2a- l IIX[o,Slll y lIi!l IT(aiX[(i-lJ2 -n, i2 -n»)11Ir

~2a-lIIX!o,Sllly Ill.t ri(u)aiTXW_l) 2- n,i2- nj ll du ° 1= 1 Y

~ 211 T lla -lllx!o, slily /litl aiXW-l)2 -n, i2 -nt

~ 2NIITlla- l IIX!O,Sllly Ilgllx '

This proves our assertion for functions g as above. For a general step function of the form

N2 n

f= L aiX[(i-ljN-12-n,iN-12-nj i= 1

we split the interval [0, IJ into [N2nm-lJ+I(~Np-l+l) intervals each having measure ~mN-12-n and then use the estimate established above in each ofthese intervals. It follows that Ilflly~Dllfllx, where D=2NIITlla- 11IX!o,Sllly(Nß- 1 + I) is independent of n. This completes the proof since the step functions f as above, with {a;}f:~ and n arbitrary, are den se in X. 0


Prool 01 2.e.l. Let To be an isomorphism from Xinto Yand, foreverymeasurable subset E of [0, 1] with Jl(E) >0, let XE denote the subspace of X of all functions supported by E. Let 'tE be an invertible transformation of [0, 1] onto E so that Jl('ti 1 lJ) = Jl(lJ)/Jl(E), for every measurable subset lJ of E. Then the mapping I ~ j( 't i 1) defines an order isomorphism SE from X onto XE'

We distinguish now between two cases.

Case l. There exist a measurable subset E of [0,1] with Jl(E) >0, a transformation 'tE as above and numbers s> land R< 00 so that if, for n = 0, I, 2, "', we put

then

inf Ilz:XC7~lly>O. n

In this case assertion (i) of 2.e.l holds. This is proved by applying 2.e.2 to the operator T= TOSE' In order to verify that T satisfies the hypotheses of 2.e.2, fix q <p < 2 and let {gn}:'= 1 be a sequence of identically clistributed independent p-stable random variables so that Ilglllq = 1. The properties ofthe p-stable random variables described above imply that the functions

satisfy

Ilwn ll y=llgll111 II! IJl gi(W)TX[(i-lJ2-n ,i2-n )1 dV(W)lIy

:;;; IlgI11 11 11TII! IIJl gi(W)X[(i-l)2- n ,i2-n )1Ix dv(w)

:;;;CllgI111111TIIÜ IIJl gi(W)X[(i-l)2- n ,i2-n )lI: dv(w)y/q

(1 ( 2n )q/P )l/q

=cllgllll11lTII r i~l IX[(i-l)2- n ,i2-n )IP dt

=Cllgllll11I TII·

Let y and lJ have the same meaning as in the statement of 2.e.2. Then, by l.d.2 n n

and the fact that (1: c lJn for all n, we get that


i.e.

inf IIYnxö.lly>O. n

Case Il. This is the case when the assumptions of Case I are not satisfied. In this case we construct inductive1y a system ofvectors {X~} ~ 1 in X, which is obtained from the Haar system {xn}~ 1 by a suitable automorphism of [0, 1], and a family {h,n}?= 1,:'= 1 of elements of Y such that, for each n,

(a) IIJ;, n - ToX;;lIx; IIx 11 y < 1/2i + 211 Tö 111, i = 1,2, ... , n. (b) {J;, n}? = 1 are pairwise disjoint. (c) J;,n+1=J;,nlsuPPfi,n+l' i=I,2, ... ,n.

Once this construction is completed, we put J; = lim J;, n (the limit exists since Y is n---oo

order continuous by l.a.5, l.a.7 and the conditions imposed on Y) and observe that {J;LOO= 1 is a sequence of mutually disjoint functions in Y which is equivalent to {X: /lIx:llx} i~ l' This, of course, shows that in Case 11 assertion (ii) of 2.e.l holds.

The possibility of constructing inductively the {X~}~ 1 and {J;,n}?= 1, ~ 1 as above follows immediately from the next lemma.

Lemma 2.e.3. Under the hypotheses oJ Case II above, if {J;}?= 1 is a sequence oJ pairwise disjoint Junctions in Y then,Jor every I: > ° and every measurable subset E oJ [0, 1] with j,t(E»O, there exist a sequence {gJ?; t oJpairwise disjoint elements oJ Y satisJying IIJ; -gilly< I: and gi = J; I SUPP9i,Jor i= 1,2, "', n, and a partition oJ E into two disjoint measurable subsets EI and E 2 , each oJ measure j,t(E)/2, so that

Proof We have already pointed out that the conditions imposed on Yensure that it is an order continuous lattice. Actually, by I.f.7 and I.f.12, Y is even r-concave for some r < 00. It follows that, for every I: > 0, there exist reals s ~ 1 and p > ° such that

lIJ;xally< 1:/2, i=I,2, ... ,n,

whenever ac es, 00) or j,t( a) < p. Furthermore, by the assumptions characterizing Case 11, for every sub set E of [0, 1] with j,t(E) > 0, there exist an integer m and a subset D c [0, s] with j,t(D) > s- p so that

where z; is the expression appearing in the definition of Ca se 1, M the r-concavity constant of Yand Br the Khintchine constant in Lr(O, 1). (Use the fact that, e.g. by 1.f.14, sup IIZ;lIy<oo and thus if R is large enough j,t(a;»s-p for every m.)

m


Let SE have the same meaning as in the proof of Case I and set

2 m - 1

h1(u, t)= L ri(U)SEXW-1)2-m,iz-mj(t) i= 1 2 m - 1

hz(U, t)= L ri(U)SEXHzm-l+i-1l2-m,(2m-l+i)2-mj(t). i= 1

Then, by the r-eoneavity of Yand Khintehine's inequality, we get, forj= 1,2, that

G IIXbTohjll~ du y/r ~ MIIG IXbTOhX du y/rll y

~MBrIIXb(~ IToSEXW-1)2-m, i2- mjlz) l/zL =MBrllXbZ;;IIy<e/4.

Thus, it is possible to find an Uo E [0, 1] so that

simultaneously forj = 1 andj= 2. We are now able to define the functions {gi}7: I. Put 9i=hXb' for i=I,2, ... ,n, and gn+1=(l-Xb)Toh, where h(t)=h1(uo,t}h2(uO' t). It is easily seen that {g;)7: f are mutually disjoint funetions in Y with g ·=r. 1 and

I J i suppgi

for i = 1,2, ... , n, sinee ,u([0, s] ",J) < p. Moreover, by its definition h(t) takes only two values, namely + 1 on a subset E 1 and -Ion a subset E2 , eaeh having measure Il(E)/2. This eompletes the proof sinee

Corollary 2.e.4. Let Y be an r.i. function space on [0, I] which does not contain uniformly isomorphic copies of Ir:., for all n. If Y eontains a subspace isomorphie to L 1 (0, I) then Y itself eoineides, up to an equivalent norm, with L 1 [0, 1].

Proo! Sinee the Haar basis of L 1 (0, I) is not uneonditional, assertion (ii) of 2.e.l eannot hold in the present ease. Henee, there is a D<oo so that IIflly~Dllflll' for every fE L 1(0, 1). Sinee IIfll1 ~IIflly, for every f, (by 2.a.l) this eoncludes the proof. 0

N. J. Kalton [136] has independently proved 2.e.4 under the less restrietive assumption that Y does not eontain an isomorphie eopy of co'

A weaker version of 2.e.1 was proved in [78] for the ease when both X and Y are Orlicz function spaees on [0, 1]. More precisely, it was shown there that if


LM(O, 1) is isomorphic to a subspace of LN(O, 1) and ßN, 00 <2 (the indices rxN, 00 and ßN,oo have been defined in 2.b.5 and, as proved there, they coincide with the Boyd indices p LN' respectively q LN; moreover, the interval [rxN, 00' ßN, 00] is the set of all numbers r for which the unit vector basis of Ir is equivalent to a sequence of mutually disjoint functions in LN(O, I)) then N(t) ~DM(t), for some constant D< 00 and every t ~ 1. This result was used in [78] to show that if LM(O, I) ~ LN(O, I) and I < rxN, 00 ~ ßN, 00 < 2 then M and N are equivalent at 00 i.e. LM(O, 1) =

LN(O, I), up to an equivalent norm. The proof of this fact is obvious once we show that also ß M 00 < 2. In view of the reflexivity of LN(O, I), which follows from the fact that I < ~N, 00 and ßN,oo < 00, and the duality relation between rxN*, 00 and ßN, 00

this is equivalent to showing that 2< rxN* 00 implies that also 2< rxM * 00' This later assertion is a consequence of I.c.lO an'd the characterizations or' the intervals [rxN" 00' ßN', 00] (respectively [rxM*, 00' ßM*, 00])' Indeed, by I.c.lO, every symmetrie basic sequence in a Köthe space 01 type 2 is equivalent either to a sequence 01 disjointly supported lunctions or to the unit vector basis in 12 , Hence, if rxN" 00 > 2 then any symmetric basic sequence in LN,(O, 1) is equivalent to the unit vector basis of an Orlicz sequence space and Ip is isomorphic to a subspace of LN*(O, 1) if and only if p = 2 or pE [rxN*, 00' ßN*, 00].

Wehave described in some detail the preceding argument since it indicates how to proceed in order to prove the uniqueness of the r.i. structure on [0, 1] of an r.i. function space X on [0, 1] which is reflexive and q-concave some q<2: passing to the dual X* of Xwhich, by I.dA, is r-convex with l/r+ l/q= I, we would have to check whether any r.i. function space Z on [0, I], whieh embeds isomorphically into X*, is also r-convex or it is isomorphie to L 2(0, 1) (recall that, by remark 2 following 2.b.3, the q-concavity of X implies that 11/11x~ Cll/llq, for some constant C< 00 and every I E LiO, 1)).

The preceding remarks lead us naturally to the study of r-convex r.i. function spaces for r ~ 2. The study of such spaces in facilitated by the fact that symmetric basic sequences in such spaces can be fully described. We have already seen above that l.c.lO gives much information on infinite symmetric basic sequences in such spaces and that such sequences have a relatively simple form (e.g. in LiO, 1), 2 <p < 00, they are equivalent to the unit vector basis in either 12 or Ip ). Let us point out that, for spaces whieh are not r-convex for r ~ 2, the situation is considerably more complicated. As shown in the beginning of this section, if 1 ~r<p~2 then Ip is isometric to a subspace of L,(O, 1). There are even symmetrie basie sequences in Lr(O, I) with I < r < 2 which are not equivalent to the unit vector basis of an Orlicz sequence space. We shall treat this question in detail in Vol. IV. For our purposes here we need a quantitative description of all the finite symmetrie basie sequences in a Banach lattice of type 2 (i.e., by l.f.17, a Köthe space whieh is 2-convex and q-concave for some q< (0). Such adescription cannot be derived from l.c.I0. The following result (cf. [58] Section 2) has, despite its somewhat complicated statement, several interesting consequences.

Theorem 2.e.5. For every K~ I, M~ 1 and every integer m there exists a constant D=D(K, M, m) so that if {X;}?~l is a K-symmetric normalized basic sequence 01 finite length in a Banach lattice X, which is 2-convex and 2m-concave with both


2-eonvexity and 2m-eoneavity eonstants ~M, then, for every ehoiee of sealars

{ai}~= l'

D- 111it1 aixill ~max {( ~ lIiY1 la~i)xiI1l2m I n ,Y/2m, W n(t1 lail2 Y/2}

~D lIit1 aixill '

where wn = Itt1 xilll n1/2 and ~ refers to summation over all the permutations 1t of

the integers {I, 2, ... , n}.

Before proving 2.e.5, we present some of its consequences. For instance, if X=Lp(O, 1),p>2 then, for every choice ofscalars {ai}~=1' we clearly have

On the other hand, since {x;}~= 1 is also K-unconditional and Lp(O, 1) is ofcotypep, we get that

Combining these two estimates with the statement of 2.e.5 we 0 btain the following result (for which a simple direct proof can be found in [58] Section I).

Theorem 2.e.6. For every K> I andp>2 thereexistsa eonstant D=D(K,p)so that, for every K-symmetrie normalized sequenee {Xi}~=1 in Lp(O, 1) and every ehoice of sealars {aJ~= l' we have

This theorem can be used to characterize those r.i. function spaces which embed isomorphically into Lp(O, 1),p>2.

Theorem 2.e.7. Let X be an r.i. funetion spaee on an interval I, where I is either [0, 1] or [0, (0), and suppose that X is isomorphie to a subspaee of Lp(O, I), p > 2. Then, up to an equivalent norm, Xis equal to either L/I), L 2(I) or L/I)nL2(I).

Clearly, when 1= [0, I] only the first two possibilities are of interest since LiO, 1)nL2 (O, 1)=LiO, 1).


Praa! Let T be an isomorphism from X into Lp(O, 1), p> 2. We treat first the case 1=[0,1]. Since, f()r every n, the sequence {TX[(i-l)2-n,i2-n)};:l is K-symmetric with K~ !lT!I !lT-111 it follows from2.e.6 that there exists a constant D < 00 so that

D -lllJl aiX[(i-l)2 -n'i2- n)1Ix

~max {IIX[o'2-n)IIxC~l laiIPY/P, C~l lai I2/2"Y/2}

~ D 11 i~l aiX[(i_1)2 -n, i2 -n)lIx '

for every n and every choice of scalars {ai};;l' Hence, for any simple function f which is measurable with respect to the algebra generated by the intervals [(i-I)2-n, i2-n), i= 1,2, ... ,2n , we get that

where IXn=2"/Pllx[O,2- n)llx' Since O~lXn~D for all n (set f= I in the above inequality) we deduce that, for any simple functionf over the dyadic intervals,

where IX = lim inf IXn • This completes the proof for the case 1 = [0, 1 J since, up to an n -+ 00

equivalent renorming, X is L 2 (0, I) if IX = ° or L/O, I), otherwise. The case 1=[0, (0) can be deduced from the previous one. Fix s ~ 1 and

observe that the map f(t) ----> f(t/s)llxro,sJllx I induces an order isometry U between an r.i. function space on [0, IJ and the restriction Xs of X to the interval [0, s). This isometry has the additional property that there exist numbers ßs and Ys' depending only on s, so that, for every simple functionfon [0, IJ, we have

Thus, it follows from the case 1=[0, I J that

for every simple function gwhich is supported by [0, sJ and for some ° ~lXs ~D. Passing to lim inf as s ----> 00, we get that there are non-negative constants ß and y such that

for every simple function 9 on [0, (0) which has a bounded support. The alternatives y = ° and ß = ° yield that, up to an equivalent norm, X is equal to L/O, (0),


respee1ively L2(O, (0). When both ß and y are striet1y positive we get that X is LiO, (0)nL2 (O, (0), up 10 an equivalent nonn. 0

Remark. The spaee LiO, 00 )nL2 (O, (0) ean be realized as the subspace of all the pairs (f, j) E LiO, (0) EB L2(0, (0). This proves that Lp(O, 00 )nLzC0, (0) is aetually isomorphie to a subspace of LiO, I) for every I ~p~ 00 sinee LiO, (0) EB L 2(0, (0) is isomorphie 10 a subspace of LiO, (0). In Seetion 2.f it will be shown that Lp(O, 00 )nL2(0, (0) is even isomorphie to LiO, (0) if 2 ~p < 00.

Corollary 2.e.8. (i) For every I ~p ~ 00, the spaee Lp(O, I) has a unique representation as an r.i.junetion spaee on [0, I] i.e. every r.i.junetion spaee Yon [0, I], whieh is isomorphie to L/O, I), is already equal to Lp(O, 1), up to an equivalent renorming.

(ii) For every 1 <p< 00, p =f. 2, the spaee L/O, 00 ) has exaetly two representations as an r.i. junetion spaee on [0, (0), namely L/O, (0) and L/O, 00 )nL2(0, (0) if p > 2 or Lp(O, (0) and Lp(O, (0) + L2(0, (0) if 1 <p < 2. The spaees L 1 (0, (0), L 2(0, (0) and Loo(O, (0) have a unique representation as r.i.junetion spaees on [0, (0).

Proof Both assertions follow in the ease 2<p< 00 from 2.e.7 and the preeeding remark. The ease 1 <p<2 is obtained by duality. That L1(O, I) has a unique r.i. strueture on [0, I] follows immediately from 2.e.4.

Suppose now that Y is an r.i. funetion spaee on [0, I] whieh is isomorphie to Loo(O, I). Sinee the images in Loo(O, I), under isomorphism, of the subspaees [X[(i-l)n-I.in-I)]?=l' n=I,2, ... of Yare unifonnly eomplemented in Loo(O, I) it follows from the argument used to prove the uniqueness of the uneonditional basis of eo (cf. 1.2.b.9) that there exists a eonstant K< 00, depending only on the distanee between Yand Loo(O, 1), so that, for every n and every ehoiee of seal ars

{aJ?=l'

Henee, K- 1 ~ IIX[O,n-I)lly, for all n, and this shows that Y is equal to Loo(O, I), up to an equivalent norm. Similar arguments ean be used to show that L 1 (0, (0), L 2(0, (0) and Loo(O, (0) have a unique r.i. strueture on [0, (0). 0

We now present 1he prooj oj 2.e.5. Sinee {X;}?=l is a K-uneonditional basis there exists a new norm 111·111 on [x;J?= 1 so that [x;J?= 1 endowed with this norm beeomes a Banaeh Iattiee and

By I.f.I7, this Iattiee is 2-eonvex with 2-eonvexity eonstant Mo depending only on K, M and m. Henee, by the argument used in the proof of the assertion of remark 2


following 2.b.3, we obtain

Ilitl aiXill~K-IIIIJl aiXilll~K-IMo-lCt la;j2Y/211litl xilll/nl/2

~K-2M(;t wnCtl1ail2Y/2 ,

for every choice of scalars {a;}f=l' On the other hand, it is easily seen that, for any permutation 1t ofthe integers {I, 2, ... , n} and any {a;}f= 1 as above,

IIJl aixill~K-l I Ilitl an(il'i(U)Xill dU~K-lll! litl an(il'i(U)Xil dull

~K-tYl lan(i)Xdll·

This proves the right-hand side inequality of 2.e.5. The proof of the opposite inequality is more difficult. Fix the scalars {a;}~=l' By l.d.6, there is a constant C, depending only on m and M, so that, for any permutation 1t of {I, 2, ... , n}, we have

IIJl aiXill~Kllitl an(i)Xill~KCIICtl lan(i)xd2Y/211

= KC II( (tl lan(i)xi 12 r) 112m"

=KC 11 (tl ml+ .. ~m,=m it,.~'i' lan(il)XiJml ... lan(illXi,12m}/2mll· mIt'" t m, ~ 1 distinct

Hence, by averaging in the sense of li~ over all possible permutations 1t of the integers {I, 2, ... , n} and by separating the term corresponding to 1= m from the other terms, we get that

where

Sl = (I 11(. I. lan(itlXitI2 ... lan(im)Xim 12)1/2mI12m)1/2m '11: It, ... ,lm

distinct


and

S2=( ~ /letll ml+ .. ~m,=m.it,~.'i' la,,(il)xilI2ml ... la,,(idxi,12m)I/2mI12mY/2m

ml •.•• fm,~l distinct

Suppose now for simplicity of notation that n is an even integer such that nI2>m. We first evaluate the expression SI by using the 2m-concavity of X.

distinct

distinct

distinct distinct

Since n - m + 1 > n - nl2 + 1 > nl2 it follows that

In order to evaluate the expression S2' we fix 1 ::;;1 <m and ml, ... , m, and put

S(/; ml , ... , m,) =( ~ IIC, .~, i, la7t(il)XilI2ml .. . la"(i dXi,12m) 1/2mI12mY/2m .

distinct

( n )1/2m j

Let ()j=m)m and zj(n)= i~lla"(i)xiI2mj ,j=I,2, ... ,1. Then, by l.d.2(i)

applied in the Banach lattice I~~(X) for the vectors Zj=(zj(n))" and {)j as above, we get that

Since l<rn at least one of the integers ml, ... , m" say ml, is ~2. Therefore, by


estimating from above the norm in I;ml by that in I~ and the norm in I;m j' 1 <j:::; I by that in I;, it follows that

Note that, by l.d.2, we have that

and consequently,

( II n 11 2m )1/2m where R= ~ i'!lla"(i)Xil In!

Combining the preceding estimates we get that there exist finite constants Cl = C1(K, M, m) and C2 = C2(K, M, m) so that

Il i aiXill:::;C1max{(n!)-1/2m max s(/;m1, ... ,m/),wn(t laiI2)1/2} i=l l"/<m .=1

ml,· ... ml

and therefore

In order to present further applications of 2.e.5, we have to study the I;~ average appearing in its statement.

Lemma 2.e.9. Let X be an r-convex Banach lattice with r-convexity constant :::;M, Jor some r>1 and some M<oo. Let q~r and let {XJ?=l be afixed sequence oJ vectors in X. Then, Jor every 1 < k:::; n, the Jormula

with L meaning summation over all permutation 7t oJ {1, 2, ... , n}, defines a norm " on R k such that R k , endowed with this norm and the pointwise order, is an r-convex


lattice whose r-convexity constant is ~ M. The same assertion holds if r-convexity is replaced by the existence of an upper r-estimate.

Proof Sinee a direet sum in the sense of lq, q ~ r of a sequenee of r-eonvex Banaeh lattices, eaeh with r-eonvexity eonstant ~ M, is clearly r-eonvex with r-eonvexity eonstant ~ Mit suffiees to prove the assertion for a norm on R k having the form

Let aj=(a{, ... , aD,j= 1, 2, ... , m be a sequence ofveetors in R k • Then

IIle~l lailry/rillo = Ili~l et la1xJy/rll ~ Ilet i~l la1xJy/rll

~Me~l Ili~l la/xdIDl/r =Me~l IIlailll~y/r The ease when X satisfies an upper r-estimate is treated in the same manner. 0

We are prepared now to present the result needed in addition to 2.e.I for proving the uniqueness ofthe r.i. strueture.

Proposition 2.e.l0 ([58] Section 2). Let Y be a Danach lattice of type 2 which is r-convexfor some r>2. An r.i.function space X on [0, 1], which is isomorphic to a subspace of Y, must also be r-convex unless it is equal to L 2(0, 1), up to an equivalent renorming.

Proof Sinee a Banaeh Iattiee of type 2 is, by 1.f.I3, also q-eoneave for so me q< 00 there is no Ioss of generality in assuming that Y is 2m-eoneave for so me integer m. Let M be a number exeeeding both the r-eonvexity and 2m-eoneavity eonstants of Yas weH as its type 2 eonstant. Let T be an isomorphism from X into Yand put K=max {IITII, IIT-1II}·

S· l" • {T }2 n • K 2 . mee, lor every mteger n, x[(i _ 1)2 - n, i2 - n) i = 1 IS a -symmetne sequenee in Y it foHows from 2.e.5 that there exists a eonstant D, depending only on K, M and m, so that

D -lllJl ai TX[(i-l)2 -n, i2 -n)11 y

~ max {(~ 11 i~l lan(i)TX[(i_1)2 -n, i2 _n)lll:m/(2n) !Y/2m, C~l lai 12 /2n Y/2}

~DIIJl ai TX[(i_1)2-n,i2- n)lly ,


for every choice of scalars {a;}f~l' In other words, for any simple function of 2n

the formJ= L aiX[(i-lj2-n,i2-n), we have i~ 1

(*) (DK)-IIIJllx :::;max {( ~ Ili!1 la1t(i)TX[(i-1)2 -n,i2 -n) 111:m j(2n) Y/2m, 11/112}

:::;DKll/llx·

The idea of the proof is to show that if X is not L 2(O, 1) then the norm of any simple function 1 as above, which is supported by a sufficiently small interval, is given by the first term of the inner expression in (*) which, by 2.e.9, defines an r-convex norm. We distinguish between two possible alternatives, as folIows.

Case I. Suppose that, for every n, we have

Then, for every simple functionJas above,

On the other hand, it follows from (*) that

1I/IIx ~ (DK) - 111/112 which shows that Xis, up to an equivalent renorming, equal to L 2 (O, 1).

Case II. There exists an integer k such that

In this case, it is clear that when we evaluate the norm of

2 n - k

X[O,2- k )= L X[(i-lj2-n,i2-n), n~k, i= 1

by using formula (*) then the maximum in the inner expression is necessarily attained in the first term, i.e.

(DK) -lllx[o, 2 -k)llx:::; (~ II:Ylk ITX[(1t(i)-1)2 _ n, 1t(i)2 _ n)lll:m j(2n)!) I/2m

:::;DKllx[O,2-k)llx'


By 2.e.9, tor every n ~ k, the expression

where L means summation over all permutations 11: of {I, 2, ... , 2"}, defines an " r-convex symmetric norm on R 2n - k with r-convexity constant ~M. Thus, by

remark 2 following 2. b.3, we get that

for every a E (R 2n-\ 111·111). It follows that, for any simple function of the form

2 n - k

t/I= L biX[(i-l)2-n.i2-n), Le. supported entirely by the interval [0, 2- k), we have i= 1

This inequality shows that if we restrict ourselves to simple functions t/I as above the maximum in the inner expression of (*) is always attained by the first term. In view of 2.e.9, it follows that the restriction of X to the interval [0, r k ) is r-convex. This, of course, implies that Xis r-convex too. . 0

We state now the result on the uniqueness ofthe r.i. structure (cf. [58J Section 5). As in 2.e.8, we shall say that an r.i. function space X on an interval I has a unique representation as an r.i. function space on I if any r.i. function space Yon I, whieh is isomorphie to X, is already equal to X, up to an equivalent norm.

Theorem 2.e.ll. Every r.i.function space X on [0, IJ, which is q-concavefor some q<2, has a unique representation as an r.i.function space on [0, 1].

Proof We present here a simplified proof which requires however the additional assumption that X does not contain uniformly isomorphic copies of lf for all n.

Suppose that an r.i. function space Y on [0, 1 J is isomorphie to X. By 1.f.12, both X and Y satisfy an upper p-estimate for some p > 1 and, by l.d. 7, Yis 2-concave. Thus, by 1.f.18, their duals X* and Y* are r.i. function spaces on [0, I J of type 2 and, by l.d.4, X* is r-convex for r satisfying l/r+ l/q= 1. It follows from 2.e.l0


that also Y· is r-convex (Y· cannot coincide with L 2(0, 1) since it is isomorphic to an r-convex lattice) and, therefore, Y is q-concave, too. By remark 2 following 2.b.3, there is a constant M such that

for every fE Lq(O, I). The Haar basis of X or of Y cannot be equivalent to a sequence of mutually disjoint functions in Y, respectively X, since it contains the Rademacher functions as a block basis and this would contradict the q-concavity of X and Y. This means that assertion (ii) of 2.e.1 does not hold for any isomorphism from X into Yor, vice-versa, from Yinto X. Consequently, assertion (i) of2.e.l is satisfied by any such isomorphisms and this, obviously, completes the proof. 0

Remarks. 1. It is interesting to compare the behavior of r.i. function spaces on [0, I] with that of spaces having a symmetric basis from the point of view of the uniqueness of the symmetric structure: in 1.4.c, we presented several spaces with a non-unique symmetric basis and it can be easily checked that those examples can be constructed as to be, for instance, q-concave for any 1 < q < 2 given in advance.

2. As we have seen in 2.e.8, even the spaces LiO, 00), 1 <p #- 2 < 00 do not have a unique representation as an r.i. function space on [0, 00). Additional information on this matter as well as on the class of spaces which have simultaneous representations as r.i. function spaces on both [0, 1] and [0, 00) will be given in 2.f.

The proof of 2.e.ll in the general case is more complicated. Actually, it is a particular case of a more general result from [58] Section 5 which is reproduced here without a proof.

Theorem 2.e.12. Let X be an r.i. funetion spaee on [0,1] whieh is q-eoneave for some q<2. Then every r.i. funetion spaee Yon [0,1], whieh is isomorphie to a eomplemented subspaee of X, is equal, up to an equivalent norm, to either X or L 2(0, 1).

The results presented above apply mostly to r.i. function spaces X sitting on "one side" of 2. These theorems can be extended to arbitrary r.i. function spaces but, since their proofs are too complicated to be presented in the book, we shall limit ourselves to the presentation of the statements. The following theorem is (almost) a generalization of2.e.1. (The word almost is used since in 2.e.l we did not assumepx> 1.)

Theorem 2.e.13 ([58] Section 6). Let X be a separable r.i.funetion spaee on [0, 1] sueh that I <Px and qx< 00. Let Y be an r.i. funetion spaee on [0, 1] or on [0, 00) whieh does not eontain uniformly isomorphie eopies of I:, for all n. If X is isomorphie to a subspaee of Y then either


(i) there exists a eonstant D< 00 so that

Jor every J E X, or (ii) the Haar basis oJ Xis equivalent to a sequenee oJ pairwise disjoint Junetions

in Y,or (iii) Xis equal to L 2(0, 1), up to an equivalent renorming ..

In view of2.c.14 and the observation made thereafter it follows that the alternative (ii) of 2.e.13 cannot hold if Y is an Orlicz function space (unless, of course, Xis L 2(0, 1)). We also recall that, for an Orlicz function space X, the conditions 1 <Px and qx< 00 are equivalent to the reflexivity of X.

CoroUary 2.e.14 ([58] Section 7). Let X be a separable r.i.Junetion spaee on [0, 1] whieh is different Jrom L 2(0, 1), even up to an equivalent renorming. Let M be an Orliez Junetion satisJying the .1 2-eondition both at ° and at 00.

(i) If 1 <Px and qx < 00 and X is isomorphie to a subspaee oJ LM(O, 00) then

IIJIILM(O.l)~DIIJllx , Jor some D< 00 and every J E X.

(ii) If X is isomorphie to a eomplemented subspaee oJ LM(O, (0) and LM(O, (0) is reflexive then, up to an equivalent norm X =LM(O, 1). In partieular, any reflexive Orliez Junetion spaee on [0, 1] has a unique representation as an r.i. Junetion spaee on [0, 1].

Let Xbe a uniformly convexifiable r.i. function space on [0, 1] and suppose that another r.i. function space Yon [0, 1] is isomorphie to X. By the discussion preceding l.e.4, 2.b.7 and 2.b.2, both X and X* have non-trivial Boyd indices (Le. 1 <Px,Px* and qx, qx* < (0). Thus, it follows from 2.e.13 that if Xis not equal to Y, up to an equivalent norm, then the Haar basis of X or of X* must be equivalent to a sequence of pairwise disjoint functions in Y, respectively, Y*. However, it can be shown with some additional effort that the following generalization of 2.e.11 is true (cf. [58] Section 6).

Theorem 2.e.15. Let X be a uniformly eonvexifiable r.i. Junetion spaee on [0, 1]. Then either X has a unique representation as an r.i. Junetion spaee on [0, I] or the Haar basis oJ Xis equivalent to a sequenee oJ mutually disjoint Junctions in X.

Remark. In Volume I V we shall present, for every 1 <p< 2, an example of an r.i. function space X on [0, I] which embeds isomorphically into L/O, 1), does not have a unique representation as an r.i. function space on [0, 1] (actually, it has uncountably many mutually non-equivalent such representations) and the Haar basis of X is equivalent to a sequence of pairwise disjoint elements in X.


We have seen in 2.d.5 that a complemented subspace Y of a separable r.i. function space X on [0, I] whose Boyd indices are non-trivial is isomorphie to X provided that Y contains a eomplemented copy of X. The complementation of X in Y is actually redundant if we exclude the case when the Haar basis in Xis equivalent to a sequence of pairwise disjoint vectors in X (the example mentioned in the remark above will also show that we have to exclude this case). This is a consequence of the following theorem from [58] Seetion 9 whieh, again, is stated without a proof.

Theorem 2.e.16. Let X be an r.i.funetion spaee on [0, I] sueh that

(a) 1 <Px' (b) X is q-eoneave for some q< 00, and (c) the Haar basis of X is not equivalent to any sequenee of disjointly supported

funetions in X. Then every subspaee of X, whieh is isomorphie to X, eontains a further subspaee whieh is eomplemented in X and still isomorphie to X. In partieular, the theorem is validfor L/O, I) spaees, I <p < 00, or, more generally, (by 2.e.14)for any reflexive Orliez funetion spaee on [0, 1].

The conclusion of 2.e.16 is valid also for X =L1(0, I) but this fact, which is due to Enflo and Starbird [38], does not follow from the general assertion of 2.e.16.

The following corollary is an immediate consequence of 2.d.5 and 2.e.l.

Corollary 2.e.17. Let X be an r.i. funetion spaee on [0, I] whieh satisfies the eonditions (a), (b) and (c) of 2.e.16. Then every eomplemented subspaee Y of X, whieh eontains an isomorphie eopy of X, is already isomorphie to X.

We conclude this section by presenting a result proved independently in [30] for subspaces of L/O, 1), 2<p< 00, with an unconditional basis and, later on, in [58] Section 2 for general Banach lattices. An interesting thing about this theorem, which actually generalizes 2.e.1O to the case when X is an arbitrary lattice, is the fact that, though its statement has nothing to do with r.i. spaces or other symmetrie structures, the proof from [58], which is reproduced here, is based on the characterization of symmetrie basie sequenees in a Banaeh lattiee of type 2, presented in 2.e.5.

Theorem 2.e.18. Let Y be a Banaeh lattiee of type 2 whieh satisfies an upper r-estimate for some r> 2. Then every laltiee X, whieh is isomorphie 10 a subspaee of Y, satisfies itself an upper r-estimate or it eontains uniformly isomorphie eopies of l~ on disjointly supported veetors for all n.

Proof Let {XJ7=1 be a sequence of mutually disjoint vectors in X such that n

L Ilxi 11' = n. For every I ~ i ~ n, eonsider the sequence Xi = (x"(i)/(n !)1/')", where 1t i= 1

ranges over all permutations of {I, 2, ... , n}, as an element of 1;!(X). Since, for arbitrary values of i andj, the veetors Xi and X j consist actually of the same sequence


of vectors in X arranged in a different manner it follows that {~;}7= 1 forms asymmetrie basie sequence in I:!(X) (with symmetrie eonstant equal to one). Observe also that the faetor (n !)1 /' appearing in the definition of the ~/s has been chosen to ensure that II~i" = 1 for all 1 ~ i ~ n.

Sinee I;!(Y) is also of type 2 and, therefore, 2m-eoneave for some integer m and sinee there clearly exists an isomorphism t from I:!(X) into I;!(Y) we ean apply 2.e.5 and conclude the existenee of a constant D, independent of the x/s, so that, for every ehoice of sealars {a;}7= l' we have

(*) D -ttl ai~i 11 ~ max { ( ~ 11 i~l lan(i)t~i 1112m/n Y12m, Wn(tl la;j2 Y/2}

~Dllitl ai~ill,

where Wn = Ilitl ~i II/n 1/2 .

Suppose now that x does not eontain uniformly isomorphie copies of I~ on disjoint veetors, for all n. Then the 2-eoneavifieation X(2) of X does not eontain uniformly isomorphie eopies of I; on disjoint veetors, for all n, and thus, by I.f.12, X(2) satisfies a non-trivial upper estimate. It follows that X, whieh is order isomorphie to the 2-eonvexifieation of X(2)' satisfies an upper p-estimate for some p>2. (We may assurne thatp<r; ifp~r there is nothingmore to prove.) Thus also I:I(X) satisfies an upper p-estimate and, therefore, for sufficiently large n we have wn < 1. For sueh n it is possible to ehoose integers hand k so that 1 ~ hw;,/(,- 2) < 2 and kh~n«k+ l)h.

Let M be the (joint) upper r-estimate eonstant of Y and I;!(Y) and, for j = 1, 2, ... , k, put

jh

uj = L ~i' i=(j-l)h+ 1

By (*) and 2.e.9, we get, for every 1 ~j~k, that (assuming, as we clearly may, that

M"tll~l)

"Uj" ~D max {(L 11. . V It~n(i)1112m/n !)1/2m, Wnh1/2} n I={J-l)h+ 1

~DM"tll max {C=(j~)h+lll~iIl'Y/', Wnhl/2 }

~DMlltll max {h1/" wnh1/2 }

and, in view of the eonditions imposed on h, it follows that hl/'~wnh1/2, i.e. IIUjll~DMIITllwnhl/2 for all 1 ~j~k. Since the {~i}i=l and therefore also the


{Uj }j= 1 are mutually disjoint vectors in 1:!(X) we get that

where Mi is the upper p-estimate constant of 1:'(X). On the other hand, again by (*), we have that

Hence, by the choice of hand k above,

where C is a constant independent of the choice of the sequence {Xi}~ = 1. This completes the proof since

In the special case when Y is an Lr(O, I) space, r> 2, Theorem 2.e.l8 can be restated in a stronger form.

Corollary 2.e.19. A Banaeh lattiee X, whieh is linearly isomorphie to a subspaee of L.(O, 1) for some 2<r< 00, is itself order isomorphie 10 an Lr(v) spaee for a suitable measure v, unless it eontains uniformly isomorphie eopies of li on disjointly supported veetors for all n.

Assertion 2.e.19 follows immediately from 2.e.18, the fact that a lattice X as above is of cotype rand l.b.13.

f. Applications of the Poisson Process to r.i. Function Spaces

The principal motivation for this section is to provide a proof for the claim made in 2.e.8(ii) that Lp(O, 1) is isomorphie to Lp(O, 00) + LiO, 00) or to Lp(O, oo)n LiO, 00) when 1 2 and, more generally, to study those r.i. function spaces on [0, I] which admit also a representation as an r.i. function space on [0, (0). As we shall see below, this class contains in particular all the r.i. function spaces on [0, I] whose Boyd indices are non-trivial. This representation of r.i. function spaces on [0, I] as suitable r.i. function spaces on [0, 00) is used

f. Applications of the Poisson Process to r.i. Function Spaces 203

later on in this seetion for proving that if t -11' belongs to some r.i. funetion spaee X on [0, 1] for some 1< r < 2 then X eontains an isometrie eopy of L,(O, 1). In partieular, L,(O, 1) is isometrie to a subspace of LiO, 1) if p<r<2. The section ends with a diseussion of Rosenthal's spaees Xp ,2 (which were introdueed in I.4.d) and proper generalizations of them in the eontext of arbitrary r.i. funetion spaees on [0, (0).

Considerations based on 2.e.13 show that if we want to find an r.i. function spaee :!( on [0, (0) whieh is isomorphie to a given r.i. funetion spaee X on [0, 1] then it is very natural to look for an :!{ whose restriction to [0, I] eoineides with X. The problem is, of course, to determine a way to define :!( on the entire half line [0, (0) so that it is isomorphie to X. We define:!( so that it behaves like L 2(0, (0) in the neighborhood of 00. More preeisely, the norm of a funetion J E :!{ will be equivalent to the expression

[J] ~= IIJ*x[o, dlx+ IIJ*X(1, oo)lb '

which, in general, is not a norm.

Theorem 2.f.l ([58] Seetion 8). Let X be an r.i.Junetion spaee on [0, 1] and let:!( be the r.i. Junetion spaee on [0, (0) oJ all the measurable J Jor whieh J*X[o, 11 E X andJ*X(l, (0) E L 2(0, (0), endowed with the norm

{ ( 00 ("+1 )2)112} IUII ~=max IIJ*x[o,llllx, n~o ! J*(u) du .

(i) If qx < 00 then X eontains a subspaee isometrie to an r.i. Junetion spaee :!( 0 on [0, (0) whieh, up to an equivalent norm, is equal to :!(.

(ii) IJ 1 <Px and qx < 00 then f![ is isomorphie to X.

Before proving 2.f.l we would like to make some eomments. First, we point out that 11·11 ~ is indeed a norm sinee both expressions inside the maximum defining 11·11 ~ ean be written as suprema in terms off F or instanee, it is easily verified that the seeond expression is equal to

where the supremum is taken over all partitions of [0, (0) into pairwise disjoint subsets {'1n}:'=O each having measure equal to one. It is also easily ehecked that

[J] ~/2~ IIJII~~ [J] ~,

for every JE :!C. For example, the left-hand side inequality follows from the fact


that, for any f E f!{, we have

( 00 n+ 1 )1/2 (00 )1/2 Ilf*X(1, 00)112 = n~l ! f*(U)2 du :::; n~l f*(n)2

( 00 (n+ 1 )2)1/2 :::; n~o ! f*(u) du

The actual form of the norm in f!{ is not really important and we may use the equivalent expression [']!'I for all practical purposes. The fact that II·II!'I is equivalent to [']!'I yields immediately that f!{ is a minimal or a maximal r.i. function space on [0, 00) according to whether Xis minimal, respectively, maximal.

Our approach to prove 2.f.l is to embed each of the subspaces Xn of f![ obtained by restricting f!{ to [n -1, n) into "independent" copies of X in X. A simple . 00 example of such an embedding is given by Sf(u, v) = L rn (u)f(n - I + v),

n= 1

0:::; u, v:::; I,fE f![, where {rn }:."'= 1 denote as usual the Rademacher functions and Sf is considered as an element in X([O, I] x [0, I]) (recall that X([O, I] x [0, 1]) is isometric to X and Ilgllx([o, 1] x [0,1 n = Ilg*llx)' It is easily verified, by using Khintchine's inequality, thatiffisa decreasingnon-negative element inf!{ then C -lllfll:::; IISfll:::; cllfll for some constant C independent off The difficulty arises if we consider general (i.e. not necessarily decreasing) functions fE f!{. In order to deal with such functions we need that not only {X[n-l,n)}:."'=l are mapped into independent random variables (as in the case with the S above) but that {X[Sj, tj )}~= 1 for any choice of disjoint intervals {[Sj' t)}~=l are mapped into independent random variables and that the distribution function of the image of X[s, t) depends only on t-s. These considerations lead us to replace S by a more sophisticated operator which is built by using a stationary random process {Nt}o';t<oo with independent increments. Let us first explain this probabilistic terminology. A random process {Nt} 0'; t < lfJ is just a family of measurable functions on some probability space. The process is called stationary if the distribution function of Nt - Ns depends only on t - s. The process is said to have independent increments if {Ntj - NsJY= 1 are independent random variables whenever {[Sj. tj)}~=l are mutually disjoint intervals.

We shall employ here the Poisson process which is one of the simplest and most important examples of a stationary process with independent increments. Let us recall that an integer valued random variable Non (Q, L, v) is said to have the Poisson distribution with parameter a > ° if

v({w E Q; N(w)=k})=e-aak/kl, k=O, 1,2, ....

The characteristic function of such a random variable (i.e. the Fourier transform of its density N) is given by

00 F~(x)= L eikx-~ak/kl=e~(e;X-l), -oo<x<oo.

k=O

f. Applications ofthe Poisson Process to r.i. Function Spaces 205

Since FaJx)Fp(x) = Fa. + p(x) it follows that if NI and N2 are independent random variables with the Poisson distribution with parameter 1X 1, respectively 1X2, then Nl + N 2 has the Poisson distribution with parameter IX l +1X2 . Consequently, ifwe consider the requirements that {Nt}o,"t<oo be a random process with independent increments and that Nt - Ns have a Poisson distribution with parameter lX(t - s) for some fixed IX> 0 and every 0 ~ s < 1 < 00 then, for every choice of a finite number of reals {/j}~=l' these requirements (conceming only {NtJ~=l) are mutually consistent.1t follows therefore from a general consistency theorem ofKolmogorov (see Doob [29] for a detailed discussion) that such a process (calIed the Poisson process with parameter IX) really exists on a suitable probability space (Q, E, v). In our case it is however simpler (and of advantage to the proof of 2.f.l presented below) to replace the general existence theorem by a concrete representation ofthe Poisson process on Q=[O, 1] x [0,1] with the usual Lebesgue measure.

Let {u k} ~ 0 be a partition of [0, I] into dis joint sets so that

and let {qJ j } 1= 1 be a sequence of independent and uniformly distributed random variables on [0,1] (a variable qJ is said to be uniformly dislribuled on [0, 1] if its distribution function is the same as that ofj(v)=v). Consider the function

00 k

Nt(u,v)= L Xak(u) L X[O,t)(qJ/v» , O~u,v,/~I. k= 1 j= 1

The family {Nt}O,"t,"l forms a Poisson process with parameter IX (restricted to O~ I~ 1) on [0, I] x [0, 1]. In order to verify this we have just to note (since the characteristic function of a random variable determines its distribution) that for every choice of 0= 10 < 11 < ... < Ih = 1 and reals {XI}~=l we have that

• 1 1 i L x,(N,,(u, v)-N,.-,(u, v»

J J e '=1 dudv. 00 • •

00 1 i L x, L x(lI_"'/l(cpiv» =jl(uo)+ L jl(uk ) Je 1=1 )=1 dv

k= 1 0

In order to define N, for every 1 ~ 0 we construct a sequence {Nin)} 0," I < 1 , n = 0, 1, 2, ... ofindependent copies ofthe family {N, }O,"I," 1 defined above and put, for every integer m and 0 ~ 1 < 1,

m-I

Nm+ t = L N<t) + N,m) . n=O


(In order to avoid conflicting notation we let NiO) be equal to the N, defined above forO~t~l.)

We are now prepared to give the prooJ oJ 2/.1. We start with assertion (i). Let {N;} ° ~ I< 00 and {Nt} ° ~ I < 00 be two independent co pies of the Poisson process with parameter 1. For ° ~s < t< 00, put

The mappings T' and TI! can be extended by linearity to linear operators defined on all integrable step functions on [0,00). We shall show that if qx< 00 then T= T'- TI! extendsto an isomorphismfrom!!C ontoasubspaceof X([O, 1] x [0,1]). In the language of probability theory (which we shall however not use below) TJ

00

is the so-called stochastic integral J J(t) deN; - N;') with respect to the symmetrie

° stationary process with independent increments {N; - Nt} ° ~ I ~ 00' (Recall that a random variable G is said to be symmetrie if G and - G have the same distribution.)

Step I. We shall first prove that the restrietion of T to the step functions on [0, 1] is an isomorphism. By the concrete representation of the Poisson process given above, we have that

00 k

N;(u, v)= L Xak(u) L X[O,tj(o/j(v» , k= 1 j= 1

for every t, u, V E [0, 1], where {a~}k=O is a partition of [0,1] into mutually disjoint subsets such that ll(aD=llek!, k=0,1,2, ... and {o/j}i=l a sequence of uniformly distributed independent random variables on [0, 1]. We shall use an analogous notation for N;'(u, v). It is readily verified that, for every step function gon [0, 1], we have

00 k

(T'g)(u, v)= L Xak(u) L g(o/j(v» , u, v E [0,1] . k= 1 j= 1

Observe now that if Ds denotes, as in 2.b, the dilation operator (acting on functions defined on the unit square, say via a fixed measure preserving isomorphism

k

between the unit interval and the unit square), then, for each k, XaL(u) L g(o/j(v» j= 1

k

has the same distribution function as DIl(ak) L g(o/j(v». Hence, j= 1

By using now the assumption that qx< 00, we can estimate the norm of T' g. Fix qx<q<oo and recall that there exists a constant Kq so that IIDsllx~KqSl/q, for

f. Application of the Poisson Process to r.i. Function Spaces

every 0 ~ S ~ I. It follows that

00

IIT'gllx~Kqllgllx L kj(ek!)l/q k= 1

from which we get

00

where K=Kq L kj(ek!)l/q. On the other hand, we also have k=l

Notice that, since T'g and T"g are independent, we have

and thus

Jl({(u, v); I Tg(u, V)I>A})~Jl({(u, v); IT'g(u, V)I>A and u E (To})

=e-1Jl({(u, v); IT'g(u, V)I>A})

Consequently,

207

Step Il. We shall now consider the action of T on an arbitrary integrable step function fon [0, (0). We note firstthat, by the properties of the Poisson process and the definition of T, the functions Tf and Tf* have the same distribution function. Thus, in order to get estimates on IITfll, we may assurne without loss of generality thatf=f*·

The sequence {T!"}:'=l' where!..=fX[n-l,n)' n=l, 2, ... is a sequence ofsymmetric independent random variables and therefore forms an unconditional basic sequence (with unconditional constant one) in every r.i. function space. Note also that, by 2.b.3, for every hEX

for some C, and that there is no loss of generality to ass urne that q was chosen to be ~ 2. Since Lq is of type 2 we get that


By Step I (applied in L q ) and the faet thatfis non-inereasing, we deduee that

Consequently,

( 00 (n+ 1 )2)1/2

~2Kllfx[O,1)1Ix+2KCBq n~o f f(u) du

~2K(l +CBq)llfll~ .

Similarly, sinee L 1 is of eotype 2 we get by Step I (applied to L 1) that

Also, by Step I (applied to X),

11 Tfllx ~ 11 Tf X[O, llllx ~ e - 211f X[O, llllx and henee

It is clear from the preeeding inequalities that if X (and thus g() is minimal then Textends to an isomorphism from g( into X. If X is a maximal r.i. funetion spaee the extension of T from the closure of the simple integrable funetions to all of g( is obtained as follows: Let f~ ° belong to g( and {hn }:'= 1 be an inereasing sequenee of simple integrable funetions whieh eonverges to f a.e. Sinee T' and TU are positive, the sequenees {T'hn}:'=l and {T"hn}~l are inereasing and norm bounded. Henee, sinee X has the Fatou property, they eonverge a.e. to some elements in X denoted by T'f and T''f, respeetively. It is easily verified that these elements do not depend on the ehoiee of {hn }:,= 1 and that T = T' - T" extends to a bounded operator on g(. Onee we know this, the eomputation done in Steps land 11 above is valid if fis an arbitrary element of g( and henee T is an isomorphism on g(.

In order to eomplete the proof of 2.f.l (i), put

Then g( 0= (g(, II'II~) is isomorphie to g( and isometrie to a subspaee of X. It remains to verify that g( ° is an r.i. funetion spaee on [0, 00). As already mentioned in this proof, if f is a step funetion on [0, 00) with bounded support then Tf

f. Applications of the Poisson Process to r.i. Function Spaces 209

and Tf* have the same distribution funetion and, thus, fand f* have the same 11·11 ~o-norm. It is easily dedueed from this, by an approximation argument, that Tf and Tf* have the same norms in X, for every fEX. This proves 2.f.1 (i).

The proof of 2f1 (U) uses the deeomposition method. It is evident that eaeh of the spaees X and X is isomorphie to its square and that fI eontains a eomplemented subspace isomorphie to X. Therefore, in order to prove that X and X are isomorphie, it suffiees to show that if, in addition, I <Px then the image Zx of:r under T is a eomplemented subspaee of X([O, I] x [0, 1]). This is proved by interpolation as follows.

Let Zr denote the spaee Zx when X=Lr(O, 1) and let P be the orthogonal projeetion from L 2 ([0, 1] x [0, I]) onto Z2' This projeetion has evidently norm one. If P were also bounded as an operator in Lr([O, 1] x [0, 1]) for every r> 2 then, sinee P eoineides with its adjoint, we would get that P extends to a bounded operator in every Lr([O, 1] x [0,1]) spaee, 1 <r< 00. It would then easily follow from 2. b.ll that P is a bounded projeetion in X whose range is preeisely Z x'

Fix r ~ 2 and, for n = 1, 2, ... , let z;n) be the c10sed linear span of the sequenee Wk = TX[(k- 1)2 - n. k2 - n), k = 1, 2, ... in Lr([O, 1] x [0, I ]). Let Pn be the orthogonal projeetion from L 2 ([0, 1] x [0, I]) onto zin) whieh is given by

Sinee PJ= Th, where

we get that

(use the faet that the spaee X eorresponding to X=Lr(O, 1) is isomorphie to Lr(O, 00 )nL2(0, 00 )). Clearly,

We also have that

11 (00 ) ::::;2n/r'IIT-ll1~ sup!!f k~1 CkWk du dv

::::;2n/r'IIT-ll1~lIflir sup IIJI CkWkllr, ,


00

where both suprema are taken over all ehoices of {Ck}k'=l satisfying L Ickl r '::;; I k=l

and I/r+ I/r' = 1. Sinee Lr,([O, 1] x [0, I]) is of type r' we obtain

from whieh it follows that

This means that we have just proved that

Mr=sup Ilpnllr< 00 . n

Sinee this holds for every r > 2 it follows from Hölder's inequality that, for eaeh simple funetion 9 and all integers m and n,

IIp mg - Pngllr::;; IIp mg - PngIW(r- 1l llp mg- Pngll~r-2l/(r-1l

::;; (2M2rllgI12rP-2l/(r-1lllp mg - Pngll~/(r-1l

and this implies that {Pn }';:l eonverges in the strong operator topology of Lr([O, I] x [0, I]) sinee it obviously tends to P in the strong operator topology of L 2 ([0, 1] x [0, 1]). This proves that P aets as a bounded operator in Lr([O, 1] x [0, 1]). ·0

Remarks. 1. It is clear that 2.f.1(ii) does not hold for every r.i. funetion spaee X on [0,1] with qx< 00 (take, e.g. X=L 1(0, 1)). However, the eonditions imposed in 2.f.I(ii) that I <Px and qx< 00 are not always needed in order to prove that X is isomorphie to f!{. It is proved in [58] Seetion 8 that X is isomorphie to f!{ if, for exampIe, Xis the Orliez funetion spaee LF(O, I) with F(t)=et-I, for whieh Px=qx=oo.

2. In general, the spaee f!{ associated to a given r.i. funetion spaee X on [0, I] by 2.f.1 is not the only r.i. funetion spaee on [0, 00) which is isomorphie to X. For some spaees, however, as e.g. X=LF(O, I) withF(t)= tP(1 +llog tl), I <p< 00,

the eorresponding spaee f!{ is, up to an equivalent norm, the unique r.i. funetion spaee on [0, 00) isomorphie to X (cf. [58] Seetion 8).

3. There are also Orliez funetion spaees on [0,1], as e.g. LG(O, 1) with G(t) = t(1 + Ilog tl)1 /4, whieh are isomorphie to no r.i. function spaee on [0,00) (cf. [58], Seetion 8).

4. The spaee Y = Lp(O, 00) + Lq(O, 00) with 1 <p< q < 2 is not isomorphie to an r.i. funetion spaee on [0,1]. Indeed, assume that Y~X with X being an r.i. function space on [0, 1]. Since Y is p-convex and q-concave it follows, byapplying 2.e.lO to X* and Y* and by l.d.4(iii), that X is also q-eoneave. Hence, by the

f. Application of the Poisson Process to r.i. Function Spaces 211

seeond remark following 2. b.3, Ilfllx:S C!lfllq for some C< 00 and all fE LiO, I) and thus, by 2.e.l, Ilfllq=llflly:SDllfllx for some D< 00 and all fE X. Consequently, X is equal to Lq(O, I) up to an equivalent norm. This however is impossible sinee Yeontains a subspace isomorphie to lp whieh does not embed into LiO, I) (sinee e.g. lp is not of type q).

As a first applieation of 2.f.l, we present some results on the possibility to embed isometrieally the spaee L,(O, I) into an r.i. funetion spaee on [0, I]. We begin with the ease of r.i. funetion spaees on [0, (0) (cf. [58] Seetion 8).

Theorem 2.f.2. Let Y be an r.i. funetion spaee on [0, (0). Jf, for some 1 :Sr< 00,

the funetion t - 1/, belongs to Y then L,(O, (0) is order isometrie to a sublattiee of Y.

The proof of 2.f.2 is based on a general method for the embedding of r.i. funetion spaees whieh is deseribed in the following simple proposition.

Proposition 2.f.3. Let Y be an r.i.funetion space on some interval I, where I is either [0, I] or [0, (0). Let I/J be a positive element of norm one in Y and let Y", be the spaee of all measurable funetions f on I sueh that f(s)l/J(t) E Y(Ix 1). Then Y"" endowed with the norm

Ilfll y", = Ilf(s)I/J(t)11 Y(l x I) ,

is an r.i. funetion spaee on I whieh is order isometrie to a sublattiee of Y. If Y is minimal, respeetively, maximal then so is Y",.

The proof of 2.f.3 is straightforward.

Praof of 2 /2. In view of 2.f.3, it suffiees to show that the spaee Y"'r' where 1/J,(t)=t- l /'/llt- l /'lly, is equal to L,(O, (0).

We evaluate the distribution funetion of f(s)1/J ,(t) on [0, (0) x [0, (0), where f is an arbitrary function in L,(O, (0). We have, for every A>O,

,u({(s, t) E [0, (0) X [0, (0); If(s)II/J,(t»A}) 00

= J ,u({s E [0, (0); If(s)II/J,(t»)~}) dt o 00

= J ,u( {s E [0,(0); !f(s)1 > u })ru'-l), -'llt- 1/'11 y' du =)~ -'llt-l/'lly'llfll~ o

= ,u({t E [0, (0); 1/J,(t»A})·llfll~·

Henee, if Ilfll,= 1 then !f(s)II/J,(t) and I/J,(t) have the same distribution funetion whieh implies that Ilflly = I, thus eompleting the proof. 0 "'r

We pass now to the ease of r.i. funetion spaees on [0, I].


Theorem 2.f.4 ([58] Seetion 8). Let X be an r.i.lunction space on [0, 1]. IJ,lor some 1 < r < 2, the lunction t - 1/" 0< t ~ 1 belongs to X then L,(O, 1) embeds isometrically into X.

Proof Suppose first that qx < 00. Then, by 2.f.l(i), X eontains a subspace isometrie to the r.i. funetion spaee!!C 0 on [0, (0), defined there. Sinee 1< r < 2 we have that Ilt- 1/'xu,oo)112<00 i.e. the funetion t- l/', O<t<oo belongs to!!Co and therefore, by 2.f.2, L,(O, (0) embeds isometrieally into !!Co. This proves that X eontains an isometrie eopy of L,(O, 1).

In the ease of a general r.i. funetion spaee X on [0, 1] we eould have defined as well the r.i. funetion spaee Yo on [0, (0) of all measurable I on [0, (0) for whieh

where T is the operator introdueed in the proof of 2.f.l(i) with the aid of the symmetrized Poisson proeess, provided we know that TXro, 1] E X. In this ease, Yo would be isometrie to a subspace of X and the present proof would be eompleted, by 2.f.2, onee it is shown that t- 1/' E Yo'

That TXro, 1] EX is easily verified direct1y. Indeed, by its definition, T'X[O, 1] takes the value k on a subset of measure liek!, k=O, 1, ... , and thus (T'Xro, 1])*(t)~ At-li, for some eonstant A. Consequently, T'X[O, l] and henee also T"X[O, 1] and TXro, 1] belong to X. In order to show that Tg E X, where g(t) = t- 1/', it suffiees again to prove that (Tg)*(t)~Aot-1/' for some eonstant A o, i.e. that Tg belongs to the spaee L"oo defined in 2.b.8 (reeall that 11/11" 00 = sup t 1/'I*(t)). Sinee obviously

O~t~ 1

qL r , 00(0, 1) = rand 9X(O, 1] E L" 00 (0, 1) we get from the first part of the proof that indeed Tg E L" 00(0,1). We have only to c1arify one point whieh was mentioned already in 2.b.8. The expression 11·11,,00 is not a norm sinee it does not satisfy the tri angle inequality. However,

t

111/111,,00= sup t 1/'-1 J I*(s) ds o <t'" 1 0

is a norm and satisfies

11/11" 00 ~ 111/111" 00 ~ rll/ll" oo/(r -1) ,

for every I E L" 00 (0, 1). The Boyd indices are c1early not affeeted by the passage from 11 11" 00 to 111 111,,00' 0

The most interesting c1ass of r.i. funetion spaees on [0, 1] whieh eontain the function t- 1/' for some 1 < r < 2 is, of course, that of Lp(O, 1) spaees with 1 ~p < r.

Corollary 2.f.5. For 1 ~p < r< 2, the space L,(O, 1) embeds isometrically into LiO,I).

f. Applications ofthe Poisson Process to r.i. Function Spaces 213

This result was first stated explicitly by J. Bretagnolle, D. Dacunha-Castelle and J. L. Krivine [18] who obtained it by using sequences ofindependent r-stable random variables and ultrapowers ofBanach spaces. We shall discuss in detail this method of embedding L,(O, 1) into L/O, 1) in Volume IV. Essentially speaking, there is no difference between the present approach and that which uses r-stable random variables: the embedding of L,(O, 00) into an r.i. function space X on [0, 1], given by 2.f.4, consists of a composition between the map generated by the symmetrized Poisson process in 2.f.1(i) and the map fes) ~ f(s)t- 1/r /llt- 1/'11 !!l"'

If Rw denotes the image of the characteristic function X[O, w) under this composition then {Rw}o",w<oo is an r-stable random process i.e. a stationary process with independent increments so that

1 1

S S eiARw(S,t) ds dt=e- cwIW , o 0

for some c> 0. The verification of this fact is direct. Corollary 2.f.5 holds also for r = 2 and 1 ~p < 2 but this fact does not obviously

follow from 2.f.4. Of course, we have already used several times the fact that Lp(O, 1), 1 ~p<2 contains an isomorphie copy of L 2(0, 1) (e.g. the span of the Rademacher functions). The existence of an isometrie embedding of LiO, 1)=/2 in Lp(O, 1) is obtained by considering the span in Lp(O, 1) of independent 2-stable random variables (i.e. variables having the normal distribution).

We turn now OUf attention to another applieation of 2.f.l which concerns the spaces X p ,2,w' p>2 of H. P. Rosenthai [114]. These spaces were introduced in 1.4.d as the closed linear span in (lp EB 12)00 of the sequence {J" + wngn}:'= l' where {J,,}~1 and {gn}:'=l denote the unit vector bases of Ip , respectively, 12 and W = {Wn}:'= 1 is a sequence of positive reals satisfying the condition

00 (*) " w2p/(p-2) = 00

L... n ' lim wn=O and wn< 1 for all n.

n=l

We recall that, by 1.4.d.6, the isomorphism type of the space X p , 2, w does not depend on the particular sequence W = { W n}:'= 1 satisfying (*), which appears in the definition of X p,2,w' and, thus, we use the notation X p,2 instead of Xp,2,w (sometimes, this space is even denoted by X p). It is easy to check direedy that X p, 2, w is not eomplemented in (Ip EB 12)00' It follows from I.2.c.14 that X p,2 is not even isomorphie to a complemented subspace of (lp EB 12)00' On the other hand, it was proved by H. P. Rosenthai [114] that LiO, 1), p > 2 has a complemented subspace isomorphie to Xp . 2 . This assertion can also be deduced from the fact, which follows from 2.f.l(ii), that Lp(O, 1) is isomorphie to the space Lp(O, 00)nL2(0, 00) for every p~2. Indeed, fix p > 2 and let {17n};;"= 1 be a sequence of pairwise disjoint subsets of [0, 00) so that

00 I P(17n) = 00, lim P(17n)=O and ° <P(17n) < 1 for all n .

n=l


Put wn=/l('1n)(p-2l/2p and observe that the sequence w= {Wn}:"1' defined in this way, satisfies the eondition (*), stated above, and, for every ehoice of sealars {an},~')= 1 , we have

i.e. the sequenee {/l('1n)-l/PX..,J:"l in LiO, (0)nL2(0, (0) is isometrieally equivalent to the unit veetor basis of X p , 2, W. Moreover, by 2.a.4, [x"J:'= 1 is a eomplemented subspaee of Lp(O, 00 )nL2(0, (0).

Let us state this result explicitly.

Theorem 2.f.6 [114]. The spaee Xp, 2' P > 2 is isomorphie to a eomplemented subspaee 01 LiO, 1).

The analysis of the way in whieh Xp,2 embeds isomorphically in LiO, 1) via the operator T, defined in the proof of 2.f.l (i) with the help of the Poisson proeess, leads to the eonclusion that the unit vector basis of Xp, 2, w is realized in LiO, 1) as a sequence {I/I n}:"1 of symmetrie and independent random variables with Poisson

00

distribution sueh that lim III/InIl2 =0 and L: III/Inll~ = 00. Rosenthal's embedding, n-+ao n= 1

however, is based on a sequenee of symmetrie and independent random variables satisfying the same eonditions but taking only three values: + 1, ° and -1.

The arguments used here to prove 2.f.6 have aetually a more general eharaeter. Let Y be an r.i. funetion spaee on [0, (0) and, for every sequence 0"= {O"n}:'= 1 of pairwise disjoint subsets of [0, (0) of finite measure so that

(:) I {/l(O"n);/l(O"n)<e}=oo, foreverye>O, n

let X Y,a be the subspaee of Yeonsisting of all I E Y which are eonstant on each of the sets O"n' n= 1,2, ... and zero elsewhere. Observe that {xaJ:"l is an uneonditional basis for Xy,a whenever Yis separable. Though, a-priori, Xy,a depends on the partieular ehoice of 0", the spaees X y,a generate the same isomorphism type X y provided we eonsider only sequenees 0" satisfying the eondition (:).

Proposition 2.f.7 ([58], Seetion 8). For every r.i.lunetion spaee Yon [0, (0), the spaee X y,a is unique, up 10 isomorphism, i.e. if 0"' and 0"" are two sequenees 01 disjoint sets 01 finite measure whieh satisly the eondition (:) above then X y, a' and X y, a" are isomorphie. Moreover, eaeh olthe spaees Xy,a is eomplemented in Y.

g. Interpolation Spaces and their Applications 215

Proof The proof of 2.f. 7 resembles that of I.4.d.6. By 2.a.4, each space X Y ais the range of a conditional expectation in Y. This fact also implies that they h~ve the same isomorphism type. Indeed, if u' = {U~}:= 1 and u" = {U~}:= 1 are two sequences of subsets of [0, 00) as above then, by (:), one can find, for each m, a subsequence

ro

{u~(m,i)}r:.l of {U~}:=l so that Jl.(u~)= L Jl.(U~(m,i») and n(m1, i1) i: n(mz , iz) i= 1

unless m1 =mz and i1 =iz · Put 17= {01 U~(m,i)}:=l and notice that the space

X Y, ~, which is c1early isometrie to X Y, a" is, by the remark made in the beginning ofthe proof, a complemented subspace of Xy,a'" Hence, Xy,a,,~Xy,a' E!1 Vfor a suitable space Vand, of course, also Xy,a,~Xy,a" E!1 W for a suitable W. It remains to show that each of the spaces X y,a is isomorphie to its square.

Let u=[Un }:=l be a sequence satisfying (:) and let <5={(\Jj;.l,:'=t be a double sequence of mutually disjoint subsets of [0, (0) such that Jl.(<5n,)=Jl.(un)

for allj and n. It follows direcdy from the definition of<5 that

while from the first part of the proof we get that

for a suitable Z, since also <5 satisfies the condition (:). Hence,

i.e. X y,a is isomorphie to its square. 0

g. Interpolation Spaces and their Applications

The main purpose of interpolation theory is to prove results of the following general nature (stated here in an imprecise way).

( +) For suitable tripies Xl' X, Xz and Y1 , Y, Yz of Banach spaces every linear operator T, which is bounded as a map from Xi to Y i , i= 1, 2, is also a bounded operator from X to Y.

By 2.a.10 this is, for example, the case if Xl = Yl =Ll(O, 1), X 2 = Yz = Lro(O, 1) and X = Y is any r.i. function space on [0, 1]. Several other such cases were exhibited in Section 2.b. The importance ofstatements ofthe form (+) sterns from the fact that it is ollen much simpler to verify the boundedness of T from Xi to Yi , i = 1, 2, than to verify directly the boundedness of T from X to Y. In the preceding sections we encountered very many examples of this nature.

In general (and again, imprecise) terms we have that if (+) ho1ds then X is in "between" Xl and X 2 and Y is in "between" Y1 and Y2 • Interpolation theory


contains many "interpolation methods" of constructing spaces in between given spaces Xl and X2 (or, more precisely, a given interpolation pair (Xl' X 2 ) cf. Definition 2.g.1 below). If X is obtained by such a construction from (Xl' X 2 ) and if this same construction applied to the pair (YI , Y2) yields the space Y then it is usually easy to verify that (+ ) holds. In order to get in this way a useful result for applications to hard analysis it is crucial to identify concretely the space obtained in this procedure from (Xl' X 2 ). In Banach space theory there is another aspect ofthis construction ofinterpolation spaces which is very useful. It is usually quite simple (and often even straightforward) to establish some geometrical properties of the spaces obtained by various interpolation methods even if these spaces cannot be described or represented explicitly. The interpolation methods often enable us therefore to produce interesting examples of Banach spaces and also lead to useful theorems of a general character. This section is devoted mainly to an exposition of this latter aspect of interpolation theory.

After establishing some preliminary notations of interpolation theory we present a quite general method of constructing interpolation spaces. This construction is a particular case of the so-called real method in interpolation theory. We apply this construction for giving examples of r.i. function spaces which illustrate some of the results of Section 2.e and clarify the role of the conditions appearing in the statement of these results. As other illustrations of the use of the general construction of interpolation spaces we prove a result concerning the notion of equi-integrable sets in Köthe function spaces and a theorem concerning factorization of weakly compact operators. In the discussion of the general construction all that we use from interpolation theory is just the definition of the spaces. The proofs that these spaces have the desired properties are easy direct verifications.

After treating the general construction we pass to a special case which was introduced and studied in depth by Lions and Peetre [81]. In this special case much more can be said than in the general case and, actually, Lions and Peetre were able to produce a well-rounded theory of the interpolation spaces they introduced. We outline briefly their theory and then discuss in some more detail the Banach space properties ofthe Lions-Peetre interpolation spaces. At the end of this section a typical application of the Lions-Peetre interpolation spaces to Banach space theory is presented, namely a "uniform convexification" of the example of Maurey and RosenthaI introduced in I.I.d.6.

Definition 2.g.1. A pair (Xl' X 2) of Banach spaces is called·an interpolation pair if these Banach spaces are given as subspaces of a common Hausdorff topological vector space Z so that the embeddings of Xl and of X 2 into Z are continuous.

Note that, by definition, the interpolation pair depends not only on Xl and X 2 themselves but also on the space Z and the specific embeddings of Xl and X 2

into Z. It is however convenient not to include Z explicitly in the notation. Typical examples of interpolation pairs are the following. I. Xl and X 2 are Köthe function spaces on the same a-finite measure space

(Q, 1:, J1.). In this case we may take as Z the space of alllocally integrable functions


on Q and define the topology on Z by requiring that f~ ----> 0 in Z if and only if S h(w) d/1----> 0 for every 0" E 1:' with /1(0") < 00. In partieular, if Xl and X 2 are Köthe

" sequenee spaees (i.e. Q = the integers with the diserete measure) Z will be the spaee of all sequenees of seal ars endowed with the topology of pointwise eonvergenee. In ease where Xl and X 2 are r.i. funetion spaces on an interval I we may take as Z also the spaee Loo(J) + LI (J).

2. Given a continuous one-to-one linear mapj from a Banach spaee Xl into a Banach spaee X 2 we can take the spaee X 2 itself as the Z corresponding to the pair (Xl' X 2 )·

Another possibility, whieh however in the present eontext is trivial and uninteresting, is to eonsider an arbitrary pair of Banach spaces (Xl' X 2 ) taking as Z the direet sum Xl EB X 2 . This is uninteresting since interpolation theorems derive their importance from the existenee of a big overlap between Xl and X 2 • In a typical situation the set theoretical intersection Xl nX2 is dense in either Xl or X 2

(usually, in both).

Definition 2.g.2. (i) Let (Xl' X 2 ) be an interpolation pair. The space Xl + X2 is defined to be the linear span of Xl and X2 in Z, i.e. {z E Z; Z=X I +X2 , Xl E Xl' X 2 E X 2 }, normed by

The space Xl nX2 is the set theoretical intersection of Xl and X 2 normed by

(ii) A linear operator T is said to be a bounded operator from the interpolation pair (Xl' X2) into the interpolation pair (YI , Y2 ) if T is defined on Xl + X2 and acts as a bounded operator from Xl into YI and from X 2 into Y2 • The norm II I1k 2 of T as an operator between pairs is defined to be

where IITII;, i= 1,2, is the norm of T: X; ----> Y;.

It is easily verified that both Xl + X 2 and Xl nX2 are Banach spaees. Let us check e.g. that 11·llxl +X2 is an actual norm (and not only a semi-norm) on Xl + X 2 •

Assurne that Ilx l +x2 11xI +X2 =0. Then there exist {XI,n}:~1 in Xl and {X2,n}:~ I in X2 so that Xl +X2 =XI,n+X2,n for every n and IIX I ,nllxl ----> 0, IIX2 ,nllx2 ----> O. Hence, Xl -XI,n= -X2 +X2,n tends in Xl (and thus in Z) to Xl and tends in X 2 (and thus in Z) to -X2 • Sinee Z is Hausdorff we deduee that Xl +X2 =0.

Clearly,

and all the inc1usion mappings are operators of norm one. Note that every operator


Tbetween the pairs (Xl' X 2 ) and (Yl , Y2 ) defines an operator (also denoted by T) from Xl + X2 and Xl nX2 into Yl + Y2 , respectively Yl n Y2 • The norms IITllx! +X2 and IITllx!nX2 are both dominated by the norm IITlk2 ofTas an operator between pairs. The notation in 2.g.2 is ponsistent with the notation Lp(l) + Lq(l) and Lil)nL/l) used in previous sections. The only role played by the space Z of Definition 2.g.l in interpolation theory is in the definition of Xl + X2 and Xl nX2 •

If Xl and X 2 are r.i. function spaces on some interval I then the two possible choices of Z mentioned in example I above lead to exact1y the same spaces Xl + Xz and Xl nXz (with obviously the same norm). Unless stated otherwise, we shall assurne in the sequel that if Xl and Xz are Köthe function spaces on (Q, L, Ji) and (Xl' X 2 ) is considered as an interpolation pair then Z is chosen so that it yields the same Xl + Xz and Xl nXz as the choice indicated in example 1 above. Also, if Xl is given as subspace of Xz with a continuous embedding we shall assurne (unless stated otherwise) that Z=Xz ' In this case Xl +X2 =X2 and XlnXZ=Xl , up to equivalent norms. The norms will coincide in case IIxllx2 ~ Ilxllx! for every x E Xl i.e. if the embedding of Xl into X2 is of norm at most one. (If Xl and X2 are Köthe function spaces on (Q, L, Ji) and Xl C X2 as sets both conventions made above are in agreement.) Another convention we make is the following: If Xl and Xz are Banach lattices we say that they form an interpolation pair (Xl' X2 ) 01 Banach lattices if the space Z is a vector lattice and Xl and Xz are both ideals in Z. In this case, of course, also Xl + X2 and Xl nX2 are Banach lattices.

We present now a general definition of interpolation spaces which is useful in Banach space theory. I t is a particular case of a more general definition introduced by Peetre [109].

Definition 2.g.3. Let (Xl' Xz) be an interpolation pair ofBanach spaces. For every choice of positive scalars a, biet k( . , a, b) denote the equivalent norm on Xl + Xz defined by

Let, in addition, Ybe aspace with a normalized unconditional basis {Yn}:= 1 whose unconditional constant is one and let {an} ~ land {bn}:= 1 be sequences of positive

00

numbers so that L min (an' bn)< 00. The space K(Xl , X2 , Y, {an}' {bn}), respec-n=l

tively K(Xl , X 2 , Y, {an}, {bn}), is defined to bethe space ofall elements z E Xl +X2

such that n~l k(z, an' bn)Yn converges, respectively L~l k(z, an' bn)Yn}:= 1 is

bounded, normed by

_ Obviously, if {Yn}:= 1 is boundedly complete then K(Xl , Xz, Y, {an}, {bn}) =

K(Xl , X2 , Y, {an}, {bn}). It is also clear that the spaces defined in 2.g.3 are Banach


spaces satisfying

and 00

min (al' b1)llzllxl +X2~ IlzIIK(x..X2)~ L min (an, bn)llzllxlnX2 . n=l

Notice that if (Xl' X2) is an interpolation pair of Banach lattices then both K(Xp X2 , Y, {an}' {bn}) and K(Xp X2 , Y, {an}' {bn}) are also Banach lattices.

It is a trivial, but important, fact that the spaces defined in 2.g.3 are interpolation spaces between Xl and X2 in the following sense.

Proposition 2.g.4. Let (Xl' X2 ) and (Wl , W2 ) be two pairs of interpolation spaces and let Y, {Yn}:'= l' {an}:'= 1 and {bn}:'=l be as in 2.g.3. Then every bounded operator Tfrom the pair (Xl' X2 ) into (Wl , W2 ) maps K(X1 , X2 , Y, {an}, {bn}), respectively K(X1 , X2 , Y, {an}' {bn}), into K(Wl , W2 , Y, {an}' {bn}), respectively K(Wl, W2 ,

Y, {an}' {bn}), and the norm ofT on these spaces does not exceed IIT111,2'

Proo! It is enough to note that for every z E Xl + X2 and every choice of scalars a,b

k(Tz, a, b)~ IITlll, 2k(Z, a, b). 0

It follows in particular from 2.gA that if Xl and X2 are minimal (respectively maximal) r.i. function spaces on an interval I the same is true for K(X1 , X2 , Y, {an}, {bn}) (respectively K(Xl , X2 , Y, {an}, {bn})) provided we normalize the norm in the space by putting

By applying 2.gA to the dilation operators D. of Section 2.b, we obtain in this situation the following estimates for the Boyd indices

The spaces introdueed in 2.g.4 were already defined in a slightly modified form (and with a somewhat different notation) on page 126 of Vol. I. They were used there in partieular to show that the universal space Ul for all uneonditional basie sequenees (defined in I.2.d.l0) has uneountably many mutually non-equivalent symmetrie bases. We shall show now that exaetly the same argument as that used in Vol. I proves that Ul has also uneountably many mutually non-equivalent representations as an r.i. funetion spaee. Reeall that U1 , by its definition, has a normalized basis {en}:'= 1 whose uneonditional eonstant is one and every normalized uneonditional basis in an arbitrary Banaeh space is equivalent to a subsequence of {en}:,= l' This property characterizes U1 in the following strong sense.


Every Banach space with an unconditional basis which has a complemented subspace isomorphie to U I is already isomorphie to U I .

Theorem 2.g.5. The universal space UI Jor all unconditional basic sequences has uncountably many mutually non-equivalent representations as an r.i. Junction space on [0, 1] and on [0, (0).

Proo! Let XI and X 2 be two minimal r.i. function spaces on [0,1] with nontrivial Boyd indices (i.e. min (PX1'Px) > 1 and max (qXl' qx)< (0) so that

(i) XI c:X2 and IIJllx2 ~ IIJllxJor JE XI' (ii) lim Ilx[O,tlllx,/llx[O,tlIIXl =0

t-+O

and let {mn }:,= I be an increasing sequence of numbers satisfying the following lacunarity condition

n-I 00

m;1 L mi+mn I mi- I <2-n-t, n=I,2, ... i=1 i=n+1

Consider the space W=K(XI, X 2 , U I, {mn-I}, {mn}) with the norm normalized so that Ilx[o, Illlw = 1. By the preceding remarks W is a minimal r.i. function space on [0, 1] with non-trivial Boyd indiees and thus, by 2,c.6, the Haar basis is an unconditional basis of W.

Exactly as in the proof of 1.3.b.4 it follows from the lacunarity condition that there exists a sequence {O'n}:'=1 of pairwise disjoint measurable subsets of [0,1] so that {en}:'=1 is equivalent to {x .. jl Ix .. JIw }:'=I' (The sets O'n are chosen so that k(Xtln' m-I, m) attains its maximum at m=mn. The lacunarity condition implies then that k(x"n' mi I, m i ) for i -# n is negligible i.e. that IIx .. ",lw can be estimated very well by using only k(X"n' m;; I, mn),) By 2.a.4 there is a projection of norm one from W onto [X .. J:'= I' Hence, W has a complemented subspace isomorphie to UI and therefore W itself is isomorphie to UI .

This proves that UI can be represented as an r.i. function space on [0, 1]. That UI has uncountably many different such representations follows from the fact that, for every 1 <p < 00 given in advance, the W above can be chosen so that Pw=qw=P (take e.g. XI =LM(O, 1) with M an Orliez function equivalent at infinity to tp log t and X 2 = LiO, 1)). By 2.f.l, it follows that UI has also uncountably many non-equivalent representations as an r.i. function space on [0, (0). 0

It is often easy to prove that if XI' X 2 and Yhave a nice property then the same is truefor K(XI, X 2 , Y, {an}' {bn}). For example, in the proof ofl.3.b.2 we verified that if XI' X 2 and Yare uniformly convex then K(XI, X 2 , Y, {an}, {bn}) has an equivalent uniformly convex norm. (In 1.3.b.2 we used instead of k(z, a, b) of 2.g.3 the equivalent expression inf {(a2I1xlllil +b2I1x2I1iY/2, Z=XI +X2 } and showed that if this expression is used in 2.g.3(ii) the equivalent norm obtained in this manner on K(XI, X 2 , Y, {an}, {bn}) is uniformly convex.) We prove now a similar result concerning p-convexity and concavity in lattices.


Proposition 2.g.6. Let l:>;;p:>;; 00, let (Xl' X2 ) be an interpolation pair oj p-convex (respectively p-concave) lattices. Assume also that the lattice structure on Y induced by {Yn},',"'=l is p-convex (respectively p-concave). Then,jor every choice ojpositive

00

{an},',"'= land {bn},',"'= I with L min (an' bn) < 00, the spaces K(XI, X 2 , Y, {an}, {bn}) n=l

and K(XI, X2 , Y, {an}, {bn}) are p-convex (respectively p-concave).

Proof The verification of 2.g.6 is essentially a straightforward computation. We carry it out in detail in the case of p-convexity, 1 <p< 00, and K(XI, X2 , Y, {an}, {bn }). Let {Z;}r=l be vectors in Xl +X2 and let a, band 8 be positive numbers. Let Xl. i E Xl' X2 • i E X2 be so that, for 1:>;; i:>;;m,

Since

we get that

k( Ct IzyYIP, a, b ):>;;aIICtl IX1,iIPYIPllxl +bllCt IX 2 • i IPYIPllx2

:>;;aM(p)(Xt)(tt Ilxt,&t.YIP +bM(P)(X2)(tl Ilx2,&t YIP

:>;;(M(P)(XI)+M(P)(X2)) ((tl k(Zi' a, b)PYIP +m I/P8).

Hence, since 8>0 was arbitrary, we deduce that

The proof of the p-concavity assertion in 2.g.6 is similar. We just have to note that by applying the decomposition property in the p-concavification (Xl + X 2 )(P)

of Xt +X2 (considered just as a vector lattice) we deduce that, whenever

( m )I/P

{zJr=t E Xl +X2 , Xl E Xl and x 2 E X2 satisfy i~l Izy =xI +x2 , then there

exist {XI,Jr=1 E Xl and {X2,Jr=1 E X2 so that IZil:>;;Xl,i+X2,i for every i, and


and consequently,

We shall apply 2.g.6 in proving a variant of 2.g.5 which is of interest in connection with the results of Section 2.e. We have seen in 2.e.ll that every r.i. function space on [0, 1], which is q-concave for some q< 2, has a unique representation as an r.i. function space on [0, 1]. That this fact is not true for q = 2 is shown by the following example from [58] Sectiön 10.

Example 2.g.7. For every 1 <p<2 there exists an r.i. junction space X on [0,1] which is p-convex and 2-concave but has uncountably many mutually non-equivalent representations as an r.i.junction space on [0, 1].

The construction of this example is similar to that presented in the proof of 2.g.5 except that U1 must be replaced by a suitable space whose existence is asserted in the following proposition.

Proposition 2.g.S. For every 1 ~p ~ q ~ 00 there exists aspace U p, q (universal jor all unconditional bases which are p-convex and q-concave), with a normalized basis {e~' q}:,= 1 having an unconditional constant equal to one, such that

(i) the laltice structure on U p, q generated by { e~' q}:,= 1 is p-convex and q-concave, (ii) every normalized unconditional basis {y;};': 1 with unconditional constant

equal to one which generates a p-convex and q-concave lattice structure is equivalent to a subsequence {e!' q} ~ 1 oj { e~' q}:,= 1 .

Proo! If p=q then lp (respectively co, if p= (0) has the desired property. We can assurne therefore that p < q. If p = 1 and q = 00 we of course take the universal space U 1 as U 1. 00 • In order to treat the general case, fix r> 1 and consider the r-convexification U1') of U l' The vectors {en }:'= I form an unconditional basis in U1') and if Y is an arbitrary Banach space with a normalized r-convex unconditional basis {yJ~ 1

then {y;}f= l' considered as an unconditional basis in the r-concavification Y(r) of Y, is equivalent to a subsequence {enJ~1 of {en}:'=l (considered in UI)' Hence, as readily verified, {Yi}~1 in Y is equivalent to {enJ~1 in ul'). This proves that Uir ) is universal (in the sense of 2.g.8) for all the unconditional ba ses which are r-convex, thus completing the proof if q= 00. If q< 00 we choose r so that prj(r-l)=q. Then, V= [e:]:,= I c(u~r»)*, where {e:}:,= I are the biorthogonal functionals corresponding to the basis {en }:'= 1 of Uir ) is universal for all the unconditional bases which are rj(r-l)-concave. Hence, we get, in the same way as above, that V(p) is universal for all the unconditional bases which are p-convex and q=prj(r-l)-concave. 0

Remark. By using the decomposition method we get easily the following strong uniqueness assertion. If W is a Banach space with an unconditional basis which generates ap-convex and q-concave lattice structure and if Whas a complemented subspace isomorphic to Up q then W itself is isomorphic to U . , ~q


Proof of 2.g.7. Fix 1 <p<2 and let Xl and X 2 be two minimal r.i. funetion spaees on [0, 1] whieh are p-eonvex and 2-eoneave and satisfy in addition the requirements (i) and (ii) appearing in the proof of 2.g.5. Let {m n };:'= 1 be a sequenee of numbers satisfying the laeunarity eondition stated in the proof of 2.g.5. Consider the spaee W=K(Xl , X 2 , Up ,2' {m;l}, {mn})' where Up ,2 is the spaee eonstrueted in 2.g.8. By 2.g.6, W is p-eonvex and 2-eoneave. It follows from l.d.7 that the lattice strueture generated by the Haar basis of W is also p-eonvex and 2-eoneave (to be preeise we first have to renorm W so that in the new norm the uneonditional eonstant of the Haar basis beeomes one). As in the proof of 2.g.5, we have that W eontains a eomplemented subspace isomorphie to Up,2 and henee, by the remark following 2.g.8, W is isomorphie to Up ,2' This fact is true for every ehoice of Xl and X 2 as above. By letting r vary betweenp and 2 and taking Xl =LM(O, 1) with M equivalent at infinity to t log t and X 2 = L,(O, 1), we obtain that U 2 has p,

uneountably many mutually non-equivalent representations as an Li. funetion spaee on [0, 1]. 0

Remark. If in the proof above we would have used Up,q with 1 <p<q<2 instead of Up,2 then we would have obtained an r.i. funetion spaee Won [0, 1] whieh is p-eonvex and q-eoneave (provided, of course, that both Xl and X2 are p-eonvex and q-eoneave). This spaee W still eontains Up,q as a eomplemented subspace but W is not isomorphie to Up,q (which would eontradiet 2.e.ll), since the Haar basis of W is not q-eoneave but only 2-eoneave.

We pass now to another situation where the spaees defined in 2.g,3 are used in Banach spaee theory. This applieation involves the following notion.

Definition 2.g.9. A set F in a Köthe funetion spaee X on a probability spaee (Q, 1:, p,) is said to be equi-integrable if for every e > ° there is a A< 00 so that

where

er(J, A)={W E Q; If(w)I~A} .

Note that every equi-integrable set is bounded. Equi-integrable sets enter in several arguments in funetional analysis, usually in eompaetness arguments. Reeall e.g. that a set F in L l (Q, 1:, p,) is equi-integrable if and only if its w closure is w eompaet (cf. [32] IV.8.ll). The next proposition enables sometimes to reduee the study of equi-integrable sets to that of bounded sets.

Proposition 2.g.10 ([58] Seetion 6). Let X be a Köthe function space on a probability space (Q, 1:, p,) and let F be an equi-integrable subset of X. Then there is a Köthe function space Won (Q, 1:, p,) so that

(i) The unit ball of W is equi-integrable in X (and thus in particular W.:: X with a continuous embedding).

(ii) Fis bounded (even equi-integrable) in W.


Proo! Let {An J:'= I be such that

We claim that W = K(L 00 (Q, 'E, /1), X, Iz, {2-n A; I}, {2n}) has the desired properties. Note first that since Loo(Q, 'E, /1) is contained in X with a continuous embedding the same is true for W. For f E Fand n = 1,2, ... we have

and therefore Fis a bounded subset of W. Moreover, if m ~ n

and consequently, for m= 1,2, ...

m 00

IIfXa(f.;.~)II;;'< L 2Zn4- Zm + L 2- Z(n-I)<r Zm +3

n=l n=m+l

and this proves (ii). To prove (i) note that if Ilfllw < 1 then k(j, 2 -nA; I, 2n) < 1 for every n and hence f=gn+hn with rnA;lllhnlloo +2nllgnllx< 1. In particular, Ilhnll 00 <An2n and thus whenever If(w)I~An2n+1 we have If(w)1 < 2Ign(w)1 and hence

Remarks. 1. It follows immediately from 2.g.1O that the convex hull of an equiintegrable set is again equi-integrable.

2. If X is p-convex then, by 2.g.6, the space W constructed above is also pconvex provided p < 2. If 2 <p < 00 the same is true provided we replace Iz in the definition 0 f W by 1 p •

As a further application of the spaces defined in 2.g.3 we present the following factorization theorem.

Theorem 2.g.tl [24]. Let V and X be Banach spaces. A bounded linear operator T: V - X is weakly compact if and only if T can be factored through a reflexive Banach space i.e. there exists a reflexive space Wand bounded operators TI : V - W, Tz: W - X so that T= TzTI .

Proo! The "if" assertion is trivial so we have to prove the "only if" assertion. Assurne that T is weakly compact and let BI = TBv . Then BI is a w compact

00

sub set of X. Consider the subspace Xl = U nBI of X. We norm Xl by requiring n=l

that BI be its unit ball. Then Xl becomes a Banach space which is continuously embedded in X. Let {an}~l and {bn}:'=l be sequences of positive reals so that


co L an< 00 and bn i 00, and consider the space W=K(X1,X,12,{an},{bn}).

n= 1

Clearly, Xl eWe X, the identity mappings j1: Xl -+ Wand j2: W -+ X are continuous and T=j2U1T). Thus in order to prove the theorem it suffices to show that W is reflexive.

Let {ZJj=l be a sequence ofvectors in Wwith IIZjllw<1 for allj. Then, in particular, for everyj and n, k(zj' an' bn) < 1, i.e.

By the weak compactness of BI = Bx !, we may assurne without loss of generality (by passing to a subsequence) that, for every n, the sequence {Yj,n}~l converges weakly in X to so me vector Yn EX. Since Ilxj,nllx ~b; 1 it follows that

and thus {Yn }:= 1 converges in norm to some vector Y in X. Clearly, the sequence {Zj}~ 1 converges weakly in X to Y and k(y, an' bn)~lim inf k(zj' an' bn) for every

J

n (recall that k(-, a, b )is a norm on X equivalent to the original one). Hence, y E Wand IIYllw ~ 1. In order to show that Zj tends to y also weakly in W (and thus verify the reflexivity of W) we have to prove that

co co lim L x:(z) = L x:(y) j-+oon=l n=l

co whenever {x!}:= 1 c X* and L Ilx:ll; < 00, where

n=l

Ilx*lln = sup {lx*(x)l; k(x, an' bn) ~ I}, n = 1, 2, ....

This however, is an immediate consequence ofthe facts that IlzJw< 1, for everyj, and ~im x:(z)=x:(y) for every n. 0

J~ co

We pass now to the most important special dass of the interpolation spaces defined in 2.g.3, namely to the Lions-Peetre interpolation spaces. Let us recall that if X is a Banach space we denote by L/R, X) the space of all the measurable

(

CO )l/P functions f from the real line R into X with Ilfll p = J

co Ilf(t)IIP dt < 00,

respectively, Ilfllco=ess sup Ilf(t)<oo if p= 00. (In ca se it is worthwhile to point out explicitly the underlying Banach space X we shall denote Ilfll p also by

IlfIILp(X)')

Definition 2.g.t2 [81]. Let (Xl' X2) be an interpolation pair ofBanach spaces. Let O<{}< 1 and 1 ~p~ 00, The space ofall Z E Xl +X2 which admit a representation


as Z=Xl(t)+X2(t), tE R, with

e8tx l (t) E Lp(R, Xl) and e -(1-8)tX2(t) E Lp(R, X2) ,

will be denoted by [Xl' X2]8,p' Thenorm in this space is defined by

IIzI18,P= inf {max (lIe8tx l(t)IIp, Ile-(l-8)txit)lIp); z=xl (t) +xit), tE R} .

It is easily verified that [Xl' X2]8,p coincides with the space of all Z E Xl + X2 so that

00

L k(z, e8", e-(l-8)")P < 00, n= -00

respectively, sup k(z, e8", e-(l-8)") < 00 if p = 00, and is thus, in particular, one -oo<n<oo

of the spaces defined in 2.g.3 (up to an equivalent norm). The "continuous" definition given in 2.g.10 is sometimes more convenient than the discrete version of 2.g.3 and it leads to better estimates of numerical constants which enter into the theory. One advantage ofthe continuous version is demonstrated by the following simple but useful observation.

Lemma 2.g.13. With the notation of2.g.I2 we have

for every Z E [Xl' X2]8,p'

Proo! We have

= inf {inf max (e8t 11 e8tx l (t)lIp, e-(1-8)tlle-(1-8)txit)lIp)} tER

= inf {lIe8tx l (t)II! -8I1e-(1-8)tx2(t)"~} , where the outer infima are taken over all possible representations of Z as Xl (t) + x2(t), tE R. 0

CoroUary 2.g.14. For every 0<0<1 and l~p~oo there is a constant C(O, p) so that,for x E X l nX2,

Proo! Pick a function 0 ~ I/I(t) ~ 1 so that

e8tl/l(t) E Li - 00,00) and e-(l-8)t(1_I/I(t» E Li - 00, 00)

and apply 2.g.13 to the representation x = I/I(t)x + (1- I/I(t»x, tE R. 0


Another consequence of 2.g.13 is the following sharper form of 2.g.4.

Proposition 2.g.15 [81]. Let T be a bounded operator Jrom the interpolation pair (Xl' X 2 ) to the interpolation pair (Yl , Y2 ). Then,for every 0<8< 1 and 1 ";;p";; 00,

T maps [Xl' X 2 ]e,p into [Yl , Y2]e,p and

Proo! Apply 2.g.13 and observe that

IIe9t Tx l (t)II~ -elle-(l-9)tTx2(t)II:

,,;; IITII~ -eIITII~lle9tx 1 (t)II~ -elle-(I-e)tx2(t)II~· 0

We state now without proof some of the main results of Lions and Peetre on the properties of [Xl' X 2 ]e, p' A detailed exposition of these results can be found (besides the original paper [81]) in [21] and especially in [10].

Theorem 2.g.16 [81] (cf. also [10] 3.7.1). Let (Xl' X 2 ) be an interpolation pair so that X I"X2 is dense in both Xl and X 2 • Then,Jor every 0<8< 1 and 1 ";;p< 00, the dual oJ [Xl' X 2 ]9,P is equal (up to an equivalent norm) to [Xi, XiJe,q' where l/p+ l/q= 1.

Note that the pair (Xi, Xi) becomes in a natural way an interpolation pair if we consider these two spaces as embedded continuously in the Banach space (X1"X2 )*. An element in [Xi, X'i]e,q is thus, by definition, a continuous linear functional on X l "X2 . One ofthe assertions of2.g.16 is that this element extends to a continuous linear functional on [Xl' X 2 ]e,p' The extension is unique in view of the density assumption in 2.g.16. A major step in the proof of 2.g.16 is the proof of the following proposition.

Proposition 2.g.17 [81] (cf. also [10] 3.3.1). Let (Xl' X 2 ) be an interpolation pair and let 0< 8 < 1 and l";;p,,;; 00. The space [Xl' X 2 ]9,p consists exactly oJ those

00 elements Z E Xl + X 2 which can be represented as Z = S x(t) dt with x being a

-00 measurableJunctionJrom R to Xl "X2 so that eetx(t) E Lp(R, Xl) and e-(1-9)tx (t) E

00 Lp(R,X2 ) (these conditions imply inparticular that J Ilx(t)llxl+X2dt<00). The

-00

norm on [Xl' X 2 ]e,p is equivalent to

IIIZII19, p= inf {max (1Ie9tx(t)lk,(xtl, Ile-(l-e)tx (t)IIL p (X 2 )); z= ]00 x(t) dt} .

Remark. It is easily verified that analogues of2.g.13 and 2.g.l5 hold for 111·1119,p·

In the applications of the Lions-Peetre interpolation to analysis it is of course important to identify explicitly the spaces [Xl> X 2 ]9,p if Xl and X 2 are concrete


function spaces. Much work has been done in this direction (cf. [10] and its references). We mention here the following result.

Theorem 2.g.18 [81] (cf. also [10] 5.2.1 and 5.3.1). Let (D, E, j,l)bea measure space. (i) The space [Lp,(Q, E, j,l), L p2(Q, 1:, j,l)]9,q with 1 ~Pl <P2 ~ 00, O<(}< 1,

1 ~ q~ 00 is equal, up to an equivalent norm, to the space Lp,q(Q, E, j,l) (introduced in 2.b.8), where I!p = (1- (})jP1 + (}!P2'

(ii) The space [Lp"q,(Q, 1:, j,l), L p2,q2(Q, E, j,l)]9,q with 1 ~P1 <P2 ~ 00, 1 ~ q l' q 2' q ~ 00, 0< (} < 1, is equal, up to an equivalent norm, to the space Lp,q(Q, 1:, j,l), where I!p = (1- (})!P1 + (}!P2'

The Marcinkiewicz interpolation theorem 2. b.I5 is an immediate consequence of 2.g.18. If an operator T is of weak types (pp P2) and (qp q2) then it is, by definition, a bounded map from L pl ,l to L p2,oo and from L q" 1 to L q2 , 00' Hence, ifp1 #-P2 and q1 #-q2 it follows from 2.g.18(ii) that, forevery (} and q, Tis a bounded map from Lr"q into Lrl,q' where I!ri=(}!Pi+(I-(})!qi' i= 1, 2. Hence, if r1 ~r2 we get, by taking q=r1, that T is bounded as an operator from Lr, into L r2 ,rl and thus in particular into Lr2 (use 2.b.9). This argument applies even if T is only quasilinear (observe that 2.g.4 is valid in general also for quasilinear T provided we replace IITII1,2 by ClIT111,2' where Cis the constant appearing in the definition of quasilinearity).

An important tool in the proof of 2.g.18 is the following general reiteration theorem.

Theorem 2.g.19 [81] (cf. also [10] 3.5.3). Let (Xl' X2) be an interpolation pair, let 0< (}i < 1, 1 ~Pi ~ 00, i = 1, 2, 3 with (}1 #- (}2. Then, up to an equivalent norm,

This theorem enables the reduction of the proof of 2.g.18(i) to the case where P2 = 00. Furthermore, 2.g.18(ii) is clearly a consequence of 2.g.19 and 2.g.18(i). The iteration theorem 2.g.19 is also used in the proof ofthe following proposition.

Proposition 2.g.20 [81] (cf. also [10] 5.6.2). Let (Q, E, j,l) be a measure space and (Xl' X2) an interpolation pair 0/ Banach spaces. Let 1 ~P1 <P2 ~ 00, O<(}< 1, and l!p = (1- ()!P1 + ()!P2' Then, up to equivalent norms,

We pass now to the study of some geometrical properties of the Lions-Peetre interpolation spaces. We have already mentioned above that, whenever Xl' X 2

and Yare uniformly convex, then K(X1 , X 2 , Y, {an}, {bn}) has an equivalent uniformly convex norm. It turns out that in the case ofthe Lions-Peetre interpolation spaces it is enough that one of the spaces Xl and X2 be uniformly convex for the


interpolation spaee to be uniformly eonvex, too. This faet is of importanee for several applieations of the Lions-Peetre interpolation spaees in Banaeh spaee theory (see e.g. 2.g.23 below). Before stating this result precisely we mention that if X is uniformly eonvex the same is true for the spaee L/R, X) whieh is isometrie to L/[O, 1], X) (this spaee was denoted in l.e also by L/X)), provided 1 <p< 00. Forp=2 weproved this in l.e.9. Aetually, we showed therethatb L2(x)(6) is equivalent to bx (6). A similar argument to that given in l.e.9 shows that also for 1 <p<2, bLp(x)(6) is equivalent to bx (6) and that, for 2<p< 00, bLp(x)(6) is equivalent to inf {t- Pbx(t6); t~ I} (ef. [40]). The faet that L/X) is uniformly eonvex whenever X is and 1 <p< 00 follows also from the eomputations done on pages 128-129 of Volume I (but this proof gives a rather poor estimate for bLp(x)(6)). We state now the result of Beauzamy [5] eoneerning the uniform eonvexity of the Lions-Peetre spaees.

Theorem 2.g.2I. Let (Xl' X 2) be an interpolation pair and let 0< 8 < 1, 1 <p< 00.

Then b[x,.X21 •. /6) dominates a function equivalent to max (bLp(XIl(6 l/(l-O»), bLp(x2)(6 l /0)). In particular, [Xl' X2 ]O.P is uniformly convex whenever Xl or X2 is uniformly convex.

Proof Let u, v E [Xl' X 2]o.P satisfy Ilullo.p< 1, Ilvllo.p< I and Ilu-vllo.p~B. Then there exist representations u=ul(t)+U2(t), v=V l(t)+V2(t), tE R so that

Sinee u-v=(ul(t)-V l(t))+(u2(t)-vit)), tE R we get, by 2.g.13, that

B::::'; Ilu - vllo. p::::'; IleOt(u l (t) - vl (t))ll~ -Olle-(l-O)t(u2(t) - v2(t))II~ .

Consequently,

By using 2.g.13 onee again, we deduee that

Ilu + vllo. p::::'; Ile8t(u l (t)+ v l (t))ll~ -Olle-(l-O)t(u2(t) + v2(t))II~ ::::.; (2 - 2bLp (x Il«B/20)l/(l-O»))l-O(2- 215 L p(xi(B/2 l - 0)l/O))O .

Henee,

and from this the desired result follows immediately. 0

Remark. It follows from 2.g.21 by duality (in view of l.e.2 and 2.g.16) that if either Xl or X 2 is uniformly smooth then [Xl' X2]o.P with 0<8< 1 and 1 <p< 00

has an equivalent uniformly smooth norm whose modulus of smoothness can


be estimated in terms of the moduli of smoothness of Xl and X2. The restriction in 2.g.I6 that X lnX2 be dense in Xl and X2 need not concern us in the present context. Indeed, for every interpolation pair (Xl' X2), the space X l nX2 is den se in [Xl' X2J8,p (this is most easily verified by applying 2.g.17). Hence, [Xl' X2J8,p = [xf, XfJ8,P' where xp denotes the c10sure of X lnX2 in Xi' i= 1,2.

We consider next the type of interpolation spaces.

Proposition 2.g.22 [5]. Let (Xl' X2) be an interpolation pair 0/ Banach spaces with Xi o/typePi , i= 1, 2. Then [Xl' X2J8,piso/typepprovidedthat 1/p=(1-()/PI + ()/P2'

Proof Let {Z;}?=l E [Xl' X2J8,p and let Zi=XI,i(t)+X2,i(t) with XI,i(t) E Xl' X 2, i(t) E X2, tE Rand 1 ~ i ~n. By 2.g.l3 and Hölder's inequality

J Iit ri(S)Zill ds~J Iit ri(S)e8txI,i(t)111-81It r;(s)e-<1-8)tX2 ,i(t)118 ds o ,= I 8, pO, = I p ,= I p

~G Ilit ri(s)e~XI,;(t)lIp ds Y-8

xGllit r;(s)e-<1-8)tx2 ,i(t)lIp dSY

~G ]00 lIitl ri(s)e8tXI,i(t)[ dt dSYI-8)IP

x G ]00 lIitl

ri(s)e-<1-8)tx2 , i(t)II:2 dt ds TP

Hence, since Xl is of type p 1 and X2 of type P2 it follows from l.e.13 that, for some constant M (dependent only on Xl' X2, PI' P2 and (),

J II.± ri(s)zi 11 ds o ,=1 8,p

x (Joo (tl lIe-<1-8l tX2,i(t)llf;YIP2 dtYIP

= M Ile8t(x l , 1 (t), Xl, 2(t), ... , Xl, n(t»11 i~f;,(X,»

x Ile-(1-8)t(X 2,1 (t), X2 ,2(t), ... , X2,n(t»II~p(l;'(X2»'

Since X1,;(t)+X2,i(t) are arbitrary representations ofzi, i= 1, ... , n it folIo ws from 2.g.I3 that


By 2.g.20, the space [/;,(XI), 1;2(X2)]o.P coincides with 1;([Xt> X2]o,p), up to an equivalence of norm, and the constant C of this equivalence is independent of n. Consequently,

Remarks. 1. Since, by 2.g.18, we have

up to equivalent norms, ifl/p = (1 = (J)/PI + ()/P2' it follows that 2.g.22 gives a sharp result in this case if 1 ~PI <pz ~2.

2. There seems to be no known result of a similar nature concerning the cotype. Note that in view of the lack of duality in the general case between type and co type we cannot dualize 2.g.22.

3. By an argument very similar to the proof of 2.g.22 the following can be proved. If (X I' X 2) is an interpolation pair of Banach lattices and if Xi satisfies an upper Pi-estimate, i= 1,2 then [Xl' X 2 ]O,p satisfies an upper p-estimate provided that l/p=(I-())/pl + ()/P2' By duality we infer that the same results are true for lower estimates. Also, from 1.f.7 we deduce that [Xl' X2 ]o,p is P-8 convex for every 8>0.

Further geometrie properties of [Xl' X2 ]o,p of an isomorphie as well as an isometrie nature can be found in [6].

We conc1ude this section by presenting an application of the Lions-Peetre interpolation spaces. In Ld.6 we described an example of B. Maurey and H. P. Rosenthai of a normalized basic sequence {en }:'= I in a Banach space E which tends weakly to 0 so that no subsequence of {en }:'= I is unconditional. We shall now "uniformly convexify" this example.

Example 2.g.23 [97]. There is a uniformly convex Banach space E with a normalized monotone basis {en }::,= I so that no subsequence of {en } ':= I is an unconditional basic sequence.

Observe that, since E is uniformly convex and thus reflexive we have automatically in this case that en ~ O.

Proo! We recall first the pertinent facts concerning the space E defined in Ll.d.6. The definition of E depends on a collection LI of sequences (j = { a j} '1= I of disjoint finite subsets of the integers. One of the properties of LI was that, for every subsequence NI ofthe integers, there is a (j = {aJ'1= I in LI so that aj e N 1 for every j. The space E was defined as the completion of the space of sequences of scalars x = (a p az' ... ) which are eventually 0, with respect to the norm


where the supremum is taken over all sequences 15 = (0) j; 1 in LI. The unit vectors {en}::"1 forma monotone basis of Eand IlenllE= IIenllE*= I for every n. The family LI was chosen so that, for every 15= {O'j}j; 1 in LI, every n and every choice of scalars {c j }j=l'

h - = - 1/2 ~ . - I 2 w ere uj-O'j ~ epJ- , , .... ieuj

Consider now the space E= [E, 12]1/2,2' By 2.g.2I, Eis uniformly convex. By 2.g.14 and 2.g.16, there is a constant C so that for every sequence x=(a1, a2 , ... )

of scalars which is eventually 0,

The unit vectors {en }:'= 1 form a monotone and normalized basis of E. (That lien llii = I for every n follows easily by a direct checking of the definition of the Lions-Peetre interpolation space. The inequalities above ensure only that C - 1 ~ lien llii ~ C for every n which is, of course, also sufficient for our purposes.) For an arbitrary 15 = { 0' j} j= 1 in LI and every integer n we have

where uj = U; 1/2 L ep j = 1, 2, .... Since, by the definition of 11 . IIE'

Ii t Ujllie,,~ 1 , J= 1 E*

for every n, we get that

and hence,

Thus, {Uj }j;1 is not unconditional. Since every subsequence {en}neNt of {en}:'=1 has a block basis equal to such a {u j} j; 1 it follows that {en }::" 1 has no unconditional subsequence. 0

Remark. By an argument similar to the one used in the comments following l.c.lO it follows that E is not isomorphie to a subspace of an order continuous Banach lattice.

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126. Szarek, S. J.: On the best eonstants in the Khintehine inequality. Studia Math. 58, 197-208 (1976). 127. Tzafriri, L.: An isomorphie eharaeterization of L p and Co spaees. Studia Math. 31, 195-304

(1969). 128. Tzafriri, L.: An isomorphie eharaeterization of L p and Co spaees, 11. Mich. Math. J. 18,21-31

(1971). 129. Tzafriri, L.: Reflexivity ofcyclie Banach spaces. Proc. Amer. Math. Soe. 22, 61--{)8 (1969). 130. Tzafriri, L.: Reflexivity in Banach lattiees and their subspaees. J. Funet. Anal. 10, 1-18 (1972). 131. Yudin, A. J.: Solution of two problems on the theory of partiaUy ordered spaees. Dokl. Akad.

Nauk SSSR 23, 418-422 (1939) (Russian). 132. Zygmund, A.: Trigonometrie series I, 11, seeond edition, Cambridge University Press 1959. 133. Davis, W. J., Ghoussoub, N., Lindenstrauss, J.: A lattice renorming theorem and applieations to

veetor-valued processes (to appear). 134. Feder, M.: On subspaces ofspaees with an uneonditional basis and spaees of operators (to appear). 135. FeIler, W.: An introduction to probability theory and its applieations, Vol. 11. New Y ork: Wiley

1966. 136. Kalton, N. J.: Embedding Li in a Banach lattiee. Israel J. Math. (ta appear). 137. Rodin, V. A., Semyonov, E. M.: Rademacher series in symmetrie spaees. Analysis Mathematika

1,207-222 (1975).

Su bject Index

absolutely summing operators (see operators) approximation property (A.P.) 33, 102, 113 - -, bounded (B.A.P.) 112 - -, compact (C.A.P.) 102, 103, 107, 111 - -, metric (M.A.P.) 33 averaging operators (see operators)

Banach lattice 1-4,6,7,13,18,22,29,31,45,79, 93, 102

- -, atom ofa 20 - -, band of a 3,9, 10, 23, 24, 35, 85 - -, canonical embedding of a (in its second

dual) 4, 17,27,28,34,35 - -, characterizations of an order continuous

7,8, 10,25,27-29 - -, complexification of areal 43 - -, definition of I - -, disjoint elements in a 2,6,7,9,20-24,38,

39, 52, 82, 84, 89, 91, 141, 157, 165, 181, 182, 188, 198-200

--,dualofa 3,4,17,24,25,29,83,84,96,97, 121

- -, functional representation of a 25, 27, 28, 35,36,39,40,52,83

- -s, interpolation pair of (see interpolation) - -, lower q-estimate (for disjoint elements) of a

79,82-85,88-92,97-99,130,132,139,140,231 - -, lower q-estimate constant of a 83 - -, monotone sequences in a 3 - -, norm-bounded sequences in a 34, 35 - -, order bounded sets in a 3-8, 29 - -, order complete 4, 6, 8, 10, 20, 35, 85 - -, u-order complete 4-10, 12, 14, 15, 20, 25,

26, 29, 34, 38, 83, 89, 114, 118 - -, order continuous (norm in a) 7,8, 10, 12,

14,15,19,21,25-29,31,32,34-40,83,89,118, 186,232

- -, u-order continuous (norm in a) 7,9, 14, 22,25,26,29,34-36,38,83, 121

- -, p-convex 40,45,46,49,51-55,57,59,73, 79-82, 85, 87-90, 95-99, 133, 181, 188, 191, 194-198,209,220-223,231

Banach lattice, p-convexification of a 53, 55, 58, 81,89,98,99,201

- -,p-convexity constant of a 46,51,81,99, 138,189,191, 194, 195, 197

- -, projection band ofa 9-11,25,34,35 - -, q-concave 40, 45, 46, 49-51, 53-56, 59,

73, 74, 79-82, 85, 88, 92, 93, 95-98, 126, 133, 162,169,170,174-176,181,186-188,195,197, 198,200,201,209,220-223

- -, q-concavification ofa 54,55,88,201,221, 222

- -, q-concavity constant of a 46, 51, 77, 81, 133,189, 193

- -, second dual of a 4, 84, 85 - -, separable 7,12,25,29,38,52,83,89, 118,

121 - -, sublattices of a 3, 19,22,51,55,59, 103 - -, subspaces of a 31,34,35,37-39,51,98,

102, 189 - -, units of a (see strong unit and weak unit) - -, upper p-estimate (for disjoint elements) of

a 79, 82-85, 87,88,90,96,97, 130, 132, 139, 140, 195, 197,200,201,231

- -, upper p-estimate constant of a 83, 202 band (see Banach lattices) basis, block basis of a 158, 161, 198 -, conditiona1 (see conditiona1 basis) -, monotone 150,151,231,232 -, perfectIy homogeneous 24 -, perturbation of a 38, 176, 179 -, precisely reproducible 158, 162, 189, 190,

195, 198, 200,201 -, reproducible 150, 158, 162 -, subsymmetrie (see subsymmetrie basis) -, symmetrie (see symmetrie basis) -, unconditional (see unconditional basis) Bohnenblust's characterization of abstract Lp and

M spaces 18 Boolean algebras of projections (see projections) - -, Stone's representation theorum for (see

Stone) - operations 114

240

Boyd indices 129-131, 134, 139, 142, 144, 147, 157,162, 167, 168, 175, 179, 188, 199,200,202, 212,219,220

- interpolation theorem (see interpolation)

C*-algebra, commutative 17 Caratheodory extension theorem 15 Cauchy-Schwarz inequality 155, 167 Central limit theorem 135, 136 concave Banach lattice (see Banach lattice, q

concave) - operators (see operators, q-concave) concavification of a Banach lattice (see Banach

lattice, q-concavification of a) condition ,12 64, 115, 120, 199 conditional basis 162 - expectation 122,128,151,179,215 cone, positive 2,3,7, 12, 17,36,44 convex Banach lattice (see Banach lattice, p

convex) convex operators (see operators, p-convex) convexification of a Banach lattice (see Banach

lattice, p-convexification of a) -, uniform 216,231 convexity, modulus of 59,63,65-72,78-80,88,

90, 97, 99, 229 -, strict 60-62 -, uniform 40,59-63,69,77,79-81,88,90,97,

102, 162, 199,202,228,229,231,232 cotype 59,72-74,77-79,82,88,90,93,96-99,

102, 111, 181, 189,208,231 - constant 73 cyc1ic space 12-16,20,26 - vector 12

decomposition method (see PeIczynski's decom-position method)

- property 2,22,26,36,88, 125,221 differentiabiJity, Frechet 61 -, Gäteaux 61 distribution function 116, 117, 132, 141, 142,

157, 160, 170,205-207,209,211 -, normal 135,213 -, Poisson 204,205,214 -, uniform 205 -, vectors with the same 140-142,157,158,160 Doob's maximal inequality 153, 155 Dvoretzky's theorem 63

Eberlein's theorem 35 equi:integrable sets 216,223 estimate, lower (see Banach lattice, lower q

estimate of a) -, upper (see Banach lattice, upper p-estimate of

a)

extreme points 121,124,159

Subject Index

factorization theorems 57,59,95 - ofweakly compact operators 216,224 Fatou property 30,118-121,123,128 Fourier transform 182 function, distribution (see distribution function) - homogeneous of degree one 40, 42, 44 -, locally integrable 29, 142 -, spaces (see Köthe, Lorentz, Orlicz and re-

arrangement invariant function spaces) -, square 126,154,156, 166 functional calculus 42, 43 -, Hahn-Banach 25 -, strict1y positive 25-27

greatest lower bound (g.l.b) 1,8,21 Grothendieck's inequality 93, 94 - universal constant 93, 94, 173

Haar system 2, 150, 151, 154-158, 160-162, 165,168,175-179,181,182,186,187,198-200, 223

Hölder inequality 44, 53, 56, 89, 106, 143, 154, 174,209

Hölder type inequalities 43, 49, 76

ideal (see Banach lattices) independent random variables (see random vari

ables) integrals (see also space X' of integrals) 29 interpolation method ofLions-Peetre 149,216,

225,227-229,231,232 - pair of spaces 216-218, 225, 227-230 - - - Banach lattices 218,219,221,231,232 -, real method in 216 -, spaces 215,216,218,219,227,228,230-232 - theorem, Boyd 144,147, 156, 168,209 - -, Marcinkiewicz 130,149,153,228 - -, Riez-Thorin 149,156,166 - -s 114, 123, 124, 126, 127, 129, 130, 142,

144,171,215

Kakutani's representation theorem for abstract L p spaces 15, 18, 24, 93, 95

- - - - - M spaces 16, 18,24,93,94 Khintchine's inequality 39,49,50,70, 134, 169,

174,186, 187,204 - -, two dimensional 173, 174 Kolmogorov's consistency theorem 205 Köthefunction spaces 14,28-31,36,38,83,89,

104,114-118.121,123,127-130,164,188,216, 218,223

Krein-Milman theorem 121,159 Krivine's theorem 72. 111, 141

Subject Index

lattices (see Banach and vector iattices) least upper bound (l.u.b) 1,3-6,8,29,35,42 Liapounoff's theorem 158 Lions-Peetre interpolation (see interpolation

method of Lions-Peetre) Lorentz spaces 64,88,97-99,120,121, 129, 132,

133,142 - -, modulus of convexity of 67,71,72 lower q-estimate (see Banach lattice, lower q

estimate of a)

Marcinkiewicz interpolation theorem (see interpolation)

- weak type, operators of (see operators of Marcinkiewicz weak type)

martingale 150-156 matrices 123, 124, 138, 173 Maurey-Rosenthal example 39,231 measure space, automorphism of a 114-116,

118, 127, 138, 149, 151, 186 measure space, complete 29 - -, separable 114 modulus of convexity (see convexity, modulus of) - - smoothness (see smoothness, modulus of)

norming subspace 29,30, 118

operators, automatie continuity of positive 2 -, averaging 122 -, compact 33 -, diagonal of 21 -, dilation 130, 131, 163, 170,206,219 -, finite rank 33 -, multiplicity theory·for spectral 1I -, norm IIT111.2 of 217-219 - of Marcinkiewicz weak type 144, 145 -ofstrongtype 144,149,150,156 - of weak type 130, 142, 144, 145, 147-149,

153, 156, 167,228 -, order preserving 2 -, p-absolutely summing 56, 57, 93 -,p-convex 45,47-49,55,57,59,95 -, positive 2,4,55-57,59,95, 115, 125 -, q-concave 46,48,49,55-57,59,95 -, quasilinear 126,127,147,149,153,228 -, simultaneous extension 21,22 -, weakly compact 216, 224 order complete Banach lattice (see Banach lattice,

order complete) - continuous Banach lattice (see Banach lattice,

order continuous) - intervaI 28 - isometry 3,4, 15-17, 19, 25, 29, 36,41,44,

58, 118, 190,211

241

order isomorphism 2,3,22,24,26,35, 121, 185 -like relation 123, 125 Orlicz functions 64, 65, 119, 120, 134, 164, 220 - spaces 67, 115, 120, 131, 134-136, 139, 140,

162, 163, 165, 181, 187, 188, 199,200,209,223 --, modulus ofconvexity of 67,71,72

p-absolutely summing operators (see operators, p-absolutely summing)

p-convex Banach lattice (see Banach lattice, pconvex)

- operators (see operators, p-convex) p-convexification of a Banach lattice (see Banach

lattice, p-convexification of a) partition of unity 94 Pelczynski's decomposition method 172, 209,

220,222 polar yJ. ofa band Y 9,10,20,26,27,36 precisely reproducible basis (see basis, precisely

reproducible) primary space 168,179 process, Poisson 202,204-207,212,213 -, r-stable 213 -, stationary 204, 206, 213 -, symmetrie 206,213 - with independent increments 204,206,213 projections, band 10, 11, 23-25, 85 -, Boolean algebra (B.A.) of 11,12,14,15,20 -, u-complete B.A. of 11-13,26 -, contractive 8, 10, 19-21, 122, 123,209 - ofthe form Px 8-10,12, 13, 15, 16 -,orthogonal 137,138,171,172,209 -, positive 8, 10, 19, 20 -, Riesz 166-168

q-concave Banach lattice (see Banach lattice, q-concave)

- operators (see operators, q-concave) q-concavification of a Banach lattice (see Banach

lattice, q-concavification of a) quotient space 3

Rademacher elements 92, 93 - functions 29,51,72-74,77,85,104,106,134,

136-138,160,162,169,198,204,213 Radon-Nikodym derivative 31 - - property 69 --theorem 27,29,122 random variables, characteristic functions of

204,205 - -, independent 85-87, 135, 160, 182, 185,

204-207,213,214 --,p-stable 181,182,185,213 ~-, symmetrie 206,207,214

242

rearrangement, decreasing 117,123,125 - invariant (r.i.) function space 40, 114, 115,

117-123,125,126,130-134,136,138,141,145, 149-151,156,158,161,162,165-170,176,178, 181-183,190,191,198-200,204,207,209,211, 220

- - - -, analytic part of a 168 - - - -, complemented subspaces ofa 168,

170,172,175,176,178,195,199,200,209 - - - -, minimal 118-121, 123, 126, 128,

133, 147, 204, 205, 211, 219, 220, 223 - - - -, maximal, 118-123, 126, 128, 136,

147,204,205,211,219 reflexivity 33,35,47,61,62,69,78,224,225,231 - in Banach lattices 4,22,31,35,37 - in r.i. function spaces 119,132,181,188,199,

200 renorming theorems 54, 55, 80, 88, 90, 115 reproducible basis (see basis, reproducible) Riemann-Stieltjes integral 12, 14 Riesz-Thorin interpolation theorem (see inter

polation)

Schauder decomposition 111 - -, finite dimensional (F.D.D.) 33, 34 - -, unconditional 33 separation theorem 30,58, 124 smoothness 60,61 -, modulus of 59,63,64,67,68,70,71,78-80,

88, 90, 229, 230 -, uniform 59-63,69,77,80,88,90,229 space, abstract LI 17,23,24,44,95 -,-Lp 14,15,18,21,22,24,58 -, - M 9,14,16-22,24,41,73 - Co 6, 7, 16,20, 22, 24, 27, 34-37, 46, 52, 57,

78,89,93,96, 119, 127, 128, 158, 187 . - co(r) 7,9,20,22,24 - co(X) 46, 47 - C(O, I) 2,4, 10, 13,36,52,62, 114, 162, 167 - C(K) 2,4-6,14,16,17,20,41-43,56,57,93,

94,96 - -, sublattices ofthe 16,20,41 -C(ßN) 6 -Hp 168 -, Hilbert (12' L 2 (0, I), etc) 52,63,73,74,96,

105,106, 111, 134-137,154, 162, 163, 165, 166, 181, 188, 190, 191, 195, 196, 198, 199,202,203, 209,213

-J 36,39 - K(Xl , X 2 , Y, {a.}, {b.}) (also K(Xl , X 2 , Y,

{a.}, {b.}» 218-221 -11 35,36,78,82,93,118,126,132,160, 161 -li 62,74,90,92-94, 124, 141, 157, 197,201 -Ip 15, 22-24,46,81,87, 107, 108, 158, 178,

182, 195, 211, 213, 222

Subject Index

space Ip , modulus of convesity of the 63, 71, 72 --,--smoothnessofthe 63,71 - I; 20, 111, 112, 194 - Ip(X) 46-48, 83, 193, 201 - I~ 63, 93, 112, 194, 200--202 -I"" 6,7, 13, 30, 36, 115, 118 -I::' 41,74,90,92-94,124,141,157,182,198 - I",,(r) 6, 7, 26 -I",,(X) 47 -Ip EB 12 213 -IM (see Orlicz spaces) - Ll(O, I) 2,4,78, 118-122,125, 127, 128, 130,

156,161, 167,187,191,200,215 - L l ( - CXJ, + CXJ) 182 -Ll(~) 17,25-27,29-31,35,38,39,44,93,94,

153,208 -Ll(X) 77 - Lp(O, I) 2, 52, 63, 75, 77, 102, 106, 129, 134~

140,151,156,161,162,166,168,169,171,172, 175,178,181,182,188-191,198-200,202,203, 209,211-214,223,231

- Lp(~) 2,4,9, 10, 12, 13, 15, 16, 19,20,22,38, 43,45,53,55,57,59,73,114,120,142-144,153, 168,182,202,207,208

--, modulus ofconvexity ofthe 63,72,81 - -, - - smoothness of the 63, 72 - -, sublattices ofthe 16 - L p(/2 ) 171, 172 - L.(X) (also L.(R, X» 170, 225-227, 229 - L 2(X) 67-69, 78, 229 - L",,(O, 1) 7, 118-122, 126-127, 130, 134-136,

150,160,161,183,191,215 - L",,(~) 17,25,30,44,73,96,159,214 - Ll(O, CXJ)nL",,(O, CXJ) 118,119,123,132 - Ll(O, CXJ)+L",,(O, CXJ) 118-120,123,125,127,

132,217 -LiO, CXJ)nL.(O, CXJ) 132-134,147,189,191,

202,209,213,214,218 - LiO, CXJ)+L.(O, CXJ) 132-134, 149, 191, 202,

209,218 - Lp •• 142-144,148,212,228 - L M (see Orlicz spaces) -, Lorentz (see Lorentz spaces) -, Orlicz (see Orlicz spaces) -, partially ordered I - Rad X 169-172 - Up,q 222, 223 - X(co) 46,47,85 - X(lp) (also XOp» 46-48 - X(/2 ) (aso X(/2» 168-176,179,180 - X(l",,) (also X(I",,» 46, 47 - X' ofintegra1s 29-31,117-121,137 - X l nX2 217,218,226,227 - Xl +X2 217,218 - [XJ , X2 ]e.p 226-228,230,231

Subject Index

space X p ,2,w (also X p ,2) 213,214 -Xy,u 214,215 Stone's representation theorem for Boolean alge-

bras 15 stopping time 152, 153 strong unit 9, 16, 17,41

- type operators (see operators) subsymmetrie basis 158 symmetrie basis 97,99,114,115,119,122,124,

127,128,160,162,188 - -, block bases generated by one vector of

a 140 - -, - - with constant coefficients of a 121

topological space, basically disconnected 4, 5 --ßN 5 - -, extremally disconnected 4, 5 - -, totally disconnected 15, 26 tree of subsets 178, 179 trigonometrie system 150, 165-167 Tsire1son's example 81 Tychonoff's theorem 20 type 59,72-74,77-79,82,88,90,92,93,96,97,

102,111,112,181,188,195,197,200,201,207, 209,230,231

- constant 73, 79, 112, 195 -, duality ofthe 79,96 - power 63, 78-80, 88, 90

(u), property 31, 32, 34 ultrapower 213

243

unconditional basic sequences in Banach lattices 37-39,50,51,174,175,188,232

- - - - rj, function spaces 126, 137, 168,

173,175-177,180,188 - basis 2,8,12,18,31,38,45,50,51,72,78,81,

150,151,155-158,160-162,168,176,191,200, 207,218-220,222,231,232

- -, projections associated to a 8, 12, 156 - constant 2,21,50,72,78,156,161,174,177,

207,222 - direct sum of disjoint ideals 9, 13, 15,36 - F,D.D. (see Schauder decomposition) unconditionally convergent series 69, 70 - - -, weakly (see weak unconditional con-

vergence) uniform convexity (see convexity, uniform) - smoothness (see smoothness, uniform) universal space V 1 for unconditional bases 219,

220,222 upper p-estimate (see Banach lattices, upper

p-estimate of a)

vector lattices 2, 14, 36

Walsh functions 104,137,138 weak compactness 35,36,216,224,225 - sequential completeness 31,34-37,47 - type operators (see operators) - unconditional convergence (w.u.c.) 31,33,34 - unit 9,12-16,25-29,35,36,38,39

Zorn's lemma 9

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