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H A N D B O O K OF

THE GEOMETRY OF BANACH S PACES

This Page Intentionally Left Blank

H A N D B OOK OF THE GEOMETRY OF B ANAC H S PACE S

Volume 1 Edited by

W.B. J O H N S O N Texas A&M University, College Station, Texas, USA

J. L I N D E N S T R A U S S The Hebrew University of Jerusalem, Jerusalem, Israel

2001

ELSEVIER Amsterdam

9L o n d o n

9N e w

York

9O x f o r d

9P a r i s

9Shannon

9T o k y o

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 EO. Box 211, 1000 AE Amsterdam, The Netherlands 9 2001 Elsevier Science B.V. All rights reserved. This work is protected under copyright by Elsevier Science, and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier Science Global Rights Department, PO Box 800, Oxford OX5 1DX, UK; phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-maih [email protected]. You may also contact Global Rights directly through Elsevier's home page (http://www.elsevier.nl), by selecting 'Obtaining Permissions'. In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) 978 7508400, fax: (+1) 978 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 207 631 5555; fax: (+44) 207 631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works: Tables of contents may be reproduced for internal circulation, but permission of Elsevier Science is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage: Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier Science Global Rights Department, at the mail, fax and e-mail addresses noted above. Notice: No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made. ISBN: 0-444-82842-7 First edition 2001

Library of Congress Cataloging-in-Publication Data Handbook of the geometry of Banach spaces / edited by W.B. Johnson, J. Lindenstrauss.-1 st ed. p. cm. Includes indexes. ISBN 0-444-82842-7 (v.1 : alk. paper) 1. Banach spaces. I. Johnson, W. B. (William B.), 1944- II. Lindenstrauss, Joram, 1936QA322.2.H36 2001 515'.732-dc21 2001023427

British Library Cataloguing in Publication Data Handbook of the geometry of Banach spaces Vol. 1 edited by W.B. Johnson, J. Lindenstrauss 1. Banach spaces I. Johnson, William B., 1944- II.Lindenstrauss, J. 515.7'32 ISBN0444828427 G The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

Preface 1. Introduction The aim of this Handbook is to present an overview of the main research directions and results in Banach space theory obtained during the last half century. The scope of the theory, having widened considerably over the years, now has deep and close ties with many areas of mathematics, including harmonic analysis, complex analysis, partial differential equations, classical convexity, probability theory, combinatorics, logic, approximation theory, geometric measure theory, operator theory, and others. In choosing a topic for an article in the Handbook we considered both the interest the topic would have for non-specialists as well as the importance of the topic for the core of Banach space theory, which is the study of the geometry of infinite dimensional Banach spaces and n-dimensional normed spaces with n finite but large (local theory). Many of the leading experts on the various aspects of Banach space theory have written an exposition of the main results, problems, and methods in areas of their expertise. The enthusiastic response we received from the community was gratifying, and we are deeply appreciative of the considerable time and effort our contributors devoted to the preparation of their articles. Our expectation is that this Handbook will be very useful as a source of information and inspiration to graduate students and young research workers who are entering the subject. The material included will be of special interest to researchers in Banach space theory who may not be aware of many of the beautiful and far reaching facets of the theory. We ourselves were surprised by the new light thrown by the Handbook on directions with which we were already basically familiar. We hope that the Handbook is also valuable for mathematicians in related fields who are interested in learning the new directions, problems, and methods in Banach space theory for the purpose of transferring ideas between Banach space theory and other areas. Our introductory article, "Basic concepts in the geometry of Banach spaces", is intended to make the Handbook accessible to a wide audience of researchers and students. In this chapter those concepts and results which appear in most aspects of the theory and which go beyond material covered in most textbooks on functional and real analysis are presented and explained. Some of the results are given with an outline of proof; virtually all are proved in the books on Banach space theory referenced in the article. In principle, the basic concepts article contains all the background needed for reading any other chapter in the Handbook. Each article past the basic concepts one is devoted to one specific direction of Banach space theory or its applications. Each article contains a motivated introduction as well as

vi

Preface

an exposition of the main results, methods, and open problems in its specific direction. Most have an extensive bibliography. Many articles, even the basic concepts one, contain new proofs of known results as well as expositions of proofs which are hard to locate in the literature or are only outlined in the original research papers. The format of the chapters is as varied as the personal scientific styles and tastes of the contributors. In our view this makes the Handbook more lively and attractive. There are many strong and intricate interconnections between the different subjects treated in the various articles, so some articles contain many cross references to other articles. But, as mentioned above, no article depends on anything past the material treated in the basic concepts article. Because of this lack of interdependence, we chose to order the chapters after the basic concepts article alphabetically according to the first author of the chapter. We included in volume 1 the chapters which came first in this order among those articles which were ready by the deadline we set for this volume. The second volume of this Handbook will contain the chapters which come later in the alphabetical order as well as those chapters which were finished only after the deadline for volume 1. This volume ends with a subject index and an author index for the material of the present volume. Volume 2 will contain a subject and author index for the entire Handbook.

William B. Johnson and Joram Lindenstrauss

List of Contributors Abramovich, Yu.A., Indiana University- Purdue University, Indianapolis, IN (Ch. 2) Aliprantis, C.D., Purdue University, West Lafayette, IN (Ch. 2) Alspach, D., Oklahoma State University, Stillwater, OK (Ch. 3) Ball, K., University College London, London (Ch. 4) Bourgain, J., Institute for Advanced Study, Princeton, NJ (Ch. 5) Burkholder, D., University of lllinois, Urbana, IL (Ch. 6) Casazza, E, University of Missouri, Columbia, MO (Ch. 7) Davidson, K.R., University of Waterloo, Waterloo (Ch. 8) Delbaen, E, E.T.H., Zurich (Ch. 9) Deville, R., University of Bordeaux, Bordeaux (Ch. 10) Diestel, J., Kent State University, Kent, OH (Ch. 11) Dilworth, S.J., University of South Carolina, Columbia, SC (Ch. 12) Enflo, E, Kent State University, Kent, OH (Ch. 13) Figiel, T., Institute of Mathematics, Sopot (Ch. 14) Fonf, V.E, Ben Gurion University, Beer-Sheva (Ch. 15) Gamelin, T., The University of California, Los Angeles, CA (Ch. 16) Ghoussoub, N., University of British Columbia, Vancouver (Ch. 10) Giannopoulos, A.A., University of Crete, Heraklion (Ch. 17) Godefroy, G., Universite Paris VI, Paris (Ch. 18) Jarchow, H., University of Zurich, Zurich (Ch. 11) Johnson, W.B., Texas A&M University, College Station, TX (Chs. 1, 19) Kisliakov, S., POMI, Saint Petersburg (Chs. 16, 20) Koldobsky, A., University of Missouri, Columbia, MO (Ch. 21) K6nig, H., University ofKiel, Kiel (Chs. 21, 22) Lindenstrauss, J., The Hebrew University of Jerusalem, Jerusalem (Chs. 1, 15) Lomonosov, V. Kent State University, Kent, OH (Ch. 13) Milman, V.D., Tel-Aviv University, Tel-Aviv (Ch. 17) Odell, E.W., The University of Texas, Austin, TX (Ch. 3) Phelps, R.R., University of Washington, Seattle, WA (Ch. 15) Pietsch, A., Jena University, Jena (Ch. 11) Schachermayer, W., Vienna University of Technology, Vienna (Ch. 9) Schechtman, G., The Weizmann Institute of Science, Rehovot (Ch. 19) Szarek, S., Case Western Reserve University, Cleveland, OH and Universit~ Paris VI, Paris (Ch. 8) Wojtaszczyk, E, Warsaw University, Warsaw (Ch. 14)

vii


Contents P refa c e List of Contributors

1. Basic concepts in the geometry of Banach spaces W.B. Johnson and J. Lindenstrauss 2. Positive operators Y.A. Abramovitch and C.D. Aliprantis 3. Lp spaces D. Alspach and E. Odell 4. Convex geometry and functional analysis K. Ball 5. A p-sets in analysis: Results, problems and related aspects J. Bourgain 6. Martingales and singular integrals in Banach spaces D.L. Burkholder 7. Approximation properties P. G. Casazza 8. Local operator theory, random matrices and Banach spaces K.R. Davidson and S.J. Szarek 9. Applications to mathematical finance F. Delbaen and W. Schachermayer 10. Perturbed minimization principles and applications R. Deville and N. Ghoussoub 11. Operator ideals J. Diestel, H. Jarchow and A. Pietsch 12. Special Banach lattices and their applications S.J. Dilworth 13. Some aspects of the invariant subspace problem P. Enflo and V. Lomonosov 14. Special bases in function spaces T. Figiel and P. Wojtaszczyk 15. Infinite dimensional convexity V.P. Fonf J. Lindenstrauss and R.R. Phelps 16. Uniform algebras as Banach spaces T. W. Gamelin and S. V. Kislyakov

V

vii

85 123 161 195 233 271 317 367 393 437 497 533 561 599 671

x

Contents

17. Euclidean structure in finite dimensional normed spaces A.A. Giannopoulos and V.D. Milman 18. Renormings of Banach spaces G. Godefroy 19. Finite dimensional subspaces of L p W.B. Johnson and G. Schechtman 20. Banach spaces and classical harmonic analysis S. V. Kislyakov 21. Aspects of the isometric theory of Banach spaces A. Koldobsky and H. Kdinig 22. Eigenvalues of operators and applications H. KOnig

707

Author Index

975

Subject Index

993

781 837 871 899 941

CHAPTER

1

Basic Concepts in the Geometry of Banach Spaces William B. Johnson* Department of Mathematics, Texas A &M University, College Station, TX 77843, USA E-mail: johnson @math.tamu.edu

Joram Lindenstrauss t Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Notations and special Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Classical spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Geometry of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Analysis in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Finite dimensional Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Local structure of infinite dimensional spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Some special classes of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 13 20 30 35 43 53 61 74 81 83

*Supported in part by NSF DMS-9623260 and DMS-9900185, Texas Advanced Research Program 010366-163, and US-Israel Binational Science Foundation. +Supported in part by US-Israel Binational Science Foundation and by NSF DMS-9623260 as a participant in the Workshop in Linear Analysis & Probability at Texas A&M University. H A N D B O O K OF THE GEOMETRY OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved


Basic concepts in the geometry of Banach spaces 1. Introduction In this introductory chapter we present results and concepts which are often used in Banach space theory and will be used in articles in this Handbook without further reference. The material we treat, while familiar to experts in Banach space theory, has not made its way into introductory courses in functional analysis. The main purpose of this article is to make the subsequent articles accessible to anyone whose background includes basic graduate courses in analysis and functional analysis. Each section of this article is devoted to one aspect of Banach space theory. Although this article can in no way be considered as an introductory course in Banach space theory, we do include indications of proof of some basic results in the hope that this will help the reader understand and get a feeling for the various concepts which are discussed. We reference only some of the (mostly introductory) books which treat the basic material we describe. Original sources are referenced in these books. We also mention some results which are not yet in elementary books on Banach spaces but which help to clarify the general picture. These generally were either discovered recently or are more difficult than the rest of the material. In these cases we refer to specific articles in this Handbook which treat the topic. In general, we do not attach names to theorems (except when experts generally refer to the theorem with a name attached, such as the Hahn-Banach theorem, the Krein-Milman theorem, Rosenthal's s theorem . . . . ) or give any historical background. Instead we refer to the books in the references as well as the articles in this Handbook.

2. Notations and special Banach spaces Banach spaces will have either real or complex scalars. When the scalar field matters (for example, in results involving spectral theory or in theorems of an isometric nature or when analyticity plays a r61e), the scalar field is mentioned explicitly, but in the notation for special spaces the scalars are not specified. Operators between Banach spaces are bounded and linear. An invertible operator T is called an isomorphism. Two norms on a vector space are called equivalent if the identity operator on X (with the two given norms) is an isomorphism. If IIT l] = 1 = I1T-1 ]], T is called an isometric isomorphism or simply an isometry and the domain and range of T are said to be isometric. We write X ~ Y to denote that the spaces are isomorphic. To denote isometry we use the equal sign. An isomorphism from a Banach space onto itself is called an automorphism. A Banach space Y is said to be a quotient of the Banach space X if Y is isometric to X / Z for some closed subspace Z of X. By the open mapping theorem, Y is isomorphic to a quotient of X if there is an operator from X onto Y. If X ~ Y, d(X, Y) denotes the Banach-Mazur distance between the spaces, defined to be the infimum of IITII lIT -1 II as T ranges over all isomorphisms from X onto Y. So d(X, Y) = 1 if X and Y are isometric; the converse is true for finite dimensional spaces but not for infinite dimensional spaces. Note that the "triangle inequality" for the Banach-Mazur distance is submultiplicative rather than subadditive; that is, d(X, Y) 0, there is a finite rank operator T on X such that [Ix - Tx l[ < ~ for every x ~ K. If a space X has the AP, then every compact operator S into X is the norm limit of a sequence of finite rank operators. Indeed, take for n = 1, 2 . . . . a finite rank operator Tn on X so that for each x in the image under of the unit ball of the domain of S the inequality IIx - Tnxll Z

1

~Y

commutes; that is, T -- V SU. Now we can make precise the statement about Ip,r" If T is a noncompact operator from a subspace of g.p into g-r, 1 2 and Bp = 1 for p ~< 2, and the exact values of Ap and Bp are known. The most elementary proof of Khintchine's inequality proceeds by checking the right inequality in (1) for p an even integer. This clearly gives the result for all 2 ~< p < oo and one then obtains the result for 0 < p < 2 from the extrapolation principle. The

Basic concepts in the geometry of Banach spaces

17

less computational modern proof of Khintchine's inequality gives a vector valued version of Khintchine's inequality called the Kahane-Khintchine inequality. This will be done in Section 8. The existence of an r-stable variable g (see Section 2) for 0 < r < 2 shows that the subspace structure of L p (0, 1), 0 < p < 2, is much more complicated than that of L p (0, 1), 2 < p < cx~. Indeed, just as in the Gaussian case, there exists for 0 < r < 2 a sequence {gn}n~l of symmetric r-stable random variables defined on (0, 1). If 0 < p < r, then these random variables are in Lp(O, 1). Now if ~ k -n- 1 I ~ l r 1, then ~ = l otkgk is again symmetric r-stable and hence has the same norm in L p(O, 1) as g. This means that for all 0 < p < r, the mapping en ~ Ilgllp-1 gn, n -- 1, 2 . . . . . extends to an isometry from er into L p (0, 1). In Section 9 we explain how to derive from this the fact that for 1 ~< p < r < 2, Lr (0, 1) embeds isometrically into L p(O, 1). It turns out that this covers all the cases in which er embeds isomorphically into Lp(iZ ) with p and r finite. The remaining cases are discussed in Section 8. In the reflexive range 1 < p < cx~, information about subspaces of Lp,(#), 1/p + 1/p* - 1, given above gives information on quotients of Lp(#). It turns out that the case p = 1 is quite different: Every separable Banach space X is isometric to a quotient of el. Indeed, if {Xn}n~=l is dense in the unit ball of X, the linear extension of the map en ~ Xn (where {en }~--1 is the unit vector basis for el) maps the unit ball of el onto a dense subset of the unit ball of X and hence extends to a quotient mapping from el onto X. Similarly, if the Banach space X has density character x, then X is isometric to a quotient of el ( F ) when F has cardinality x. Another useful and interesting property of el (which also holds for e l ( F ) for any set F ) is the lifting property: If T is an operator from a Banach space X onto el then there is a lifting S of T; that is, an operator S :el --+ X for which T S = Ie~. Indeed, by the open mapping theorem there are Xn in X with Txn = en and )~ :=- supllxn II < cx~. The mapping en ~ Xn then extends to an operator S : e 1 ---+ X with [[Sll - ;~ satisfying T S = le~. As noted in Section 3, e 1 has the Schur property; that is, weakly convergent sequences in el are norm convergent. An immediate consequence is that every weakly compact subset of el is norm compact. The weakly compact subsets of L 1(0, 1) are more complicated since L 1(0, 1) contains infinite dimensional reflexive subspaces. There is however a nice characterization of subsets of L1 (/z) which have weakly compact closure when /z is a finite measure. First, if X is any Banach space and W is a subset of X such that for each E > 0 there exists a weakly compact set S so that W C S + EBx, then W has weakly compact closure (use the fact that a bounded subset of X is weakly compact if its weak* closure in X** is a subset of X). Next, given W C L1 (#) with # a finite measure, set for k E N, a k - a k ( W ) "--sup{llxllxl~klll" x ~ W}. Clearly {ak}~= 1 decreases to some a = a(W) ~ O. The set W is called uniformly integrable if a = 0. If a = 0 then W has weakly compact closure because W is a subset of kBL~(~) + akBLl(~) and BL~(u) is weakly compact in L1 (/z). If a(W) > 0, the set W does not have weakly compact closure and in fact even contains a sequence equivalent to the unit vector basis of el. To see this, define two further numerical parameters for a subset W of L 1(#); namely, b(W) := sup l i m n ~ { Ilxn 1An II1 }, where the supremum is over all sequences {Xn}n~__l in W and {An}~=l of measurable sets with #An --+ 0; and c(W), defined the same way as b(W) except that the supremum is

18

W.B. Johnson and J. Lindenstrauss

over all sequences of disjoint measurable sets. It is an elementary exercise in measure theory to verify that a(W) -- b(W) = c(W). Now if c(W) > 0, take a sequence {Xn}n~=l in W and a sequence {An}n~=l of disjoint measurable sets so that 0 < [IXn1an II1 ~ c(W). Clearly c({xn 1Xn}) = 0, so {Xn 1X, } has weakly compact closure. The sequence {Xn 1An }nCC_--I is equivalent to the unit vector basis of e 1 since the An's are disjoint and IlXn1An II is bounded away from zero. This implies that a subsequence of {Xn}nCC=l is also equivalent to the unit vector basis of el. (Remark: If {Yn}n~ is bounded in any L1 (IX) space and there is a sequence {Bn}n~__l of disjoint measurable sets and b > 0 so that for all n, Y-~k#n llynlsk 111 ~< b/2 0, define the support o f x by S(x) := supn(nx)/x u. The supremum exists because X is order complete. The support S ( x ) is a component of u. (A vector 0 ~< y ~< u is called a component of u [or simply a component if u is understood] provided y is disjoint from u - y.) If T : X , --+ C ( K ) is a lattice isometry with T u = 1K, then T x is an indicator function if and only if x is a component. Of course, if A C K, 1a is in C (K) if and only if A is clopen (i.e., both open and closed). So the components form a Boolean algebra (a fact which is also easy to verify directly) which is complete because X is order complete. Thus the clopen subsets of K are also a complete Boolean algebra. An important fact is that the clopen subsets of K form a base for the topology of K. One way to see this is to observe that if 0 ~< x with x in Xu and t > 0 satisfies y := (x - tu) v 0 ~ 0, then 0 ~< t S ( y ) 0] contains a nonempty clopen subset. Since K is compact this implies that the clopen subsets of K form a base for the topology. Using this and the completeness of the Boolean algebra of clopen sets it is a simple exercise to prove that C (K) is itself an order complete Banach lattice. As was pointed out in Section 4, this implies that C (K) is 1-injective. From the discussion in the previous paragraph we can deduce: I f E is afinite dimensional subspace o f an order complete lattice X and ~ > O, then there is a finite dimensional sublattice F o f X and an automorphism T o f X so that E C T F and ]lI - T ]l < e. Given the subspace E, take positive vectors x l . . . . , xn in X whose span contains E and normalized so that u := max/xi has norm one. Then F C Xu and Xu is isometric to a C ( K ) space for which the clopen subsets of K form a base for the topology, which means that the span of indicator functions of clopen sets is dense in C ( K ) . Thus fixing a basis yl . . . . . yk for E and 3 > 0, we get a subspace F of Xu spanned by disjoint vectors and a vector u in F with ]lull = 1 so that for each 1 ~< i ~< k, there is a vector Xi SO that ]Xi -- Zi] ~ (~U (and hence Ilxi - zi [I ~< ~). Now apply the principle of small perturbations. A Banach lattice is order continuous if every downward directed net whose greatest lower bound is zero converges in norm (or weakly; it is the same) to zero. This is equivalent to saying that every order bounded increasing sequence converges in norm (necessarily to the least upper bound of the sequence) (see [15, 1.a.8] or [1, Theorem 12.9] or the beginning of the argument below). It is also easy to check that an order continuous Banach lattice is order complete [15, 1.a.8]. A Banach lattice is not order continuous if and only if it contains a sequence o f disjoint positive vectors which is equivalent to the unit vector basis f o r co and is bounded above. The "if" direction is clear. If X is not order continuous, one gets an upward directed net of positive vectors which is bounded above by, say, x, with ]ix ]l = 1. The net cannot converge in norm, so one gets 0 ~< x l ~< x2 ~< ... ~< x so that

24 a := infn

W.B. Johnson and J. Lindenstrauss Ilxn+~ -

Yn >~ O,

Xn

II > 0. Let Yn := Xn+l -- Xn ~ O. So we have for every n: IlYn II ~ a,

~

Yk ~< x.

(2)

k--1 (Although we do not need it here because we want to "disjointify" the yn'S, it is worth noticing that (2) implies that Yn --+ 0 weakly and hence {Yn}n~__l has a basic subsequence, and that any basic subsequence of {yn}n~__l is equivalent to the unit vector basis for co.) Take Yn* in the unit sphere of X* so that Y * ( Y n ) = [[Yn[[. By replacing y~* with [Yn*[ if necessary, we may assume that y~* ~> 0. For each n the nonnegative sum Y~m Y,~ (Ym) is at most I[x [I. It is an elementary exercise in combinatorial reasoning to deduce from this that for any E > 0 there is a subsequence {zk}~__l "-- {Ynk }~--1 of {yn}n~__l so that for each k > j , * Thus for each n, Z k*(Zj) < 2-J6 2, where z k* "-- Ynk"

(Zn+l--6-1~Zk)k=l V0

Zn+ 1 Zn+ l -- 6

1

Zk

>~a - e.

k=l

This means that the zn's have big disjoint pieces. More precisely, let e = a / 4 and set

(Z+l

and

Wn := (Vn -- EX) V O.

Then the Wn'S are pairwise disjoint positive vectors smaller than x with norms bounded away from zero. It is easy to see that {Wn}n~__l must be equivalent to the unit vector basis of co. This completes the proof, but note that if X is order complete the mapping en ~ Wn from co into X extends to a mapping from the positive cone of s to X by defining {~n6n }n~__l sup n an Wn; this mapping extends to an order isomorphism from s into X. Other useful characterizations of order continuous Banach lattices are given by the following (see [15, 1.b.16] or [1, Theorem 12.9]): X is order continuous if and only if every order interval is weakly compact if and only if X is an ideal in X**. Since c is not order continuous but is isomorphic to the order continuous Banach lattice co, there is not a linear topological characterization of order continuity. While it is only a sufficient condition for order continuity of X that co not embed isomorphically into X, this condition is only a bit too strong as arguments similar to those given above show that if co embeds into X then it embeds as a sublattice (see [1, Theorem 14.12]). The functional representation theorems for order continuous Banach lattices are stronger and more useful than the representation already mentioned for Banach lattices which have a strictly positive functional. Here we just indicate what is going on and refer to [15] for details. However, the motivated reader is encouraged to work out the details for himself or herself (partly because the discussion in [15] wanders unnecessarily). First, it is not hard to show that an order continuous Banach lattice which has a weak order unit u also admits a strictly positive functional u* (see [15, 1.b.15] or [1, 12.14]) and thus can be thought of as


25

a space of integrable functions on a finite measure space (S-2, #) with u* (x) = f x d # for x in X. One can assume that u* (u) = 1 = Ilu II and Ilu* II ~< 2. Since X is order continuous, one has for each x ~> 0 in X that (nu)/x x converges to x as n --+ cx~, so that Xu is dense in X. By replacing the underlying a-algebra with the smallest cr-algebra for which all the functions in X are measurable, it can be assumed that X is dense in L1 (#). Then necessarily u > 0 a.e. By replacing the measure # with u d # and functions f in X by f / u , one can assume u -= 1. Now it is possible to recover the/z-measurable sets and the measure #. From the density of X in L 1 (/Z) it follows that the indicator function of a measurable set is an element x in Xu which is a component of u. From this it is essentially obvious that the L ~ ( # ) norm on Xu agrees with its abstract M space norm. If we represent Xu as a C ( K ) space, then the sets of positive/z measure are mapped onto the nonempty clopen subsets of K in an obvious way so that we can represent X and L l ( # ) as function spaces on K and transfer the measure # to K. We then get that C ( K ) C X C L1 (#) with both inclusions having dense range. Moreover, C (K) = L ~ (#) and every/z-measurable set is equal/z-a.e. to a clopen subset of K. Finally, X is an ideal in L1 (/z) (use again order continuity and the fact that X contains all indicator functions). One consequence of this representation is: Let X be an order continuous Banach lattice which has a weak order unit. A closed subspace Y of X either embeds into L 1(/z) for some measure/z or contains a normalized basic sequence which is equivalent to (even equal to a small perturbation of) a disjoint sequence. For a proof see [ 15, 1.c.8] or modify the proof of the dichotomy principle discussed in Section 4. By applying L1 theory discussed in Section 4 one gets that a subspace of an order continuous Banach lattice is reflexive if and only if it does not contain a subspace isomorphic to either s or co and that a nonreflexive Banach lattice has a sublattice which is order isomorphic to g l or co. The representation can also be used to prove that a Banach lattice which does not contain a copy of co must be weakly sequentially complete [15, 1.c.4]. Although we have used here the representation theorem for abstract L 1 spaces, it should be mentioned that since an abstract L1 space is necessarily order continuous (every normalized disjoint sequence is obviously 1-equivalent to the unit vector basis of ~ 1), some of the ideas (particularly building components in X) can be used to prove Kakutani's representation of an abstract L 1 space which has a weak order unit as a space L 1 (/z) for some probability measure/z. Expressions such as ( E n L 1 ]Xn]P) 1/p, 1 0 (see [15, If.l]). Since it is easier to do analysis on a Banach space which has a norm with good geometric properties than on a general space, it is important to know which Banach spaces can be equivalently renormed so as to become strictly convex or smooth or uniformly convex or uniformly smooth, and it is useful to have in these last cases moduli which are as good as possible. The simplest, but nevertheless useful, renorming technique is as follows. Suppose that T is an injective operator from a Banach space X into a strictly convex Banach space Y. Then it is easy to check that IIIxlll := Ilxll + IITxll is an equivalent strictly convex norm on X. Moreover, if T is an isomorphism and Y is uniformly convex, then II1"III is an equivalent uniformly convex norm on X. Now if X is separable, then there is an injective operator T from X into ~2, so Ilxlll = IIx II + IITx II is an equivalent strictly convex norm on X. Also, there is an operator S from ~2 into X with dense range, so S* is an injective operator from X* into ~ --~2 and IIx*l12- IIx*ll + IIS*x*ll defines an equivalent strictly convex norm on X*. Since the adjoint operator S* is weak* to weak continuous, I1" 112 is dual to a (necessarily smooth) norm I1" 112 on X. If X is smooth and T is an injective operator from X into ~2 then (llx II2 + IIZ x II2) ~/2 is an equivalent norm which is both strictly convex and smooth. Hence every separable Banach space has an equivalent norm which is both strictly convex and smooth. For certain nonseparable spaces; in particular, ~ ( F ) with F uncountable (see [6, Chapter II.7]), there may be no equivalent strictly convex or smooth norm. If we are interested in obtaining a uniformly convex or smooth equivalent norm we have to restrict attention to reflexive spaces. However, not every reflexive space can be so renormed. For example, if ~ , = 1 g l )2 had an equivalent uniformly convex norm II II and I1" II, denotes the restriction of I1" Ilto the nth coordinate space ~ , then the expression IIIxIII := lim IIx II, (where "lim" is interpreted to be a limit over some free ultrafilter on N or a Banach limit or the limit along an appropriate subsequence) defines an equivalent norm on the finitely supported vectors in el which extends uniquely to an equivalent uniformly convex norm on ~1, but this is impossible. There is a characterization of those spaces on which there is an equivalent uniformly convex norm. These spaces, called superreflexive spaces, are discussed in Section 9. The superreflexive spaces are also the class of spaces on which there is an equivalent uniformly smooth norm. (These deep facts are discussed in [29].) If a space X has an equivalent uniformly convex norm II1"III, then the equivalent uniformly convex norms are dense in the metric space of equivalent norms (considered as bounded functions on the unit sphere of X). Take, for example, I1" II +E II1"111.The uniformly convex equivalent norms also form a G~ set since those whose modulus of convexity at 1/n is positive forms an open set. Thus the equivalent uniformly convex norms on X is a dense G~ in the space of equivalent norms on X. Since, as we mentioned, X* also admits an equivalent uniformly convex norm, it OO

/7

.

34


follows by duality that the equivalent uniformly smooth norms on X is also a dense G~ in the space of equivalent norms on X, hence so is the family of equivalent norms which are simultaneously uniformly convex and uniformly smooth. In Section 8 it is pointed out that an infinite dimensional L p (#) space cannot be equivalently renormed so as to have a better modulus of convexity or smoothness than that of the natural norm. There has long been a desire to describe reflexivity geometrically (that is, to show that a space is reflexive if and only if there is an equivalent norm on the space that satisfies some geometrical condition), and the notion of uniform convexity pushed in that direction. This problem was recently solved (at least for separable spaces) and is discussed in [29]. Early in a first course in functional analysis a student learns that a Banach space is reflexive if and only if its closed unit ball is weakly compact. There is a beautiful and useful characterization of weak compactness, called James' theorem, which does not explicitly involve the weak topology: A nonempty closed convex subset C of a Banach space X is weakly compact if and only if every x* in X* attains its maximum on C. The only if part is of course trivial. For the hard direction see [11, Theorem 79] or [26] for an accessible proof when X is separable. This theorem says that on a nonreflexive space X there exist linear functionals which do not attain their norm on the unit ball of X. Nevertheless, the functionals which attain their norm on the unit ball is a rich set: The Bishop-Phelps theorem says: Let C be a nonempty closed bounded subset of a Banach space X. Then the functionals which attain their maximum on C is (norm) dense in X*. This theorem, which is the starting point of the theory of optimization on Banach spaces, has many extensions and applications (see [25]). We outline the proof of the Bishop-Phelps theorem. First note that if f is a continuous bounded function on a complete metric space (U, d), then there is, for every E > 0, a point u0 in U so that f ( u ) 1 t}, K2 := {(x, t): x E X; t ~ x*(xo) + ~ l l x - x011}. The set K2 has a nonempty interior which is disjoint from K1, so the separation theorem gives a nonzero point (u*, c~) in X* 9 • and a fl so that u* (x) + a t ~> fl for (x, t) in Kl and u*(x) + at 0 there is ~ > 0 so that Ilr(A)]l < ~ whenever # ( A ) < 6 (see [8, p. 10]). If r is an X valued measure of finite variation it is clear that r is absolutely continuous with respect to its total variation It] and even satisfies (14) below (with # :--

Irl). An X valued measure r is differentiable with respect to a scalar measure lz provided that there is an X valued measurable function g so that r ( A ) = fa g d # for every measurable set A. We say that the Radon-Nikodym theorem holds in X provided that if r is an X valued measure of finite variation and r is absolutely continuous with respect to a finite scalar measure #, then r is differentiable with respect to #. If X satisfies this condition only for all separable finite scalar measures, we say that the separable Radon-Nikodym theorem holds in X (a measure # is called separable provided L1 (#) is separable). The usual definition is that a Banach space X has the RNP provided the Radon-Nikodym theorem holds in X and this is equivalent to saying that the separable Radon-Nikod3~m theorem holds in X (see [8, Chapter III]). Later we prove this equivalence for separable X, but first we show a general space X has the RNP if and only if the separable Radon-NikodSm theorem holds in X. Suppose that the separable Radon-Nikod3)m theorem holds in X and let f ' [ 0 , 1] ~ X be a Lipschitz function. One defines a linear mapping T from the step functions on [0, 1] into X by setting Tl[a,b] := f ( b ) - f ( a ) for a subinterval of [0, 1] and extending linearly. Since f is a Lipschitz function, the mapping T is continuous when the step functions are given the L 1(0, 1) norm, and hence T uniquely extends to an operator (also denoted by T) from L1 (0, 1) into X. The assignment r ( A ) :-- T1A obviously defines an X valued measure of finite variation which is absolutely continuous with respect to Lebesgue measure m, so we get an X valued measurable function g on [0, 1] for which r ( A ) = f a g dm for every Lebesgue measurable subset of [0, 1]. In particular, f (t) -- f0 g d m + f (0) and thus


40

by what we proved in the beginning of this section f ' ( t ) exists a.e. on [0, 1] (and is equal to g(t)). Suppose that X has the RNE To see that the separable Radon-Nikod3~m theorem holds in X, suppose first that the scalar "control measure" Ix is Lebesgue measure on [0, 1] and that the X valued measure r satisfies

[[r(A)I[ ~ 0 so that #(A) = fA f dv for every v-measurable set A. Of course, IX is then also a separable measure and, as we have already remarked, r satisfies (14), so from what we already have proved there is an X valued Ix-measurable function g so that r(A) = f A g dix for every Ix-measurable set A. Then f . g is v-measurable and r(A) = fA f " g dv for every v-measurable set A. Observe that the simple argument reducing the study of an X valued measure which is absolutely continuous with respect to a finite control measure to the case where the X valued measure satisfies (14) yields another characterization of the RNP; more precisely, that the (separable) Radon-Nikodym theorem holds in X if and only iffor each operator T

from an L 1(Ix) space with tx finite (and separable) into X there is an X valued measurable function g so that T f = f f g dix for all f in L I(IX). From this it is easy to see that if X is separable and the separable Radon-Nikod3)m theorem holds in X then the RadonNikod2~m theorem holds in X. Indeed, let IX be a finite measure on a cr-algebra B and let T'LI(Ix) --+ X be an operator. Since X is separable there is a sequence {X*}n~ in X* which separates the points of X. Let A be a countably generated sub ~r-algebra of 13 so , oc separates points of that all the Loc(Ix) functions T* x n* are A-measurable. Since {Xn}n=l X, T f = T E ( f I A) for each f in L1 (Ix). The restriction of Ix to A is a separable measure since A is countably generated. Thus we get an X valued A-measurable function g so that T f = f f . g dIx for each A-measurable function f in L l (IX). But then for a general f in L1 (Ix) we have T f = T E ( f I A) = f E ( f I A)g dIx = f f . g dIx since g is A-measurable. One of many places where the RNP arises naturally is in the study of vector valued L p spaces. There is a natural isometric identification of Lp,(Ix, X*), 1/p + 1/p* = 1, with a subspace of Lp(Ix, X)*, and for 1 ~ p < (x~, Lp(Ix, X)* = Lp,(Ix, X*) for allfinite (or cr-finite) measures Ix if and only if X* has the RNP (see [8, Chapter IV]). There are other important analytic characterizations of spaces with the RNP in terms of martingales. In particular, the RNP spaces are exactly those Banach spaces in which the martingale convergence theorem is valid in the sense that X has the RNP if and only if every L1 bounded X valued martingale converges a.e. (see [8, Chapter V]).


41

It turns out that in many places where one might assume reflexivity in order to use weak compactness of the unit ball it suffices to assume that the space has the RNP. We now discuss the differentiability of (real valued) convex continuous functions on a Banach space X. Part of the importance of this topic derives from the fact that the norm is a convex continuous function and differentiability of the norm is intrinsically related to its smoothness. The most elementary reference for the differentiability of convex functions and related topics is probably [11, Chapter 5]. An easy consequence of the definition is that a locally bounded convex function is continuous and even locally Lipschitz. By using the separation theorem in X | IR it follows that whenever f is convex and continuous in a neighborhood of a point x0 the set (called the subdifferential o f f ) Of(XO) :"- {X* E X*: x * ( x -- xo) ~ f ( x ) -- f ( x o ) for all x 6 X} is nonempty. From the theory of convex functions on R we know that for each u the right and left derivatives of the function t ~ f (xo + tu) exist at t = 0. These one-sided derivatives agree for every u (that is, all directional derivatives exist at x0) if and only if Of(xo) is a single point which is then necessarily the G-derivative of f at x0. Consequently f is G-differentiable at x0 if and only if for every u, f ( x o + tu) + f ( x o - tu) - 2 f ( x 0 ) -- o(t) as t --+ 0. By considering f ( x ) = Ilxll we recover the fact mentioned in Section 6 that the norm is G-differentiable at x0 in the unit sphere of X if and only if x0 is a smooth point of the unit ball of X. It also follows that f is F-differentiable at x0 if and only if f ( x o + u) + f (xo - u) - 2 f ( x 0 ) = o(llull) as Ilull -+ 0. I f a convex function f is F-differentiable in a neighborhood ofxo then D f (x) is continuous there; that is, F-differentiability of a convex function on an open set implies that it is C 1 there. Indeed, suppose that Xn --+ xo and set tO* " - D f ( x o ) ; u* "-- D f ( x n ) . Given E > 0 there is 6 > 0 so that f ( x o + y) - f ( x o ) w*(y) /(~/2)llu~* - w*ll for all n. Then since u,* - D f ( x n ) , we have by the convexity of f that

6 + tO* (Yn) + f (xo) >~ f (xo + y/7) >~ u* (Yn - Xn + xo) + f (Xn) or

(~/2)llun* --

tO*

II ~< (Un* --

tO*

)(Yn) Y to maximize the volume of the image of the g~ ball as above, it is obvious that n works, where xk 9-- Tek , x k* (x) "= det(T0) 1 det(Tk), and T k ' g ln ---->Y the basis {Xk, x k9}k=l is defined by letting Tkei be xi when i --/=k and Tkek = x. We turn to the proof of Lewis' lemma. Since the volume of T C for any measurable set is a constant multiple of [det(T) Ivol(C), for any operator S : X ---> Y we have

det(To+S) ot(r0 + S)

~< Idet(T0) I.

(18)

Certainly To must be invertible, so by dividing (18) by Idet(T0)l we can rewrite (18) as Idet(Ix + T o ' S ) l v o l ( T B x ) f o r every operator T on ~n [C n] f o r which T B x C Br. Then there exist contactpoints xl . . . . . XN o f B x and B r and contact points x 1 . . . . , x N o f B x , and Br', and ck ~ 0 so that I = ~ = l ckx~ | xk. Also, N 0 a subspace whose B a n a c h - M a z u r distance to g kp is less than 1 + E This implies that an infinite dimensional L p ( # ) space cannot be given an equivalent norm which has a better modulus of convexity or smoothness than its natural norm. Two notions that are very important for both the finite dimensional and infinite dimensional theories are that of type and cotype. A Banach space X is said to have type p provided there is a constant C so that for every sequence x l . . . . . xn in X,

E

• i=1

1/p 6iXi

2 or cotype q for any q < 2. For every space X, 7"1(X) = 1 and Coo(X) -- 1 by convexity of the norm. As functions of p and q, Tp(X) is nondecreasing and Cp(X) is nonincreasing. The inequalities Tp(X) 0 SO that for every sequence x, x l . . . . . Xn in a Banach space, n

2) 1/2.

X + y ~ 8iXi i=1

i=1 To prove (25), first note that there is r

> 0 so that for all 0 < t ~< 1,

11 + Crptl p + I1 - Crptl p ~ lip 2

J

(25)

~< (1 + t2) 1/2.

(26)

This follows, for example, from L'H6pital's rule. The best constant is O'p -- (p - 1) -1/2 (see [15, I.e.14]). Secondly, deduce a vector valued version of (26): (EIIx + el~pyll p) 1/p ~ (EIIx + ~lYll 2) 1/2.

(27)

51

Basic concepts in the geometry of Banach spaces One can assume that I[x + YI[ + [Ix - YI[ - 2 with t 1 ~Crp Ilx + opey II ~
nZ/P/N(p, Cp(X), ~) with d (Y, gk) ~< 1 + ~. In particular, if X is a subspace of L 1(~) then the k above is proportional to n, with a constant depending just on E. See [17, 4.15] or [16, 5.3]. A connection between type and cotype theory and Krivine's theorem is provided by the Maurey-Pisier theorem:

If X is an infinite dimensional Banach space and Po : - sup{p: X has type p} and qo := inf{q: X has cotype q}, then for any ~ > 0 and any k there are subspaces Yk and Zk of X k ) ~ n - E. In particular, taking n : 2 we see that: Every nonreflexive space contains real subspaces arbitrarily close to real s That is, a real space whose unit ball does not have a two dimensional section arbitrarily close to a square must be reflexive. See [2, 4.111] for proofs of these results. We next discuss the duality theory of type and cotype. It is simple that if X has type p then X* has cotype p* and C p , ( X * ) r it follows that S is onto. Hence the operator n-r+h

/7

~. (x e . . . 9 x)~

defined by S ( x l . . . . . Xn)

--

--, ( x e . . . 9 :~ 9 R ' m ) ~

(S(xl . . . . .

Xn),

vj(xh)),

1 ~ j X by T w * * - wi for 1 ~< i ~< r. t h a t T w i** - - w i - - w i** for 1 ~< i ~< k. If r < i ~< n then Y'~h=l # i , h W h -- wi - - the ith component of S ( w l , w e . . . . . Wn) - - the ith component of S**(w~( *, w e . . . . . w r ) = Z r h = l # i , h W h** -- W i** -- 0. Hence Tw** - - w i also for r < i ~ n. That T is an 1 + ~ isometry follows from the choice of 6 and the fact that for all i,

Note

1+3

~

11//9i 11 = IITw;*II

~ suplvff(w/) I -suplw~*(v~) I ~ J

(1 +3)-'.

J

We describe next a useful construction in Banach space theory (having roots in mathematical logic) which is related to the notion of finite representability.


55

Recall that a family b / o f subsets of a set I is called a filter if it is closed under finite intersections, does not contain the empty set, and whenever A C B with A E b / t h e n B b/. A maximal (with respect to inclusion) filter is called an ultrafilter. By Zorn's lemma every filter is contained in an ultrafilter. An ultrafilter is called free (or nontrivial) if the intersection of all sets in b / i s empty. An indexed family {Xi}i6I in a topological space is said to converge to x with respect to a filter bt (in symbols, x -- l i n ~ xi ) provided for every open set G containing x the set {i: xi ~ G} belongs t o / g . A Hausdorff space is compact if and only if every indexed family {xi }i~I converges (to a unique point) for every free ultrafilter b / o n I. Assume now that I is a set and b / i s a free ultrafilter on I; assume also that for all i, Xi is a Banach space. We define a seminorm Ill'Ill on (Y-~i Xi)ec by IIIxlll = lirr~ Ilxi II where x -- {xi }i~/with xi ~ Xi for all i. The limit exists since a closed bounded interval on the line is compact. The quotient of (~-~i Xi)ec with respect to the closed subspace of all x with ]llx Ill --- 0 with its obvious norm is a Banach space, called the ultraproduct of the Xi (with respect to b/), and is denoted by (1--IiXi)cr If all the Xi are the same space X we call the space thus obtained an ultrapower of X, denoted also by Xu. Ultraproducts of Banach spaces are treated in detail in [9, Chapter 8]. Given two families {Xi}icI and {Yi}i~I of spaces and operators 7 ) : X i ~ Yi with sup/ ]]Ti ]] < cx:), there is a natural operator T:(1-Ii Xi)lg -'+ (Hi Yi)bt called the ultraproduct ofthe operators 7). It maps an element in (1-Ii xi)cr represented by x = {xi}i6I in (~_~X i ) ~ into the element in (]-Ii Yi)Cr represented by y = {~xi}is/. The ultraproduct of one dimensional spaces is one dimensional and more generally if dim Xi = n < ec for all i then (I-Ii xi)cr is also n-dimensional. On the other hand if I = N and limu (dim Xi) = ec then (I-Ii x i ) u is already nonseparable. The ultraproduct of Banach lattices is again a Banach lattice if we take as the positive cone in (Hi xi)cr the set of all elements which have representatives x = {xi}iEl in (Y]~i Xi)oc with xi ~ 0 for all i. If all the Xi are abstract Lp spaces for some fixed p, 1 ~< p < ec, then (Hi x i ) u is again an abstract Lp space and hence is isometric to Lp(#) for some measure # by the L p version of the Kakutani representation theorem. Similarly, if all the Xi are C (Ki) spaces for some compact Hausdorff Ki then so is (I-Ii xi)cr However, for other families of Banach spaces Xi (even, e.g., if all are Orlicz spaces with the same Orlicz function q)) the determination of the nature of (1-[i Xi)u is not an easy task. As a first application of ultraproducts we shall prove now a fact mentioned already in Section 4: If 1 0. Introduce a partial order on I by (El, F1,61) < (E2, F2, 62) if El C E2, F1 C F2, and 61 > 62, and let/g be an ultrafilter on I which refines the partial order filter. For every (E, F, 6) E I let TE,F,e :E ~ X be the operator given by the principle of local reflexivity. Using these operators we define as above an isometry T from X** into XU. Define a map S from X u into X** by S({xi }) = w*-limu xi. From the properties of {TE,F,~} one deduces easily that ST is the identity on X**, so that T S is a projection of norm 1 from XU onto T X**. A property (P) of Banach spaces is called a super property provided that if X satisfies (P) and Y is finitely representable in X, then Y satisfies (P). In particular, a super property passes from a space X to all closed subspaces of its ultraproducts. So if (P) is a hereditary property (i.e., passes to closed subspaces), a Banach space X has super (P) if and only if every ultrapower of X has (P). For example, X is superreflexive if every ultrapower of X is reflexive. An explicit local property which characterizes superreflexivity is the following: A Banach space X is superreflexive if and only if for every 6 > 0 there is an integer N(6) so that any 6-separated dyadic tree in the unit ball of X has height ~< N (6). By an 6-separated dyadic tree of height N we mean a set of points {Xi,n: 1 gp(E), S E ' g ~ (E) ---> E so that IITEII ~< 1, IISEII ~< ~ and SETE -- Ie (n(E) - - d i m E ) . L e t L / b e an ultrafilter on I which refines the order filter and put Y -- (FI gp(E))z4- Define T from X to Y by mapping x to the class represented by {TEx}Ecl (in view of the choice o f / 2 it does not matter that TEx is defined only for E which contain x). Define S: Y -+ X by S{yE} = w-lim~ SEyE (recall that X is reflexive). Then ST = Ix and Y is an L p ( # ) space. Consequently: Every E,p space X, 1 < p < oo, is isomorphic to a complemented subspace ofan L p ( # ) space. If X is separable we deduce that X is isomorphic to a complemented subspace of L p (0, 1). Similar considerations (starting with the fact proved in Section 4 that g2 is isometric to a complemented subspace of Lp(O, 1)) yield that every Hilbert space is isometric to a complemented subspace of some L p (#) space when 1 < p < oe. In the cases p -- 1 and p -- cx~ we can reason similarly, but now define S ' Y --+ X** by S{yE} -- w*-limu SEyE and get that ST -- Jx, the natural inclusion of X into X**. By considering T**" X** ----> Y** and S**" Y** ----> X (iv) and recalling that there is a norm one projection from X (iv) onto X** (see Section 2), it follows that if X is an 121 space then X** is isomorphic to a complemented subspace of an L l ( # ) space. Since L1 (#)* is

58


injective and since there is a norm one projection from X*** onto X* we deduce that if X is an E1 space then X* is an injective Banach space. Similarly, if X is an s space then the constructed ultraproduct Y is a C ( K ) space and X** is isomorphic to a complemented subspace of the injective space Y**. That is, if X is an 12~ space then X** is an injective Banach space. The converse of the previous statements essentially hold. First we show: Let 1 < p < ~ . A Banach space X is isomorphic to a complemented subspace of an L p ( # ) space if and only if X is an s space or X is isomorphic to a Hilbert space. To see the "only if" direction, assume that there is a projection Q from Y = L p ( # ) onto a subspace X which is not isomorphic to a Hilbert space. By the dichotomy principle for L p spaces, 2 0. There is a finite dimensional subspace F of Y (a small perturbation of the span of disjoint indicator functions) containing E so that d(F, ep) ~ 1. Hence for every x ~ X, IITx[Ip / . . . . If the sequence is infinite, then necessarily )~n(T) --+ O. For the Banach space ~2, the spectral theorem for compact op-

66


erators says that if T ~ K ( H , H), then there is an orthonormal basis {Xn}nC~1 SO that the self-adjoint operator T*T is represented as

T * T x -- Z )~n(T*T)(x,xn)Xn nEP

(31)

with )~n(T*T) > 0 for n 6 P and T * T x n - 0 for n r P. Now if X is a Banach space with a 1-symmetric basis S - {en}n~__l, let S(62) be those compact operators on 62 for which ~ ~/)~n(T*T) en converges in X, and, for T 6 S(62), set

Then (S(62), o's) is a Banach space (verifying the triangle inequality is a bit tricky). It is less difficult to check that S(62) satisfies the unitary ideal property

crs(UTV) s 1 as above, and put !A! := [IA | I H ' s --->s We have to find an estimate on !A! independent of n and A. Note first that for all ui and vj in a Hilbert space

aij (ui, vj) ~< !A! max Ilui II max Ilvj II. i

i,j=l

j

Since all Hilbert spaces are created equal, we can use for H any (infinite dimensional or even 2n-dimensional) Hilbert space. For the purpose of proving Grothendieck's inequality, some Hilbert spaces are more equal than others! We use for H the subspace of L2(0, 1) mentioned in Section 4 consisting just of functions having a Gaussian distribution with mean 0. To prove Grothendieck's inequality it is enough to consider norm one vectors {Xi}in=l a n d {YJ}j=I in H. Given 0 < 6 < 1/2 there is an M -- M(6) so that for any norm one function f in H , Ilf - f M 112 --" • where

fM (t) - ] f (t)

if If(t)l ~< M, if If(t)[ > M.

[ M sign f(t)

Note that by our assumption on the matrix A, for any choice of functions f/ and g j in L2 (0, 1) which are uniformly bounded by M,

Zaij(fi,gj) i,j

f01

Z aij f i ( t ) g j ( t ) t,j 9

dt ~< M 2.

.

Hence

Z aij (xi, yj )

Zai,j(XiM, y M} + Z a i j ( x i ,

i,j

i,j

Yj -- Y M)

i,j

+ Z aij(xi- x M, yM) i,j

Loo(/z) 12> L2(#) v > X, with J an isometry, 12 the formal identity, and II vii- v/if, Since L ~ ( # ) is 1-injective the operator J can be extended to a norm one operator T from Y into L ~ ( # ) . The operator P :-- V12 T is a projection from Y onto X with norm at most v/ft. Notice also that the factorization used above gives another proof of the result proved in Section 8 that d ( X , ~ ) ~ x/~. The estimate of x/~ for the projection constant of an n-dimensional space is essentially sharp. This is discussed in [33]. The p-integral operators form a class of operators which are closely related to the psumming operators. An operator T :X --~ Y is said to be p-integral, 1 ~ p ~ cx) (in symbols T E Z r ( X , Y)), provided that the composition J r T of T with the canonical embedding Jy :Y --~ Y** factors through the formal identity I~,p : L ~ ( # ) ~ L n ( # ) for some probability measure #: I~, p

L~(#)

AI X

T

> Lp(#)

JY >Y

~B > Y**

(34)

72


The p-integral norm ip(T) is then defined to be the infimum over all such factorizations of [[A [[ [[B [[. By taking ultraproducts one sees that this infimum is really a minimum. The space (Zp(X, Y), ip) is easily seen to be a Banach space and ip satisfies the ideal property i p ( S T U ) ~ []S[[ip(T)[[U[[. If T is in Zp(X, Y) and X is a subspace of C ( K ) , with K a compact Hausdorff space, then there is a probability measure v on K and an operator S : L p ( v ) ---+ Y** with [[S[[ ---ip(T) which makes the following diagram commute:

Ip

C(K)

> Lp(v) (35)

X

T

>Y

Jr

> Y**

Indeed, if (34) holds, A can be extended to an operator A" C(K) --+ L ~ ( # ) because L ~ ( # ) is 1-injective. By the Pietsch factorization theorem there is a probability measure v on K so that for each x E C (K),

[[BI~,pAx[] ~ 7rp(Bl~,pA)

[x[ p dv

and the desired conclusion follows from

7rp(BIc~,pA) ~ IIBllTrp(l~,p)llS, II =

IIBlllIAII.

It is evident that zrp(T) ~ ip(T) and from the Pietsch factorization theorem it follows that Zcp(T) = ip(T) if the domain of T is a C ( K ) space. As was mentioned implicitly in the discussion of 2-summing operators, 7r2(T) -- i2(T) for any operator. One reason for defining p-integral via a factorization of Jy T rather than T is that this forces i p(T) -ip(T**) (use the fact that a dual space is norm one complemented in its bidual). The 1-injectivity of C(K)** gives that a p-summing operator T into a C ( K ) space is p-integral with ip(T) = 7rp(T). The equality il (T*) = il (T) follows from the observation that the adjoint I~ of 11 "C(K) --+ Ll(v) is l ~ , l ' L ~ ( v ) --+ Ll(v) followed by the identification of L1 (v) with the norm one complemented subspace of C(K)* consisting of the finite signed measures which are absolutely continuous with respect to v. For other values of p, the adjoint of a p-integral operator need not be strictly singular (see [9, 5.12]) and hence need not be q-summing for any q < c~. For each 1 ~< p ~< cx~, p -r 2, there exist p-summing operators which are not p-integral (see [9, 5.13]). The case p -- 1 is particularly easy to deduce from the theory we have presented. We saw that every operator T :~1 --+ ~2 is 1-summing. If T :~1 --+/~2 is 1-integral, A

I~, 1

B

then T has a factorization ~1 > L ~ (#) > L1 (/z) > ~2. But then B and also I~, 1 are 1-summing, hence B I~, l, whence also B I~, 1A = T, are compact (use the fact that L 1(/z) has the DP property). For p = c~ and Y reflexive the infinity integral operators from X to Y are exactly those which factor through some L ~ ( # ) space (with the integral norm equal to the best factorization). In particular for X reflexive the infinity integral norm of the identity of X is


73

finite if and only if X is finite dimensional (and is equal to the projection constant of X in that case). Thus also for p -- cx~ it is evident that summing and integral norms can be very different. The main reason for introducing p-integral operators is that they are needed for the duality theory of lip(X, Y). For simplicity, we restrict to the case where X and Y are finite dimensional. Following the notation used in Section 8, for finite dimensional X, Y and ot a norm on B(X, Y) we represent the dual of (B(X, Y), or) as (B(Y, X),ot*), where the pairing is given by (S, T) = trace TS ( = t r a c e S T ) . Then for all 1 ~X

one has for E > 0 an operator A~ :Y --~ L ~ ( # ) with [[A1 - Aell < ~ so that A~Y is contained in the simple functions. This gives a factorization of B11~,l A~ of the form (36) with IIBIIIIAIIIIAII ~< IIB~ II III~,~ IIIIA~ II. Setting N := dim Y, we get A[(S - BII~,IAE) ~ trace Ix = n is clear. To see the reverse inequality, take T E B ( X , X) so that trace T = n and ~ ( I x ) ~ * ( T ) = n. Let G be the (compact) group of isometries of X and/z normalized Haar measure on G. For each S in G, trace S - 1T S = trace T = n and, by the ideal property of c~*, or* (S - 1 T S) = or* (T). The operator

To "= fG S - I T S d l z ( S ) satisfies trace To = trace T, ot*(T0)~< ot*(T), and T commutes with all elements of G. Hence To = )~Ix and since trace T o - n we conclude that )~-- 1. It follows that a ( I x ) a * ( I x ) but is better behaved. For example, (Xo,p, I1" II0,p)


80

is uniformly convex (rather than just superreflexive) if either X0 or X 1 is uniformly convex and 1 < p < cx~ [15, 2.g.21]. Moreover, I]" IIO,p is very good for interpolation purposes, for if Y is another Banach couple then (Xo,p, ]]. ]]0,p) and (Yo,p, ][" ]]0,p) are an exact interpolation pair of exponent 0 with respect to X and Y. It is of course important to identify Xo,p when X is a concrete Banach couple. The spaces that arise in this connection when X is a couple of L p ( # ) spaces are the L p,q spaces. Given 0 < p < ~ , 0 < q < cx~, a measure #, and a Banach space X, Lp,q (#, X) is the space of X valued strongly measurable functions x for which I]Xllpq "--

( f0 q/p

[tl/px.(t)]q dt t

)

1/q < o~,

(42)

where x*(t) is the decreasing rearrangement of IIx(t)llx. For 0 0

W h e n X is the scalar field we write Lp,q(#). E v i d e n t l y Lp,p(#, X) = Lp(#, X). Note that for p > q ~> 1, the space Lp,q(#) is the Lorentz function space Lw, q(#) defined in Section 5 with W(t) - - q t q/p-I and hence is a Banach space. W h e n q > p ~> 1 the expression II 9I]p,q does not satisfy the triangle inequality, although II 9Ilp,q is equivalent to a norm when p > 1. In any case Lp,q (#, X) is a metrizable topological vector space. The main result about spaces obtained from L p and L p,q spaces via t h e / C - m e t h o d is the following (see [4, 5.3.1]): Let 0 < Po, Pl, qo, ql 0, C - 1 ] ] 9 ]]p,q ~ ]] 9 ][O,q C II 9 II p,q. W h e n the expressions II 9 II Pi ,qi' i -- 0, 1, are equivalent to norms one can deduce from (44) and earlier comments an interpolation theorem for operators. In fact, there are cases where interpolation is valid even when the spaces are not all Banach spaces. In particular, assume that T" Lpi,r i (l~, X) -+ Lqi,s i (19, X ) is continuous for i -- 0, 1, with P0 ~: Pl, q0 ~ ql, 0 < 0 < 1, and define p, q by 1 / p - (1 - O ) / p o + O/pl, 1/q - (1 - O ) / q o -O/ql. If p ~< q, then T ' L p ( l Z , X) ~ Lq(19, Y) is continuous and for 0 otl/Pl[f*(t)l[x < co (Section 11). Here 0 < p ~< co and f * is the decreasing rearrangement of 11f ]Ix. Lp,q (/z, X) when X -- ]R. The closed unit ball of the Banach space X. The closed ball of radius r with center x in the Banach space X; denoted also B(x, r) when X is understood. All linear functionals which vanish on S (when S is a subset of a Banach space). The intersection of the kernels of all linear functionals in S (when S is a subset of the dual of a Banach space). The Banach-Mazur distance from X to Y (Section 2). The space X is isomorphic to the space Y. The modulus of convexity of the space X (Section 6). The modulus of smoothness of the space X (Section 6). When X is a lattice and u ~> 0, the abstract M-space which has the order interval [ - u , u] as the unit ball (Section 5). When X is a lattice and u* ~> 0 in X*, the abstract L 1-space which is the completion of X under the seminorm [[x ]]u. -- u* ([x[) (Section 5). A Rademacher sequence (Section 4). The identity operator on the space X. The canonical embedding of X into X**. The formal identity operator from C(K) to Lp(/z) (when/z is a finite measure on the compact Hausdorff space K). The formal identity mapping from Lp(/z) to Lq(/z). The bounded operators from X to Y. The compact operators from X to Y. The weakly compact operators from X to Y. The strictly singular operators from X to Y (Section 10). The Fredholm operators from X to Y (Section 10). The nuclear operators from X to Y (Section 8). The nuclear norm of the operator T (Section 8). The p-convexity constant of the operator T (Section 5). The p-concavity constant of the operator T (Section 5). The type p constant of the Banach space X (Section 8). The cotype p constant of the Banach space X (Section 8).

Basic concepts in the geometry of Banach spaces ITp(X, Y) 7rp(T) I n ( X , Y) ip(T)

The The The The

83

p - s u m m i n g o p e r a t o r s f r o m X to Y ( S e c t i o n 10). p - s u m m i n g n o r m o f the o p e r a t o r T ( S e c t i o n 10). p - i n t e g r a l o p e r a t o r s f r o m X to Y ( S e c t i o n 10). p - i n t e g r a l n o r m o f the o p e r a t o r T ( S e c t i o n 10).

References [1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, New York, 1985. [2] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Mathematics Studies 68, North-Holland, Amsterdam (1985). [3] Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Coll. Pub. 48, Amer. Math. Soc., Providence, RI (2000). [4] J. Bergh and J. Lrfstrrm, Interpolation Spaces: An Introduction, Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, New York (1976). [5] Yu.A. Brudnyi and N.Ya. Kruglyak, Interpolation Functors and Interpolation Spaces I, North-Holland, Amsterdam (1991). [6] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Scientific & Technical, Essex (1993). [7] J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York (1984). [8] J. Diestel and J.J. Uhl, Vector Measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, RI (1977). [9] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Math. 43, Cambridge University Press, Cambridge (1995). [10] R. Durrett, Probability: Theory and Examples, Duxbury Press (1996). [ 11] P. Habala, P. H~ijek and V. Zizler, Introduction to Banach Spaces I, Matfyzpress, Univerzity Karlovy (1996). [ 12] P. Habala, P. Hfijek and V. Zizler, Introduction to Banach Spaces II, Matfyzpress, Univerzity Karlovy (1996). [13] S.G. Krein, Yu.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, RI (1982). [14] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Springer-Verlag, Berlin (1977). [15] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 97, Springer-Verlag, Berlin (1979). [16] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer-Verlag, Berlin (1986). [17] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge University Press, Cambridge (1989). [18] H.L. Royden, Real Analysis, 3rd edn., Macmillan, New York (1988). [19] W. Rudin, Functional Analysis, 2nd edn., McGraw-Hill, New York (1991). [20] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, Essex (1989). [21 ] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Studies in Advanced Mathematics 25, Cambridge University Press, Cambridge (1991).

H a n d b o o k articles which are referenced in this introductory article [22] [23] [24] [25] [26]

D. Alspach and E. Odell, L p spaces, This Handbook. D.L. Burkholder, Martingales and singular integrals in Banach spaces, This Handbook. EG. Casazza, Approximation properties, This Handbook. R. Deville and N. Ghoussoub, Perturbed minimization principles and applications, This Handbook. V. Fonf, J. Lindenstrauss and R.R. Phelps, Infinite dimensional convexity, This Handbook.

84


[27] T. Gamelin and S.V. Kislyakov, Uniform algebras as Banach spaces, This Handbook. [281 A.A. Giannopoulos and V. Milman, Euclidean structure in finite dimensional normed spaces, This Handbook. [29] G. Godefroy, Renormings of Banach spaces, This Handbook. [30] W.T. Gowers, Ramsey theory methods in Banach spaces, This Handbook. [31] N.J. Kalton and S.J. Montgomery-Smith, Interpolation andfactorization theorems, This Handbook. [32] S.V. Kislyakov, Banach spaces and classical harmonic analysis, This Handbook. [33] A. Koldobsky and H. K6nig, Aspects of the isometric theory of Banach spaces, This Handbook. [34] M. Ledoux and J. Zinn, Probabilistic limit theorems in the setting of Banach spaces, This Handbook. [35] J. Lindenstrauss, Characterizations of Hilbert space, This Handbook. [36] E Mankiewicz and N. Tomczak-Jaegermann, Quotients offinite-dimensional Banach spaces; Random phenomena, This Handbook. [37] B. Maurey, Banach spaces with few operators, This Handbook. [381 B. Maurey, Type, cotype and K-convexity, This Handbook. [39] D. Preiss, Geometric measure theory in Banach spaces, This Handbook. [401 H.E Rosenthal, The Banach spaces C (K), This Handbook. [41] L. Tzafriri, Uniqueness of structure in Banach spaces, This Handbook. [421 P. Wojtaszczyk, Spaces of analytic functions with integral norm, This Handbook. [43] M. Zippin, Extension of bounded linear operators, This Handbook. [441 V. Zizler, Nonseparable Banach spaces, This Handbook.

CHAPTER

2

Positive Operators Y.A. Abramovich Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, IN, USA E-mail: yabramovich @math. iupui, edu

C.D. Aliprantis Department of Economics and Department of Mathematics, Purdue University, West Lafayette, IN, USA E-mail: aliprantis @mgmt.purdue, edu

Contents 1. Ordered vector spaces and Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Operators between Banach lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. W h e n is every continuous operator regular? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Domination, compactness, and factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Invariant subspaces of positive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Compact-friendly operators and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Integral operators and invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All fights reserved 85

87 90 91 93 98 102 105 111 117 117 117


Positive operators

87

The theory of positive operators is a distinguished and significant part of the field of general operator theory. The extra feature of this part of operator theory is the existence of an order on the spaces involved. This ingredient is of tremendous importance and lies in the core of many specific results valid for positive operators. The two main objectives of the present work are to discuss the relationships between general operators and positive operators and to demonstrate the effects the order structure has on general operators acting between Banach lattices. We list here several books devoted primarily to positive operators between ordered spaces: [15,22,29,41,75,78-80,86,87,96,100,123,126,135,144,145]. The size constraints of this article as well as the areas of interest of the authors have inevitably influenced the selection of the material for this survey and have precluded us from mentioning many existing directions within the theory of positive operators. We especially regret having to omit the ergodic and interpolation properties of positive operators. For these topics the reader is referred to [83] and [84] respectively. Since our survey is devoted to operators, we have reduced to a bare minimum the general theory of Banach lattices presented here. Most of the books cited above, the series of notes by Luxemburg and Zaanen [93,92] and the two surveys [39,40] are excellent sources for such a theory as well. Throughout this work the word "operator" is synonymous with "linear operator".

1. Ordered vector spaces and Banach lattices Though we will be dealing mostly with operators on Banach lattices, it is fruitful to start with the general framework of (partially) ordered vector spaces. We will introduce some necessary terminology and prove a few basic results; for a systematic presentation we refer the reader to [70,94,101,107,134,143]. Recall that a real vector space X equipped with a partial order ~> is said to be a (partially) ordered vector space whenever x ~ y imply otx ~> oty for all ol ~> 0 and x + z ~> y + z for all z E X. The set X+ = {x E X: x ~> 0} is called the positive cone of X and its elements are referred to as positive vectors. The cone X+ is said to be generating whenever X -X+ - X+, i.e., whenever every vector can be written as a difference of two positive vectors. An ordered vector space X is said to be Archimedean whenever n x 0 or x ~< 0.

E]

THEOREM 2 (M. Krein-Smulian). Let X be a Banach space ordered by a closed generating cone. Then there is a constant M > 0 such that f o r each x ~ X there are x l, x2 E X+ satisfying x -- xl - x2 and Ilxi II ~< Mllxll f o r each i.

88

Y.A. A b r a m o v i c h a n d C.D. A l i p r a n t i s

PROOF. We present a sketch of the proof. For each n define the set En = {x ~ X" =tXl, X2 G_ X - t- with x - Xl - x2 and [Ixi II ~ n (i = 1, 2) }.

Clearly, each En is convex, symmetric, and 0 6 En. In addition, note that En cc_ Em whenever n ~< m. Since X+ is generating, we see that X - Un~=l En. So, by the Baire Category Theorem, some Ek contains a closed ball B(xo, r) = {x 6 X: Ilx0-xll ~< r}. The properties ofthe sets En imply that B(0, r) c_ E~. Now by imitating the proof of the open mapping theorem, we can show that B(O, r) c_ Ezk holds and the proof is done. The details can be found in [135]; see also [4]. D C O R O L L A R Y 3. Let X be an ordered Banach space whose positive cone is closed and generating. I f xn --+ x holds in X, then there exist Yn, Zn, Y, z ~ X + such that Xn - - Yn -- Zn, x = y -- Z, Yn --+ Y and Zn --+ z. C O R O L L A R Y 4. Let X be a Banach space partially ordered by a closed generating cone, and let Y be a topological vector space. Then an operator T : X --+ Y is continuous if and only if the restriction o f T to X + is continuous.

And now we introduce the central concept of this work. D E F I N I T I O N 5. An operator T : X ~ Y between two ordered vector spaces is said to be positive if T (X+) _ Y+, i.e., if x ~> 0 implies T x >~ O.

It is a remarkable fact that quite often positive operators are automatically continuous. This was first proven by M. Krein for positive linear functionals [81 ] and later was generalized in several contexts by various authors; see, for instance, [32,99,101,121]. The next result, due to Lozanovsky, is the strongest in this direction and appeared in [136]. COROLLARY 6 (Lozanovsky). Let X and Y be two ordered Banach spaces whose cones are closed. I f additionally, the cone o f X is generating, then every positive operator T : X --+ Y is continuous. PROOF. It suffices to show that a positive operator T has a closed graph. So, assume Xn --+ 0 in X and Txn --+ y in Y. By passing to a subsequence, we can also assume oo Y~n=l n ]lXn I] < cx~. By Theorem 2 there exist some M > 0 and two sequences {Yn } and {Zn} of X+ satisfying Xn = Yn - Zn, Ilynl] ~< m[Ixnll, and I[Zn [] ~< mllxnl[ for each n. Since X+ is closed, the vector z = ~-~n~=l n(yn + Zn) of X belongs to X+ and - z ~ 0: Ix[ ~< ~.u}. Clearly this norm is always Fatou; if X is a-Dedekind complete, then I]" Ilu is complete.

2. Operators between Banach lattices If X and Y are Banach spaces, then the symbol s Y) denotes the Banach space of all continuous operators from X to Y; we let s = E ( X , X). 2 If X and Y are in addition Banach lattices, then there are several very important classes of operators associated with the order structure on these spaces. We denote by 12+ (X, Y) the collection of all positive operators. This collection is a cone and it induces a natural (partial) order on the Banach space C ( X , Y). The subspace s Y) - E + ( X , Y) of/2(X, Y) generated by the c o n e / 2 + ( X , Y) is denoted b y / 2 r (X, Y), or simply/2 r, and is referred to as the space of regular operators. In other words, an operator is said to be regular if it can be written as a difference of two positive operators. An operator T" X --+ Y is order bounded if T maps order bounded sets in X to order bounded sets in Y. The symbol/2 b (X, Y) denotes the space of all order bounded operators. Clearly each regular operator is order bounded, that is, s c_ E b. The converse is not true in general (see [29] for several counterexamples, the first of which was found by S. Kaplan). Here we can ask the question: when does the equality s = F b hold? The next famous theorem of E Riesz and L. Kantorovich describes a rather general situation when this happens. THEOREM 8 (Riesz-Kantorovich). Let X, Y be vector lattices with Y Dedekind complete. Then E r (X, Y) = 12b (X, Y), i.e., each order bounded operator T : X --+ Y is regular. Moreover, in this case, Er (X, Y) is itself a Dedekind complete vector lattice whose lattice operations satisfy the Riesz-Kantorovich formulas, according to which f o r each T ~ E r (X, Y) and each x ~ X+ we have T + ( x ) -- sup{Tu" 0 i T } .

Clearly IIT II ~ IIT IIr and for positive operators both norms coincide. It is well known that (Z;r (X, Y), II" IIr) is a Banach lattice. If Y is Dedekind complete, then the modulus Irl exists by Theorem 8, and in this case IIr lit = IlITI II.

3. When is every continuous operator regular? As the title of this section indicates, we shall discuss here the pairs of Banach lattices X and Y for which the space of all continuous operators E(X, Y) coincides with the space of all regular operators s (X, Y). We should distinguish here between two closely related but different questions. The first one requires simply the set-theoretic equality IZ(X, Y) = ~,r ( x , Y),

(*)

while the second requires (,) and, additionally, the norm equality IIT II = IIT Ilr

for each T E L;(X, Y).

(**)

We will refer to these problems as the isomorphic and isometric problems, respectively. The following two theorems, which are essentially due to Kantorovich and Vulikh [76], describe the two most important classes of Banach lattices for which each continuous operator is regular. These are the familiar AM- and AL-spaces, also known under the names abstract M spaces and abstract L l spaces, respectively, and defined in [72, Section 5].

92

Y.A. Abramovich and C.D. Aliprantis

THEOREM 9. Let Y be a Dedekind complete Banach lattice with a strong unit. Then f o r each Banach lattice X equality (,) holds, that is, s Y) = s (X, Y). If the norm on Y is Fatou (in particular, if Y is equipped with a unit norm and, thus, is an AM-space), then (**) also holds. THEOREM 10. If X is an AL-space and Y is an arbitrary Banach lattice with a Levi norm, then equality (,) holds. If the norm on Y is Fatou, then we also have (**). To be precise, it was assumed in [76] that Y in Theorem 10 was a KB-space, and it was noticed in [ 129] that the original proof could be easily carried over to an arbitrary Banach lattice with a Levi norm. For a KB-space Y the proofs can be found in [29, Theorem 15.3], and [ 134, Theorem 8.7.2]. Under the assumption that Y is positively complemented in Y** (which is somewhat stronger than the Levi property) a proof of Theorem 10 is presented in [96, Theorem 1.5.11]. Two principal questions can be asked in connection with the previous theorems. (1) Do Theorems 9 and 10 characterize the AL- and AM-spaces? (2) To what extend is the Levi condition essential f o r the validity o f (,)? The first question is quite old and has various interpretations depending on whether we deal with the isometric or the isomorphic version. We refer to [3] for a brief survey of major results in this direction. Here we mention a few of them, starting with the isometric version of this question. THEOREM 1 1. Suppose that the Banach lattices X and Y satisfy both (,) and (**). Then either X is an AL-space or Y is an AM-space. The isomorphic version is much deeper and has, in general, a negative solution [2]. THEOREM 12. There exist two Dedekind complete Banach lattices X and Y satisfying (,) such that X is not order isomorphic to any AL-space and Y is not order isomorphic to any AM-space. Moreover, f o r any given e > 0 the Dedekind complete Banach lattices X and Y can be constructed in such a way that IIT IIr ~< (1 + e)II T ]lf o r each T ~ 12(X, Y). The strongest affirmative isomorphic result is also due to Cartwright and Lotz [44] and is stated next. A simple proof of this result is given in [ 17], where the technique introduced by Tzafriri in [133] of abstract Rademacher functions is utilized. The works of Fremlin [59], ~rno [ 105] and Schaefer [ 122] contained some versions of the next result. THEOREM 13. Let two Banach lattices X and Y satisfy (,), and assume that either X* or Y contains vector sublattices uniformly in n isomorphic to g np f o r some p E [1 cx~). In the f o r m e r case Y is order isomorphic to an AM-space and in the latter case X is order isomorphic to an AL-space. The case when X = Y is exceptionally nice and was studied in [2]. THEOREM 14. I f a Banach lattice X satisfies s either to an AL- or AM-space.

= s

(X), then X is order isomorphic

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PROOF. To sketch a proof, note that the following alternative is true: X either contains uniformly in n the vector sublattices order isomorphic to g ~ or it does not. In the former case, X* contains uniformly in n the vector sublattices g~ and so, by Theorem 13, X is isomorphic to an AM-space. In the latter case, X is isomorphic to an AL-space by Drno's theorem. [] Questions (1) and (2) formulated above, though quite natural, have been addressed by Abramovich and Wickstead only recently in [21 ]. THEOREM 15. A Dedekind complete Banach lattice Y has a Levi norm if and only if for every AL-space X we have s Y) = s (X, Y). This converse to Theorem 10 is somewhat partial since we assume Y to be Dedekind complete. On the other hand, as the next two theorems in [21 ] show, it is the best one can get. Note also that there are non-Dedekind complete Banach lattices (hence, without a Levi norm) which, nevertheless, satisfy (,). THEOREM 16. For a Banach lattice Y the following statements are equivalent:

(a) Y is Dedekind complete. (b) For all Banach lattices X, the space f_r (X, Y) is a Dedekind complete vector lattice. (c) For all AL-spaces X, the space s (X, Y) is a vector lattice. THEOREM 17. For a Banach lattice Y the following statements are equivalent:

(1) (2) (3) (4)

Y is ~-Dedekind complete. s (X, Y) is ~-Dedekind complete for each separable Banach lattice X. s (c, Y) is a vector lattice, where c is the space of all convergent sequences. /:r (LI [0, 27r], Y) is a vector lattice.

We close this section by mentioning one more direction of research devoted to the connections between the order and topological properties of compact operators. Namely, what can be said about the modulus of a compact operator T :X --+ Y between Banach lattices? Does this modulus exist? When is it compact or weakly compact? A well known example by Krengel [29, Example 16.6] shows that even on ~2 the modulus of a compact operator may fail to exist. We refer to [20] for some history, further references and concrete results concerning the properties of the modulus of a compact operator.

4. Domination, compactness, and factorization DEFINITION 18. Let T, B : X --+ Y be two operators between vector lattices with B positive. We say that the operator T is dominated by the operator B (or that B dominates T) provided ]T(x)l ~< B(Ixl) for each x 6 X.

It should be clear that every operator between Banach lattices dominated by a positive operator is automatically continuous, and that a positive operator T is dominated by another positive operator B if and only if 0 ~< T ~< B. When Y is Dedekind complete, an

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operator T is dominated by a positive operator B if and only if T is regular and ITI ~< B holds (use Theorem 8 to check this). We are now ready to discuss a new and interesting direction dealing with dominated operators on Banach lattices. This direction can be described collectively as the domination problem. THE DOMINATION PROBLEM. Assume that B : X --+ Y is a positive operator between Banach lattices and assume that B satisfies some property ( P ). What effect does this property have on an operator T dominated by B ? Of course, the best one can expect is that T also satisfies (P). Here are a few examples of the properties that have been studied in connection with the domination problem: compactness, weak compactness, Dunford-Pettis, and Radon-Nikodym. A first general domination result was established in 1972 by Abramovich [ 1] for weakly compact operators. He proved that if a linear operator S : X --+ Y from a Banach lattice to a KB-space is dominated by a weakly compact operator, then S is also weakly compact. Several years later Pitt [ 109] proved a domination result for compact operators, namely he s h o w e d t h a t i f B isacompactpositive operatoron L p ( # ) with 1 0 such that T x -- r ( T ) x . PROOF. We will sketch a proof. The inclusion r ( T ) E cr(T) is caused merely by the positivity of T. Indeed, if we denote by R()~) the resolvent operator of T, then clearly R0~) > 0 for each )~ > r ( T ) , Also for each )~ with 1)~1 > r ( T ) the inequality ]]R(])~[)]] ~> ]]R()~)]] 1 holds. Therefore, for )~, -- r ( T ) + -d we have []R()~n)l[ ~ oc, whence r ( T ) ~ or(T). Assume further that T is compact and r ( T ) > 0. There exist unit vectors Yn E X+ such that ]lRO~,)yn I[--~ oc. Using the vectors y,, we introduce the unit vectors xn = R()~n)yn/llR()~n)Yn][ ~ X+. Since T is compact we can assume that Txn --+ x ~ X+. Finally, using the identities r ( T ) x n - Txn -

[ r ( T ) - Xn]xn + ()~n - T)xn

= [r(T) - ~ , ] x , + y~/IIRO~)y~II and that r ( T ) > 0 we infer that x 5~ 0 and that T x -- r ( T ) x . The conclusion of the previous theorem remains valid if we replace the compactness of T by the compactness of some power of T. Indeed, assume that T k is compact for some k. Since r ( T k) = [r(T)] k > 0 the previous theorem implies that there is a vector x > 0 such that T k x = [r(T)]kx. It remains to verify that the non-zero positive vector k - I r i T k - l - i x is an eigenvector of T corresponding to the eigenvalue r ( T ) . Y -- Y~i=0 The reader is referred to [121,123,144] for complete proofs and many pertinent results concerning the Krein-Rutman theorem. Some relevant results can be found in [4]. Note that the Krein-Rutman theorem holds not only for Banach lattices but for ordered Banach spaces as well. There is an interesting approach allowing to relax the compactness assumption. Namely, as shown by Zabrel3cO and Smickih [146] and independently by Nussbaum [ 102], instead of the compactness of T it is enough to assume only that the essential spectral radius re(T) is strictly less than the spectral radius r ( T ) . A different type of relaxation is considered in [121], where the restriction of T to X+ is assumed to be compact, that is, T maps the positive part of the unit ball into a precompact set. A version of this result, given in terms of re(T), can be found in [ 102]. Another classical result by M. Krein [82, Theorem 6.3] is the following. THEOREM 34 (Krein). Let T" C(I-2) --+ C(I-2) be a positive operator, where S72 is a compact Hausdorff space. Then T*, the adjoint o f T, has a positive eigenvector corresponding to a non-negative eigenvalue.

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PROOF. Consider the set G -- { f E C(t'2)+ f ( 1 ) = 1}, where 1 denotes the constant function one on t'2. Clearly, G is a nonempty, convex, and w*-compact subset of C ( ~ ) * . Next, define the mapping F ' G --+ G by

F ( f ) --

f+T*f f+T*f = [f + T'f](1) 1 + f(rl)

A straightforward verification shows that F indeed maps the set G into itself and that F: (G, w*) --+ (G, w*) is a continuous function. So, by Tychonoff's fixed point theorem (see, for instance [22, Corollary 16.52]) there exists some q~ E G such that F(4~) = 4~. That is, q~ + T*4~ -- [1 + 4~(T1)]4), or T*4) -- 4)(T1)4~, establishing that 0 < 4~ E C(12)~_ is an eigenvector for T* having the non-negative eigenvalue 4~(T1). M A proof of Theorem 34 that does not use fixed point theorems can be found in [77]. COROLLARY 3 5. Every positive operator on a C (Y2)-space (where Y2 is Hausdorff, compact and not a singleton) which is not a multiple of the identity has a non-trivial hyperinvariant closed subspace. PROOF. Let T ' C ( S 2 ) --+ C(Y2) be a positive operator which is not a multiple of the identity. By Theorem 34 the adjoint operator T* has a positive eigenvector. If )~ denotes the corresponding eigenvalue, then the subspace (T - )~I)(X) has the desired properties. V] Recall that a continuous operator T : X ~ X on a Banach space is said to be quasinilpotent if its spectral radius is zero. It is well known that T is quasinilpotent if and only if l i m n ~ ]lTnxl] 1/n = 0 for eachx c X. It can happen thata continuous operator T: X ~ X is not quasinilpotent but, nevertheless, l i m n ~ IlTnx ]ll/n = 0 for some x ~ 0. In this case we say that T is locally quasinilpotent at x. This property was introduced in [6], where it was found to be useful in the study of the invariant subspace problem. The set of points at which T is quasinilpotent is denoted by QT, i.e., QT = {x 6 X: l i m n ~ IlTnxll 1/n = 0 } . It is easy to prove that the set QT is a T-hyperinvariant vector subspace. We formulate below a few simple properties of the vector space QT. 9 The operator T is quasinilpotent if and only if QT = X. 9 QT = {0} is possible every isometry T satisfies QT = {0}. Notice also that even a compact positive operator can fail to be locally quasinilpotent at every non-zero vector 9 For instance, consider the compact positive operator T:g2 -+ ~2 defined by T(xl, X 2 , 9 9.) - - (X l , x22 ' x33 . . . . ). For each non-zero x E ~2 pick some k for which Xk r 0 and note that IlTnxl] 1/n ~ ~]xk[ 1/n for each n, from which it follows that T is not quasinilpotent at x. 9 QT can be dense without being equal to X. For instance, the left shift S: g2 -+ g2, defined by S(xl, x2, x3 . . . . ) = (x2, x3 . . . . ), has this property. 9 If QT ~: {0} and QT ~ X, then QT is a non-trivial closed T-hyperinvariant subspace of X. The above properties show that as far as the invariant subspace problem is concerned, we need only consider the two extreme cases: QT = {0} and QT = X.

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We are now ready to state the main result about the existence of invariant subspaces of positive operators on gp-spaces. It implies, in particular, that if a positive operator is quasinilpotent at a non-zero positive vector, then the operator has an invariant subspace. This is an improvement of the main result in [6]. THEOREM 36. Let T : g~p --+ g~p (1 O, then B has a non-trivial closed invariant ideal. Moreover, if another positive operator T" X --+ X commutes with B, then T and B have a common non-trivial closed invariant ideal.

PROOF. Let B be quasinilpotent at some x0 > 0. Also, let T" X --+ X be another positive operator that commutes with B. Fix three non-zero operators R, C, K ' X ~ X with K compact such that RB-

BR,

ICxl ~ R(IxI),

and

ICxl ~ K(IxI)

for each x 6 X.

We can suppose that l[B + T II < 1 and define A -- ~ n = 0 ( B + T) n. Clearly, the positive operator A commutes with both B and T and satisfies A x >~x for each x >~ 0. Also, for each x > 0, let Jx denote the non-zero principal ideal generated by A x in X, that is, Jx -- {Y ~ X: lyl ~< )~Ax for some )~ > 0}. If Jx ~ X for some x > 0, then the ideal Jx is a non-trivial closed (B + T)-invariant ideal. This ideal Jx is, of course, also invariant under both B and T. So, we can assume that Jx - X for each x > 0, i.e., A x is a quasi-interior point in X for each x > 0. Since C r 0, there exists some Xl > 0 such that CXl ~ O. Since A l C x l l is a quasiinterior point and ICxl[ O, and in particular r(U) > 0 and r ( V ) > O. PROOF. We will outline the main points of the proof. First of all, let us show that every non-zero compact quasinilpotent positive operator T has a non-trivial closed positively hyperinvariant ideal J, that is, J is invariant under each positive operator in the commutant {T}' of T.

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Let F be the closed ideal generated by the range of T. We claim that F is the closure of a principal ideal. The compactness of T implies that the range of T is separable. So, there exists a countable subset {yl, y2 . . . . } of T (X) consisting of non-zero vectors that is norm dense in T ( X ) . Consider u -- y~'n~__l 2,I~1'~I,ii which is in F. For the principal ideal A, generated by u we clearly have A, _ F. Conversely, the inclusion {yl, y2 . . . . } _ A, implies that T (X) ___A,. Hence, F c_ Au and so A, = F. Clearly, F 5~ {0} and it is easy to verify that F is indeed a positively hyperinvariant ideal for T. We are done if F is proper. So assume that F -- A, = X, which implies in particular that X has quasi-interior points. In this case, the existence proof of a non-trivial closed positively hyperinvariant ideal is just the second part of the proof of de Pagter's result mentioned above [106, Proposition 2]. The basic steps of this proof follow. Let J + = {0~< S E s 3 R E {T}'with0~< S ~< R}. Also, consider the vector subspace ,7 = {S1 - $2: Sl, $2 E ,.7+ } of s In addition, for each x >~ 0, let J [ x ] = {Sx: S E J } . It is shown in [106] that: (1) For each x > 0 the closure J [ x ] is a non-zero T-invariant closed ideal; and (2) For some x > 0 the closed ideal J [ x ] is non-trivial. Next, let C be a positive operator that commutes with T. If S E J , then there exist S1,82 E , J + and RI, R2 E {T}' satisfying S = Sl - 82 and 0 ~< Si ~ Ri (i -- 1,2). Then, 0 ~ 0 and also that the off-diagonal entries of I - A are nonpositive numbers. The economic situation is "feasible" if and only if I - A is nonsingular and r ( A ) ~< 1. Under these conditions the system can be uniquely solved for each output vector d ~> 0 and x - (I - A ) - l d ~> 0. For details and various generalizations see [33,35].

Economics: general equilibrium. Here we shall describe the basic framework of general equilibrium theory in economics" for details see [23]. The principal mathematical tools needed in this area are functional analysis and positive functionals rather than operators. Nevertheless, it seems very appropriate to mention this area in order to demonstrate the extent to which mathematical sophistication has penetrated work in economics and finance. The economic intuition of commodities and prices is understood by means of a dual system (X, X'). The vector space X is the commodity space and X I is the price space. The evaluation (x, x I) is interpreted as the value of the "bundle" x at prices x ~. For simplicity, we shall assume here that X is a (finite- or infinite-dimensional) Banach lattice and that X ~ is its norm dual. An exchange economy consists of a finite number of consumers, say m, indexed by i (i -- 1 . . . . . m) whose commodity-price duality is described by (X, X'). Every consumer i has a preference relation >-i, i.e., a binary relation on X+, which allows the consumer to distinguish out of any two bundles x and y the one which is better for him. The relation x ___iY is read "the bundle x is preferred to y " T h e preference ___.iis assumed to be complete (i.e., any two vectors in X+ are comparable), transitive, convex (i.e., the better-than-x set {y E X+" y ___iX} is a convex set for each x E X+), continuous (i.e., for each x E X+ the sets {y E X+" y ~i X} and {y E X+" x ___iY} are both closed), and monotone (i.e., "more is better" in the sense that y > x ~> 0 implies y ___iX). Each consumer i has also an initial endowment COi > 0. The total or the social endowment (or the resources) of the economy is the vector c o - Ziml coi. An allocation (redistribution) is any m-tuple (xl, x2 . . . . . Xm) such that xi >~ 0 for each i and Ziml x i - co. The objective of the economic activity in the economy is to redistribute the resources among the consumers in such a way that each consumer becomes better off. The efficient ways of redistributing the goods among the consumers are achieved by means


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of Pareto optimal allocations. An allocation (x l, X 2 . . . . . Xm) is said to be Pareto optimal if there is no other allocation (yl, y2 . . . . . Ym) such that Yi ~-i Xi for each consumer i and Yi ~ i xi for s o m e / ( w h e r e yi ~i Xi means yi ~i Xi and xi ~ i Yi). In other words, a distribution of the resources is Pareto optimal if it is impossible to make some consumer better off without making some other one worse off. Another important property of allocations is that of individual rationality: an allocation (x l, x2 . . . . . Xm) is individually rational if Xi ~ i (-t)i holds for each consumer i; that is, the redistribution does not give any consumer a bundle which is worse than his initial endowment. If the order interval [0, co] is weakly compact (in particular, if X has order continuous norm), then an easy application of Zorn's lemma guarantees that Pareto optimal individually rational allocations exist. The above efficiency notion requires that the acceptable redistributions of the goods are Pareto optimal and individually rational. This brings us to the important decentralization problem in economics, which in mathematical terminology can be stated as follows: Is there a price system (i.e., some linear functional x ~ E X ~) that can be used to enforce a given Pareto efficient allocation? Put another way: Can we use a system of prices - like the one we see in every day life - to redistribute the resources of the economy? This problem is related to the mathematical notion of supportability of convex sets by continuous linear functionals. Recall that a subset A of a vector space is said to be supported by a non-zero linear functional f at some point a E A if f (a) ~< f (x) holds for all x 6 A. An allocation (x l, x2 . . . . . Xm) is said to be supported by a non-zero price x ~ E X ~ if x' supports the better-than-x/set of each consumer i (i.e., the set {x E X+: x ~i Xi }) at the point xi. That is, x ~ E X~supports the allocation (Xl, x2 . . . . . Xm) if and only if

x •

xi---> (x, x') >>,(xi, x').

Now the economic concept of the decentralization of the allocations finds its best interpretation in the two famous theorems of welfare economics. These theorems were first proved for finite dimensional commodity spaces by K. Arrow and G. Debreu and are stated next. 1 S T WELFARE THEOREM. Every allocation supported by prices is Pareto optimal. 2ND WELFARE THEOREM. Every Pareto optimal allocation can be supported by prices. In particular, the second welfare theorem is of fundamental importance and its validity is a "must" for any economic system. To establish the validity of this theorem is, in general, a difficult problem, especially when one deals with an infinite dimensional commodity space. Supporting a convex set at a given (boundary) point by a non-zero continuous linear functional is not always feasible and supporting m convex sets each at a specified point by the same non-zero continuous linear functional is, if not impossible, a very serious matter. To achieve this goal, we must impose some extra conditions. The existence of a supporting price for a given Pareto optimal allocation (x 1, x2 . . . . . Xm) is proved by utilizing the lattice structure of the price space X ~. It usually consists of two steps. In the first step one proves the existence of a non-zero price x i that supports the

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better-than-x/ set at Xi. Once this is done, then the second step requires a sophisticated argument to establish that the supremum linear functional

x' -- x'l v x~ v . . . vx~ is non-zero and supports the allocation (x l, x2 Xm). The lattice structure of X' is of fundamental importance. As a matter of fact, if X' is not a vector lattice, there are examples of Pareto optimal allocations that cannot be supported by prices. . . . . .

Finance: choosing the optimal portfolio.

We shall discuss here how positive operators appear in the theory of finance in the context of choosing an optimal portfolio under uncertainty. For a systematic presentation the reader can consult the excellent book [95]. All investment decisions are made under uncertainty regarding future financial conditions. To analyze the decision process under uncertainty in the theory of finance, one usually introduces a two-period model (the simplest possible model), where at period 0 (today) everything is known and at period 1 (tomorrow) everything is unknown and uncertain. To capture the notion of uncertainty tomorrow one usually introduces the set S of all possible states of the world tomorrow. The usual assumptions about S is that it is either a finite set S -- {1, 2 . . . . . S }, or a more general compact Hausdorff topological space, or a probability space (S, P). By the term security one should understand any object or property that can be traded in the markets and that has a value today. While any such object (security) has a specific value today, its value tomorrow is uncertain and depends, in general, upon the prevailing state of the world tomorrow. Thus, a security is characterized by a value (or a payoff) at each possible state of the world tomorrow. Mathematically, this naturally leads to the interpretation of a security simply as a function x : S --+ R, where x(s) is interpreted as the value of the security x tomorrow when the state of the world s has been realized. Obvious economic considerations require that the collection of all securities X is a vector space under the usual pointwise operations. This space is called the space of securities. The most common spaces of securities used in finance are C (S) and L2 (S, P); the spaces L p (S, P) also are used quite often. In sum: the vector space of securities represents every object or property available that can be possibly traded in the market today. However, in today's market not every security is available for trade. For instance, every house is a security, but not everyone's house is in the market for sale today. The collection M of all securities that are available for trading today is assumed to be a vector subspace of X, and this space M is called the marketed space. If M -- X, then the markets are called complete and if M -r X the markets are incomplete. From the practical point of view the markets are always incomplete. Clearly, the structure of M should play an important role in making decisions today given the uncertainty of tomorrow. Now let us discuss the hedging problem. In order to protect their securities, given the uncertainty of tomorrow, people are forced to buy insurance. An insurance is simply a non-negative marketed security. If a person owns a security x and wishes the value of this security tomorrow never to fall below a certain level k (called the striking price or the floor), then he must purchase some insurance a c M+ = M A X+ so that x + a ~> k, or x(s) + a(s) >7k for each state s of the world tomorrow. This implies a ~ k - x, and hence a ~> (k - x) +, where (k - x)+(s) =

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max{k - x(s), 0}. Therefore, the "smallest" desired insurance seems to be (k - x) + and it is called the put option of the security x with striking price k. Since M need not be a vector sublattice of X, the put option (k - x) + need not be available in the market today, i.e., (k - x) + need not belong to M. This is why the theory of lattice-subspaces plays an important role here along with the theory of Banach and vector lattices (see for instance [ 14,24]). And now we can state the hedging problem of the theory of finance. THE HEDGING PROBLEM. Given a marketed security x, what is the "cheapest insurance" that one can purchase in order to guarantee at least the value k for the security tomorrow no matter what the prevailing state will be? Of course, if M is a vector sublattice of X and x, k 6 M, then a = (k - x) + should be the desired answer. The general solution of the Hedging Problem is non-trivial and is related to the meaning of the "cheapest insurance" 9This is the place where positive operators and their properties make their appearance. To simplify matters further, let us assume that there is either a finite or a countable number of linearly independent securities whose linear span is M. Considering the simplest case of the two, we will suppose that there are N securities x l, x2 . . . . . XN in M+ that span M and that S = {1, 2 . . . . . S}. In this case we have N O}.

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Then a price p 6 M* is arbitrage free if and only if (p, RO) = (R* p, 0) > 0 for each 0 6 K, where R* :M* --+ ~I~u is the adjoint operator of R. This duality property is very useful in determining the arbitrage free prices. From economic considerations only arbitrage free prices are acceptable prices and the Hedging Problem can be stated now as follows: Given an arbitrage free price p 6 M*, a security x and a striking price k, find the portfolio 0 E R x that minimizes the expression (R* p, O) = (p, RO) subject to RO ~ x x/ k. The Hedging Problem is also related to the notion of a derivative in the theory of finance and has been studied in m a n y contexts. A derivative is a security of the form x ( s ) = f ( y l ( s ) , y2(s) . . . . . yk(s)), where (in the case X = C ( S ) ) f : R k --+ R is a continuous function and yl, y2 . . . . . yk are k given securities; that is a derivative is a security whose payoff depends on the payoffs of a finite number of other securities.

Acknowledgment The authors would like to express their thanks to A. Kitover and V. Troitsky for many useful suggestions.

Added in Proof Detailed proofs of the results discussed in this survey as well as an extensive treatment of many related topics can be found in the forthcoming book "An invitation to Operator Theory" by the authors that will be published shortly by the American Mathematical Society.

References [1] Y.A. Abramovich, Weakly compact sets in topological Dedekind complete vector lattices, Teor. Funkcii Funkcional. Anal. i Prilo~en. 15 (1972), 27-35. [2] Y.A. Abramovich, On spaces of operators acting between Banach lattices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Investigations on Linear Operators and Theory of Functions VIII 73 (1977), 182-193. [3] Y.A. Abramovich, When each continuous operator is regular, Funct. Analysis, Optimization and Math. Economics, Oxford Univ. Press (1990), 133-140. [4] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Positive operators on Krein spaces, Acta Appl. Math. 27 (1992), 1-22. [5] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, On the spectral radius of positive operators, Math. Z. 211 (1992), 593-607. Corrigendum: Math. Z. 215 (1994), 167-168. [6] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces of operators on e p-spaces, J. Funct. Anal. 115 (1993), 418-424. [7] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces ofpositive operators, J. Funct. Anal. 124 (1994), 95-111. [8] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Invariant subspaces for positive operators acting on a Banach space with basis, Proc. Amer. Math. Soc. 123 (1995), 1773-1777. [9] Y.A. Abramovich, C.D. Aliprantis and O. Burkinshaw, Local quasinilpotence, cycles and invariant subspaces, Proceedings of the Conference Interaction Between Functional Analysis, Harmonic Analysis, and Probability, Lecture Notes Pure Appl. Math. 175, N. Kalton, E. Saab and S. Montgomery-Smith, eds, Marcel Dekker, New York (1995), 1-12.

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[139] A.W. Wickstead, Extremal structure of cones of operators, Quart. J. Math. Oxford Ser. (2) 32 (1981), 239-253. [140] A.W. Wickstead, The regularity oforder bounded operators into C(K), Quart. J. Math. Oxford 44 (1993), 257-270. [ 141 ] A.W. Wickstead, Dedekind completeness of some lattices of compact operators, Bull. Polon. Acad. Sci. 43 (1995), 297-304. [142] A.W. Wickstead, Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Camb. Phil. Soc. 120 (1996), 175-179. [143] Y.C. Wong and K.E Ng, Partially Ordered Topological Vector Spaces, Clarendon Press, Oxford (1973). [144] A.C. Zaanen, Riesz Spaces H, North-Holland, Amsterdam (1983). [145] A.C. Zaanen, Introduction to Operator Theory in Riesz Spaces, Springer, Berlin (1997). [ 146] P.E Zabre~o and S.V. Smickih, A theorem ofM. G. Kre~n and M.A. Rutman, Funktsional. Anal. i Prilozhen. 13 (3) (1979), 81-82.

CHAPTER

3

Lp Spaces Dale Alspach Department of Mathematics, Oklahoma State University, Stillwater, OK, USA E-mail: alspach @math. okstate, edu

Edward Odell* Department of Mathematics, The University of Texas, Austin, TX, USA E-mail: odell@ math. utexas, edu

Contents 1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Global structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Sequences in ~ p and L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Subspaces of L p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. /2p-spaces, 1 < p < cx~, p-f: 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*Research supported by N S E H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 123

126 129 134 140 146

156


L p spaces

125

In this chapter we will discuss the structure of the L p-spaces and their subspaces. We are concerned mainly with the reflexive case (1 < p < ~ ) . The space L ~ and the /21-spaces are considered elsewhere [90]. A theory requires and feeds on its examples. The L p-spaces have provided much fodder for the general theory of Banach spaces because they appeared early in the theory and the study of these spaces has motivated the definitions of many properties of more general Banach spaces. For example, with its usual norm L p is a Banach lattice under the pointwise almost everywhere ordering. In the reflexive case it also has an unconditional basis and thus has another lattice structure as a sequence space (and is a Banach lattice under a different norm except in the case p - 2). These spaces naturally occur as interpolation spaces and are the simplest of the rearrangement invariant spaces. The Hardy spaces, Hp(D), of analytic functions on the unit disk with Lp boundary values are isomorphic to Lp, 1 < p < cxz, and the Bergman spaces A p ( D ) of analytic functions on the unit disk which are in L p ( D ) are isomorphic to ~p, 1 ~< p < e~, [67]. The study of the structure of the finite dimensional subspaces of L p paved the way for much of the extraordinary development of the local theory of Banach spaces in the 1980's [77]. In investigations of other Banach spaces and operators the existence and classification of operators from, into or factoring through L p-spaces provide fundamental information on the structure. We will concern ourselves primarily with the infinite dimensional isomorphic structure of the Lp-spaces. In particular we shall concentrate on separable Lp-spaces and, as noted in the basic concepts chapter, [49, Section 4], this reduces essentially to studying g p and L p(O, 1) (which we shall denote by L p). Functions are assumed to be real-valued as in general there are only minor adjustments needed for the complex case. As the theory of Lp-spaces was developed some natural questions arose. What are the complemented subspaces of Lp (or ~p).9 When does a given Banach space X embed into Lp (or gp)? If X c_ Lp, what subspaces must live inside of X? Does every Banach space contain some g p or co? What can be said about the structure of an unconditional basic sequence in L p ? We address these problems and others below. With apologies to the experts we choose to include some of the well known (to them) structural results that were mentioned in the basic concepts chapter or appear in books such as [70]. In Section 1 we review certain inequalities for sequences in L p. For example we show that by integrating against the Rademacher functions one can deduce that a normalized unconditional basic sequence in Lp (2 < p < CX~)admits upper ~2 and lower gp estimates (respectively, for 1 < p < 2, upper gp and lower ~2 estimates). In Section 2 we study the global structure of L p and in particular the Haar basis. Among the results given we examine the span of subsequences of the Haar basis and Schechtman's result that every complemented subspace of L p with an unconditional basis is isomorphic to a complemented subspace spanned by a block basis of the Haar basis. Section 3 begins with some properties and characterizations of the unit vector basis of p. The details of the argument using the Pelczyfiski decomposition method that a complemented subspace of g p is isomorphic to g p are presented to help give context and meaning to the notion of (p, 2) bounded operators discussed later in Section 5. We also discuss spreading models and types and the Krivine-Maurey theorem. This latter result gives a sufficient condition, in terms of types, for a Banach space X to contain almost isometric copies of g p for some p. Section 4 deals with subspaces X of L p. For example if

D.AlspachandE. Odell

126

C Lp

X (2 < p < cx~) then X must contain an isomorph of s or s and if X does not contain s it must embed into s We also consider a necessary and sufficient condition for a reflexive space X to embed into the s sum (1 < p < cx~) of finite dimensional spaces. The last section c o n c e r n s / 2 p - s p a c e s - those complemented subspaces of L p which are not Hilbert spaces. Here the known examples and their isomorphic classification are discussed. In particular Rosenthal's space X p and its generalizations are presented in detail. The role of (p, 2)-bounded operators is also explained. Finally we discuss some of the results for s with )~ near one and their relation to the isometric theory of Lp-spaces.

1. Preliminaries We first recall a few key properties of L p and s which are discussed throughout the basic concepts chapter. The unit vector basis for s is a 1-symmetric basis [49, Section 3]. The Haar basis (hi)~ is an unconditional basis of Lp for 1 < p < c~ [49, Section 3], [24]. It is also a monotone basis for for 1 ~< p < c~. The Rademacher functions (rn)n~=l , [49, Section 4], are equivalent to the unit vector basis of s for p < ~ , (and the unit vector basis of el for p = ~ ) . Thus for 0 < p < oc there exist constants A p, B p with

Lp

Ap(~lanl2)l/2 ~APpfo

]xi(s)] 2

ds.

1

Now

Ilxi I12 p

p

/'/

= II(iix, li~), II~2t. /7

= ~ [Ixi []Ppai

for some

(ai)~z ~ e2/(2-p)

of n o r m 1

1

~ Js

[Xi(S)I2)P/2(~I [oi12/(2-P))(2-p)/2ds

by H61der's inequality

"-- fo which completes the proof of (1.2).

iXi(S)12

ds,

D. Alspach and E. Odell

128

(1.2) and (1.3) can be viewed as generalizations of Clarkson's inequalities [29]. Since II 9IILp ~< I1" IlL2 for p ~< 2 we also have using (1.2) for p = 2 that

(So

1/2

~ri(t)xi 1

p

dt

~

ilxill 2

The technique of integrating against the Rademacher yields some useful inequalities for unconditional basic sequences in L p. If (Xn) is a A-unconditional basic sequence in L p then

A-I [f01 (Elanl2iXn(S)l

0 we partition (hji) into a countable number of subsequences (hji)iEN k so that supp hji A supp h je = 0 if i -J= e 6 Ark and so that if k >~2, i ~ Nk then there exists ~ Nk-1 with supphji c_ supphje. Set Bk = UicNk supphji" Clearly B1 ___ B2 ___ " "

and

(-]Bk - - A .

One now chooses a subsequence (kn) of N with m (Bk, \ A) < en, en .[. 0 rapidly. Define

do- L icNk o

d,- L

hji

i~Nk 1

and inductively for m odd

+ }, d2"+m -- Z hji summed over {i 6 Ark," supphji ___suppd2n_l+~(m+l)/2]] and for m even

d2n+m-Zhjisummed over {i ~ Nk," supphji

c_ d~_,+~(m+,)/2~}.

A perturbation argument yields that (dn) is equivalent to the Haar basis and it is not hard to see that [(dn)] is complemented in X, [10]. D A vector-valued version of this theorem exists [78]. There also exists a finite dimensional version of this theorem.

L p spaces

131

THEOREM 2 ([80]). Let 1 / 1 and f o r all n a rearrangement (hn~2 n

2n

n 2n

~'i 'i=0 ~ (hi)i=o so that if M E N and (zi) M is a normalized block basis o f (b i )i=0 then there exists A c {1 . . . . . M} with [A[ ~> M / 2 so that ( Z i ) i E A i s Kp equivalent to the unit Ial vector basis o f g~p .

The rearrangement is defined in terms of the supports of the Haar functions. For i < j either supp b n and supp b jn are disjointwithsuppbi n lying to the left of supp bjn orsuppbi n _ suppbj. Every s has a basis ([54], see Section 5 below) but it is still open as to whether or not it must have an unconditional basis. However those that do can be realized isomorphically as being spanned by block bases of (hn). THEOREM 4 ([95]). Let 1 < p < cx~ and let X be a complemented subspace o f Lp with an unconditional basis (Xn). Then there exists a block basis (Yn) o f the Haar basis with [(Yn)] complemented in L p and such that (Xn) is equivalent to (Yn). The first part of this theorem follows from the following proposition. PROPOSITION 5 ([95]). Let (Xn) be an unconditional basic sequence in Lp (1 < p < ~ ) . Then there exists an equivalent block basis (Yn) o f (hn). PROOF. From (1.7) and (1.8) it follows that if (yi) is another unconditional basic sequence in Lp with [yi[ -- Ixi[ then (yi) is equivalent to (xi). By a perturbation argument we may assume that there exist integers m n ~ ~ and scalars (ai) so that 2 mn - l

Xn--

Z azmn+klh2mn+k[" k=0

One defines the yn'S via this formula by removing the absolute values.

(2.1)

[3


132

Before completing the proof of Theorem 4 we recall a useful isomorphic representation of Lp as Lp(~2) for 1 < p < cxz. The latter is the Banach space of all sequences (fi) of measurable functions on [0, 1] with

II(fi)llL,r

fi2 (t)

-

dt

2 it can be shown that (fij) is unconditional and since ( f i ) is a subsequence of (fij) we obtain a projection of Lp(s onto [(fi)]. ( f i ) is equivalent ~'2mn--1 F u r t h e r m o r e (h 2mn+k) is equivalent to to (xi) and is a block basis o f (h 2mn+e)n ~ 1,k=0 (h2mn+k) and this completes the prooffor p > 2. For p < 2 a duality argument is necessary [951. D Some Banach spaces have the property that whenever they are isomorphic to a subspace of another nice enough Banach space that there is another isomorphic embedding with better properties. One property of this type is reproducibility. A basis (xi) for a Banach space X is reproducible if whenever X ___Y and Y has a basis (yi) then some block basis of (yi) is equivalent to (xi). It is easy to see that because they are weakly null and subsymmetric that the unit vector bases of s (1 < p < cxz) or co are reproducible. It is also not hard to show that the unit vector basis of s is reproducible. THEOREM 6 ([67]). The Haar basis for Lp (1 ~< p < cx~) is a reproducible basis. Moreover the same is true for any unconditional basis of Lp (1 < p < cx~). PROOF. By virtue of Proposition 5 it suffices to prove that (hi) is a reproducible basis for L p for 1 ~ 0 there exists a measurable set B c_ A with m(B) -- m(A \ B) and x*(1B) -x*(1A\B) for x* E F. Let en ,~ 0. We can use the above to produce a dyadic tree of sets (An,j a~'2n +n=0, j= 1 as in the supports of the Haar basis and a sequence (fn) as follows. Set f0 -- 1 a0,1 = 1, fl

-- 1Al,l

-- 1Al,2,

f2 -- 1A2,1 - - 1A2,2

and so on. Then ( f , ) is 1-equivalent to (hn) and for some subsequence (Pk) of N cx~

O l'~co. Then L p (#) does not embed into a space with an unconditional basis.

3. Sequences in gp and Lp In this section we look at some special subspaces of ~p and L p which are isomorphic to gr, for some r. Of course for ~p, the only possible value of r is p itself, but for Lp there is always r = 2 and for p < 2 an entire interval of possibilities. General sequences and even unconditional basic sequences in L p are probably too varied to ever be described (see Section 10, [50], for a particularly peculiar subspace with unconditional basis), however, near the end of this section we note some results for Orlicz sequences and other spaces with a symmetric basis. In the previous section we looked at the span of subsequences of the Haar system. For the unit vector basis of gp, subsequences are not mysterious, and in this simpler situation we can successfully describe the span of block bases. Let (ei) be the unit vector basis of ~p. We begin by recalling the a simple fact about block bases of (ei) which was mentioned in [49, Section 64], [71, Proposition 2.a. 1], [44, Proposition 1.5.3]. PROPOSITION 10. Suppose that (Xn) is a block basis of the unit vector basis of g.p. [Xn: n E IN] is isometric to g.p with the isometry which is induced by the basis map Xn ----> ]]Xn lien, n = 1, 2 . . . . . A normalized basic sequence is said to be perfectly homogeneous if it is equivalent to all of its normalized block bases. We have thus shown that the unit vector basis of g p has this property. It is easy to see that the unit vector basis for co is also perfectly homogeneous. It turns out that this is a very special property. THEOREM 11 ([101]). Suppose that X is a Banach space with a normalized perfectly homogeneous basis (Xn), then (Xn) is equivalent to the unit vector basis of g p or co. We refer the reader to Zippin's paper or the book [71, p. 60], for the proof. There have been some generalizations of this result in the context of symmetric spaces [9].

Lp spaces

135

Another interesting property of block bases in ~ p mentioned in [49, Section 4] is the fact that the closed span is always complemented. Indeed, let Xn = ~iEFn aiei be a normalized block basis of (ei) (here the Fn are finite subsets of I~t and F1 < F2 < ... in the sense that maxFn < min Fn+l)and define x n = IlXnll- p Y]icFn lai]P-2aie* for each n, where, as usual, e* denotes the biorthogonal functional to ei. Then (x*) is biorthogonal to (Xn). The formula OO

P x -- E

x* (x)xn

forxeep

n=l

then defines a norm one projection from g p onto the closed span of (Xn), [71, Proposition 2.a. 1], [44, Proposition 1.5.3]. This gives THEOREM 12. If (xn) is a block basis of g.p, then there is a norm one projection from g.p onto [Xn : n ~ H]. As noted in Section 2, a very similar argument yields that if (Xi) ~ Lp is a nonzero disjointly supported normalized sequence then (xi) is 1-equivalent to the unit vector basis of ep and [(xi)] is 1-complemented in Lp. The analog of Theorem 12 holds for co and, as it turns out, is peculiar to these unconditional bases. THEOREM 13 ([69]). Suppose that X is a Banach space with an unconditional bases such that f o r every permutation rc of H and every block basis (yj) of (Xjr(n)), [yj: j ~ H] is complemented, then (Xn) is equivalent to the unit vector basis of g.p or co. As was noted in [49, Section 4], it is an immediate consequence of Theorem 12 and standard perturbation arguments that COROLLARY 14. If Y is an infinite dimensional subspace of g.p, then f o r every e > O, Y has a (1 + E)-complemented subspace (1 + e)-isomorphic to g.p. We turn to some results of Petczyfiski [86] which were important in stimulating interest in classification problems for complemented subspaces. THEOREM 15. If X is an infinite dimensional complemented subspace of g.p, then X is isomorphic to ( p. A proof using Pe{czyfiski's decomposition method, [86] is given in [49, Section 4]. A more effective but more complex proof is given in [7]. That proof proceeds by directly showing that a complemented subspace of g p actually decomposes into a sum of finite dimensional subspaces each isomorphic to gnp for an appropriate n and then deducing that the space is isomorphic to g p from the decomposition. The proof using the decomposition method depends rather heavily on the ability to estimate the norm of an operator on a g p sum by norms of the restrictions to the summands. In


136

Section 5 we will see that the fact that this does not work for other types of unconditional sums leads to a rather delicate approach to understanding the complemented subspaces of C

Lp. It is not true in general that if X ~ Y and Y ~ X then X is isomorphic to Y [42]. One problem that was open until the 1970's was whether every Banach space contained g p for some p or co. If we are satisfied with finite dimensional subspaces Dvoretzky's theorem [33] tells us that for every Banach space X, ~2 is finitely represented in X. Krivine's theorem [61 ] extends this to basic sequences: Every basic sequence admits a p so that g p is block finitely represented therein. Going the other way if X is such that for some ~. < cx~, 1 ~< p < ~ , every finite dimensional subspace of X X-embeds into ~pm for some m then X embeds into L p. This is easy via an ultraproduct argument. In any event if we are seeking some collection of infinite dimensional "atoms" which must live inside of every X, these must include more than just co and g p, 1 ~< p < cxz, as witnessed by Tsirelson's example [99,28]. However we do get a positive result for subspaces X of Lp (1 ~< p < c~). In fact within a certain class of "stable" spaces the conjecture is true in a strong sense. Before we discuss these results let us define a notion that is a little stronger than finite representability. DEFINITION 16. Suppose that (Xn) is a sequence in a Banach space X and that Y is a Banach space with a basis (Yn). We say (Xn) generates the spreading model (Y, (Yn)) if for every s > 0 and k E N, there is an N such that

(1 + / 3 ) -1 ~ a i Y i k=l

aiXni ~ (1 + s) i--1

ai Yi k--1

for all N < ni < n2 < ... < nk and finite sequences of scalars (ai). (Y, (Yn)) is said to be a spreading model over X if there is a sequence (xn) in X generating the spreading model (Y, (y~)) and such that limnl 0 one considers

He,p(X) -- { (xi)~ ~ X: (xi)~ is (1 + e)-equivalent to the unit vector basis /7 Of~p}.

This is naturally a closed tree and can be given an ordinal index [ 16]. One shows that the index is Wl and so the tree has an infinite branch. In general, even for very well-behaved sequences in L p for 1 ~< p < 2, one cannot find a nice s basis by passing to subsequences. (See [43].) These results have been extended to weakly stable Banach spaces (defined as in Definition 17 except one also requires (Xn) and (Yn) to be weakly null) which allows for the case of co [12]. Thus every subspace of a stable Banach space contains a subspace (1 + e)-isomorphic to s One way to look at these results is to say that every subspace of L p, 1 l . A (possibly finite) sequence (x~) is a skipped block sequence with respect to (Gi) if there exist integers r l max(suppxl), the support being with respect to the Ei's. (II) chooses x2 E S/E i )n~ and so on. An outcome of this game is a sequence in

~ - - { ( n l , x l , n 2 , x2 . . . . )" Xj E S(Ei)nj and nj+l > max(suppxj) for j EN}. Player (I) wins if the outcome belongs to r -- { (n 1, x l, n2, X2 . . . . ) E ~" (Xi)C~ is K-equivalent to the unit vector basis of g p }. The precise meaning of (A) is that player (I) has a winning strategy while (B) means that (II) has a winning strategy. If ~ is given the relative product topology then r is a closed subset of ~. Martin's theorem yields then that this game is determined, i.e., either (I) or (II) has a winning strategy. But if (B) holds one easily constructs a block tree T E To(X) with respect to (En) so that no branch is K-equivalent to the unit vector basis of ~ p. D By employing some further blocking and perturbation arguments one can extend this result to PROPOSITION 32 ([84]). Let 1 < p < oc and let X be a reflexive space. Assume there exists K < oo so that whenever T E T~o(X) is a weakly null tree then some branch of T is K-equivalent to the unit vector basis of g.p. Then there exists a sequence of finite dimensional spaces (Fn) so that X ~ ( ~ . Fn)ep. PROOF OF THEOREM 30. One first shows that the hypotheses of both (a) and (b) imply the hypothesis of Proposition 32. This depends upon the fact that X c L p and arguments like those used to prove Theorem 27. For example (b) yields that there exists 3 -- 3(K) > 0 so that if (xi) is a normalized weakly null sequence in X then some subsequence can be written as x if -- Yi -Jr-di where

L p spaces

145

Yi /X di

- - 0 , the di's a r e disjointly supported and Ildi I1 ~> 6 for all i while (a) yields in addition that one can take []di ]1 --+ 1. Using this given T ~ T~o(X) one can select a branch of the tree exhibiting similar behavior and hence is C (K)-equivalent to the unit vector basis

Of~p.

The proof of Proposition 32 yields that if X c Y, where Y is reflexive with an FDD (En) then one can block (En) into (Fn) so that X ~ ( ~ Fn)ep. Thus one can block the Haar basis into an FDD (Fn) so that X embeds into ( ~ Fn)ep. Since each Fn ~ gp" for some m n ~ N, the latter space is a subspace of ~p. ff] From Theorems 30 and 27 it follows that a subspace X of L p for 2 < p < oc which is not isomorphic to ~2 nor to a subspace of s must contain ~2 | g p. What additional conditions on X ensure that X ~ g p ff~ g2 ? The following gives one such result. THEOREM 33 ([52]). Let 2 < p < oc. Let X be a subspace of Lp which is isomorphic to a quotient o f a subspace ofg~p • ~2. Then X embeds into g~p 9 g~2. The method of proof is a more complicated blocking argument which ultimately produces a blocking (Hn) of the Haar basis so that the natural mapping

P

is an into isomorphism. The following is open. It concerns the next layer of small subspaces of Lp. PROBLEM 34. Let X be a subspace of Lp, 2 < p < oc and assume that ( ~ s Does X ~ s 9 s

~

X.

The situation for subspaces of L p with 1 < p < 2 is of course more complicated (cf. Theorem 24). But we can say the following. THEOREM 35 ([93]). Let 1 ~ - . r

Some years after Vaaler's result, several authors, independently, found an analogue, in which the maximum length of the vi is controlled, instead of the sum of the squares of the lengths. It states: THEOREM 8. There is a constant ~ > 0 (independent of everything) so that if (l) i )r~ is a sequence of unit vectors in R n, then vol({x" I(x, vi)[ ~ 1, forall i}) 1/n ~log(1 + m) This statement can be extracted from some results of Carl, and this was done by Carl and Pajor [ 14]: it was proved independently by Gluskin [23] and by Barany and FurEdi [7]: and it was rediscovered by Bourgain, Lindenstrauss and Milman [ 12]. In fact Gluskin's method gives something a bit stronger: it is not necessary to assume a "uniform" estimate on the lengths of the vectors. THEOREM 9 (Gluskin). If (vk) is a sequence of vectors in R n satisfying

Ivkl ~< 2v/log(1 + ~) f o r each k, then

vol({x" I(x, vi)l ~ 1, f o r a l l i}) 1/n

1

>~.

176

K. Ball

It should be remarked immediately, that if one replaces the expressions log(1 + ~) by log(1 + k), then the statement becomes almost trivial. The improvement given by the theorem is of interest if the number of vectors is fairly small compared to the dimension: especially if it is no more than a fixed multiple of the dimension. This is also the range in which Vaaler's Theorem is most useful (and is typically used). Gluskin's result (in the form stated above) includes Vaaler's, apart from the value of the "constant" involved. To see why, observe that if 1~

Ivkl 2 ~ 1

n

and the vk are in decreasing order of norm, then

~log(1 + ~) Strangely enough, the proofs of Theorems 7 and 9, like the proofs of the volume ratio estimates of the last chapter, use comparisons between Lebesgue measure and an appropriate Gaussian measure. In the case of Theorem 9, the crucial property of Gaussian measure is captured by Sidak's Lemma. A symmetric slab in R n is a set of the form {x 6 R n" I(x, u)l ~< t} for some vector u and positive number t. LEMMA 2 (Sidak). If y is standard Gaussian measure on R n and (Si) is a sequence of symmetric slabs then

z(nsi)

>~l--I z(si).

Sidak's Lemma (and more especially its proof) solves a special case of a well-known and rather intriguing problem: the correlation problem for Gaussian measure. QUESTION 1. Is it true that if y is standard Gaussian measure on R n and K and L are symmetric convex bodies in R n, then

y ( K A L) >~y ( K ) y ( L ) ? For a discussion of this problem see, e.g., [36]. In his article on the central sections of the cube, Hensley not only showed that the volumes of the 1-codimensional sections are at least 1, but also that they are at most 5, independent of the dimension. At first sight this seems very surprising; but in a second article [25] Hensley showed that all convex bodies have a similar property; each convex body (after an appropriate linear map), has 1-codimensional sections with "almost constant"

Convex geometry and functional analysis

177

volume. This fact is of considerable importance in a number of situations. Hensley also asked for the sharp upper bound for the volumes of 1-codimensional sections of the cube, conjecturing the value x/2. The present author proved this in [1 ], and in a later paper [2] found the optimal upper bound for k-codimensional sections, (x/2) k . These results will not be proved here, but in the short Section 2.2, a general upper bound for volumes of sections of the cube will be deduced from the Brascamp-Lieb inequality of the last chapter. An unexpected byproduct of the upper bound, x/2, for 1-codimensional sections of the cube, was a "concrete" solution of the so-called Busemann-Petty problem. In [13], these authors asked the following question. QUESTION 2. Suppose that K and L are symmetric convex bodies in some Euclidean space with the property that, for each 1-codimensional subspace H, vol(H n K) ~< vol(H n L). Does it follow that vol(K) ~< vol(L)? At the time, there was a widespread belief that the answer should be yes: among other things, it was known that if two such bodies have sections of equal volume, then they are the same body. So it came as something of a surprise, when Larman and Rogers [29] constructed a random symmetric perturbation of the Euclidean ball in R 12, whose slices all had smaller volume than those of a ball of equal volume. However, with the knowledge that the unit cube has sections of volume at most x/2, the problem loses some if its mystique. When the dimension is large, the Euclidean ball of volume 1 has 1-codimensional sections of volume about x/-e. Thus, for large enough dimension, a cube and an Euclidean ball of slightly smaller volume, provide a counterexample for the Busemann-Petty problem. Following this observation, Giannopoulos [ 19] pointed out that, in fact, there are extremely simple concrete counterexamples, in dimensions as low as 7. It is now known that the Busemann-Petty problem has a negative answer if and only if the dimension is at least 5. The positive answer in dimension 3 was provided by Gardner [ 16] and that in dimension 4, by Zhang [39]. Recently Koldobsky [28] has developed a new approach to the problem which greatly simplifies and clarifies matters. This led to a definitive unified solution to the problem in all dimensions, which will appear in [ 18]. (The topic was also dealt with at some length, in the book by Gardner [17].) This chapter is organised as follows. Vaaler's Theorem, Sidak's Lemma and Gluskin's Theorem are proved in the first section. The second section gives a brief account of some upper bounds for volumes of sections of cubes.

2.1. Vaaler's Theorem and its relatives As explained in the introduction all the results in this section depend upon comparisons between Lebesgue and Gaussian measures. The comparison method was studied in detail by Kanter [27]. It begins with the following definition.

178

K. Ball

If # and v are two probability measures on some Euclidean space R n, say that # is more peaked than v if for every symmetric convex set K in R n ,

# ( K ) >~ v(K). In order to prove Vaaler's Theorem, the aim will be to show that Lebesgue measure on the cube is more peaked than an appropriate Gaussian measure. To begin with, l e t / z be Lebesgue measure on the interval [- 89 89 and let v be the unique Gaussian probability on R whose density

f ( t ) - - e -Jrt2 satisfies f (0) = 1. Clearly in this case, # is more peaked than v. Kanter's basic lemma, shows that peaking-order is preserved under the formation of product measures, at least in the presence of some restriction. This restriction involves a notion of unimodality: several such notions have been considered. The most versatile seems to be the one used here, which was introduced by Barthe [8]. A probability measure on R n will be called unimodal if it has a density f , which can be expressed as the increasing limit of a sequence of functions, each of which is a positively weighted sum

~jpilKi of characteristic functions, 1 gi, of symmetric convex sets. Clearly, Lebesgue measure on the cube and Gaussian measures are unimodal. LEMMA 3 (Kanter). If lZl and 1~2 are probabilities on R n, with #1 being more peaked than #2, and v is a unimodal probability on R m, then lZ l | v is more peaked than lz2 Q V. PROOF. The problem is to show that for every symmetric convex K in R n+m ,

# l | v ( K ) >~# 2 (~ v(K). By the unimodality of v, it may be assumed that the density of v is the characteristic function of a symmetric convex set C. If K is the intersection of K with the cylinder R n x C, then for any probability # on R n,

m

m

It is a standard consequence of the Brunn-Minkowski inequality that the function g, defined on R n by

g(x) -- fRm 1 ~-(X, y) dy


179

has a concave logarithm. Also, g is an even function, because the set K is symmetric. From this it follows that g can be approximated by positive combinations of the characteristic functions of symmetric convex sets in R n. The fact that #l is more peaked than #2 now ensures that

fR n g(x) d#l (x) ~> fR n g(x) d#2(x), as required.

D

By applying Kanter's Lemma repeatedly it is easy to see that if # and v are the uniform and Gaussian probabilities on R, described above, then #|174174 is more peaked on R n than

v|174174 From this one gets Vaaler's Theorem as follows. PROOF. The task is to show that if H is a k-dimensional subspace of R n and Q is the unit cube [- 89 89 ]n then the k-dimensional volume of H n Q is at least 1. As explained above, if K is any symmetric convex body in R n, then K n Q has volume at least as large as Ke Trlxl2dx. By approximating the subspace H by very thin convex "tubes", one obtains that the k-dimensional volume of H n Q is at least the integral over H (with respect to k-dimensional measure) of e -Jrlxl2", and this is 1. [3 The reader is invited to check that the two other formulations of Vaaler's Theorem, described in the introduction to this chapter, follow from the statement just proved. An application of Kanter's Lemma, to the volumes of sections of g np balls other than cubes appears in the article of Meyer and Pajor [33]. It was mentioned in the introduction, that the proof of Sidak's Lemma is closely related to the Gaussian correlation problem. Sidak's Lemma says that the Gaussian measure of an intersection of symmetric slabs, is at least the product of the Gaussian measures of the slabs. It is clear that this statement can be deduced by induction from the following lemma. LEMMA 4. If K is a symmetric convex body in R n, S is a symmetric slab and y is standard Gaussian measure on R n, then

•

n S) >1•215

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K. Ball

It was pointed out by A. Giannopoulos [20] that this lemma can readily be recast so as to fit into the framework described earlier. Define a probability/z on R n by #(A)

=

y(A AS) •

The lemma states that # is more peaked than 9/. By writing R n as the direct sum of the 1-dimensional space spanned by the vector perpendicular to the slab S and the (n - 1)dimensional space parallel to S, the measure # can be regarded as the product of an (n - 1)dimensional Gaussian measure and a 1-dimensional measure. The latter has a density of the form s w+ (constant) e - s 2 / 2

l[-t,t](s),

obtained by restricting the Gaussian density to a symmetric interval, and then renormalising. Plainly, this 1-dimensional probability is more peaked than the standard Gaussian. Therefore, by Kanter's Lemma, # is more peaked than y. Sidak's Lemma gives Gluskin's Theorem in the following way. k -1/2 and let PROOF. Assume that for each k, v~ has norm at most (4 log(1 4- n))

K = {x" I(x, vk)l (v"2-~-~)n VI(1

--e -21~

--(%/~)n

k

Vi(1 _ k

1) k)2

(1+ n

"

The product can easily be estimated since

Z

' k)2 ) ~>

log (1 -

(1+~

(

log 1 -

1 x

) 2 ) dx -- - 2n log 2.

D

(1+~

REMARK. It is clear that the preceding argument can be adapted to show that if the vk satisfy the stronger estimate

Ivkl

v/log(1 + k)

then the Gaussian measure of the intersection of the corresponding slabs is at least 1/2 (not just 1/2n). This implies that if I[. II is the norm whose unit ball is the intersection of these slabs, then the random variable II. II on the probability space (R n , V) has a median which is at most 1. In fact, it is not too hard to show that under this assumption on Irk l, even the mean

fR" Ilxlldy(x) is at most a constant. Using his remarkable majorising measure theorem, [37] Talagrand has shown that the "converse" is true: if the mean of ]].[[ is at most 1, then the unit ball contains a convex body which is the intersection of slabs, (Sk), whose widths are at least 3v/log(1 + k).

2.2. Upper bounds for the volumes of sections of cubes The purpose of this section is to explain how the Brascamp-Lieb inequality, quickly provides upper estimates which are complementary to those of the previous section. In fact, the argument here is really no more than a disguised form of the volume ratio estimate for symmetric bodies; but the statement looks a bit different. THEOREM 10 (Sections of the cube). Every k-dimensional section of an n-dimensional unit cube has volume at most kj2

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K. Ball

The estimate is sharp whenever k divides n; but even in the cases where it is not sharp, the estimate has roughly the right "shape". As mentioned above, the alternative estimate (~/-~)n-k is known, and this is sharp whenever k >~ n/2. PROOF. Let Q be the unit cube, [ - l , 89 ]n. Let H be a k-dimensional subspace of R n and let P be the orthogonal projection from R n onto H. For each i, 1 ~< i ~< n, let Vi m P e i ,

where el, e2 . . . . . en is the standard basis of R n, and let [2.

Ci - - [Vi

Since P acts as the identity I/4, on H, Citti (~ Ui

I H m Z V i @ V i : ~ i

i

where the U i are unit vectors. If x 6 H, then x 6 Q if and only if, I(x, vi)[ ~ 89for every i. In terms of the u i, H A Q is the set

{

1}

x 6 H" I(x, ui)l ~ 2x/-6-/

9

An application of Theorem 2 in the space H gives,

vol(n['~a)~H(~i) 1

(5)

Since ~ Ci k, the convexity of the function c w-~ c log c shows that the right-hand side of (5) is maximised when all Ci are equal: and hence equal to k / n . [] :

3. Plank problems In the early 1930's, Tarski asked the following question (in connection with the BanachTarski paradox). Let us call the region between two parallel hyperplanes in R n , a plank. QUESTION 3. If a convex body of minimum width 1, is covered by a union of planks, must the widths of the planks add up to at least 1? Clearly it is possible to cover with planks of total width 1, by aligning them perpendicular to the direction in which the body has minimum width. Tarski himself showed that the answer to the question is yes, if the body is an Euclidean ball in 2 or 3 dimensions. During the 40s, the problem gained a certain notoriety. It was


183

finally solved in the affirmative by T. Bang in 1951 [6]. The basic lemma used by Bang, is proved below. Though apparently simple, this lemma turns out to be the starting point for several surprising developments. At the end of his paper, Bang asked a further question, which is a good deal more natural than Tarski's original one, since it is affine invariant. Given a convex body K, the relative width of a plank, S, is the width of S, divided by the width of K in the direction perpendicular to S. Bang's affine plank problem asks:

QUESTION4. If a convex body is covered by a union of planks, must the relative widths of the planks add up to at least 1? Plainly, an affirmative answer to Bang's question, strengthens Bang's result. The general case of this affine plank problem is still open, but the most important case, in which the body is assumed to be centrally symmetric, was proved in [3]. If K is such a body, then it may be regarded as the unit ball of some finite-dimensional normed space. A plank in a normed space is a set of the form {x c R ~" lOS(x)- ml llOi f o r every i.

This formulation of the theorem makes it clear that the result simultaneously extends the Hahn-Banach Theorem and sharpens the Uniform Boundedness Principle. It also suggests a similarity between the plank problem and the "Coefficient Problem" in harmonic analysis. For any sequence of numbers (Ck) satisfying Ick[ 2 < ~ , the function t w+ ~

Ck e ikt

(6)

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K. Ball

is square-integrable on the circle. For an integrable function f on the circle, let f ( k ) denote the kth Fourier coefficient of f . The de Leeuw-Kahane-Katznelson Theorem [ 15] states that for any sequence (Ck) of positive numbers satisfying (6), there is a bounded function f on the circle for which If(k)l > ck for every k. Thus, it is impossible to distinguish bounded functions, from other squareintegrable functions, just by looking at the sizes of their Fourier coefficients. It is natural to wonder just what conditions on a sequence, (~pk), of functions will ensure that for any square-summable sequence (Ck) of coefficients, there is a bounded function f for which I(f, ~P~)l > c~ for every k. The de Leeuw-Kahane-Katznelson Theorem is proved using modifications of random functions such as

Z ekCkgrk, where (ek) is a sequence of independent choices of sign. The method requires at least some control of the functions ~Pk in a norm "bigger than 2": a uniform estimate on their Lz+e norms, for example. More crucially, it depends heavily upon the orthogonality of the characters. The method, therefore, leaves open several natural questions; for example, what happens when the 7tk are the trigonometric characters, but it is required that the function f be supported on some restricted part of the circle? The situation was resolved in a rather startling way by the following theorem of Nazarov [34]. (All functions and coefficients are, from now on, assumed to be real, since this results in no loss of generality.) THEOREM 12 (Nazarov). Let (Tzk) be a sequence of unit functions in L1 (of a measure space) which satisfy an upper-2-estimate: that is, there is a constant M so that

~ tOi

f o r every i.

PROOF. The following argument deals with the special case in which all the mi are 0. The general case is only marginally more complicated: see, e.g., [3] for a proof. Choose the sequence of signs for which the vector x is longest. Now, suppose k is between 1 and n and let y

-

-

~EiWiUi iCk

so that x -- y + ek w~uk. By the choice of e~, x has norm at least as large as that of y ek wk uk. This means that

e~:(y, u~:) >~O. This in turn guarantees that the number

(x, uk) = (y, uk) + ekwk = ek(ek(y, uk) + w~:) is at least w~ in size. This is what was wanted.

D

3.2. The affine plank theorem The aim in this section is to prove Theorem 11, at least in the special case in which all m i are zero. The argument in the general case is no different; it merely uses the more general case of Bang's lemma.

186

K. Ball

Most of the work goes into solving the problem on a finite-dimensional space. In this case it may be assumed that there are only finitely many functionals (0i)~, and it suffices to prove that if

Zwi = l , then there is a point x in the unit ball for which Iqbi(x)l ~ Wi for each i. By slicing the planks into thin "sheets" it may also be assumed that all the tOi are the same, i.e., that each is equal to 1In. For each i, let xi be a vector of norm 1 for which

~i(Xi) = 1. Let A

=

(aij) be the matrix given by aij -- q~i (x j)

so that each diagonal of A is 1. It suffices to find a sequence (~.j) for which ~ I)~j[ ~< 1 and 1

~j aij )~j / > -

n

for each i; for, given such a sequence, the vector

X = Z ~.jXj, J has the required properties. The main task will therefore be to prove the following combinatorial statement: THEOREM 13. Let A = (aij ) be an n x n real matrix with 1 's on the diagonal. Then there is a sequence (~.j ) f o r which Z

I)~jl ~< 1,

(7)

but

ZJ aij )~j f o r each i.

>>-1

n

Convex geometry andfunctional analysis The argument below will guarantee rather more; namely, that the sequence 2 1 ~j ~< - .

187

()~j)

satisfies (8)

n

This implies (7) by Cauchy-Schwarz. The advantage of aiming for (8) is that one can apply Hilbert space methods to modify the matrix A. From now on, a matrix will be called positive, if it is symmetric and positive semi-definite. The modification of a matrix needed here, is described in the following lemma. LEMMA 6. Let A be an n x n matrix, o f which each row contains a non-zero entry. Then there is a diagonal matrix 69, with non-negative entries, and an orthogonal matrix U, so that the matrix H--OAU is positive and has all diagonal entries equal to 1.

Lemma 6 can be proved, either from Brouwer's Fixed Point Theorem or, more directly, by a variational argument: see [3]. The entries in the diagonal matrix, 69, whose existence is guaranteed by this lemma, are not easily expressed in terms of the original matrix A. For the present purpose it is necessary to understand how large these entries can be. The next lemma will provide an estimate for them. LEMMA 7. Let H be a positive matrix with 1 's on the diagonal and U an orthogonal matrix. Then the diagonal entries o f the product H U satisfy

~-~(HU)2i ck for each j. Theorem 12 reduces to Theorem 15 via the following weak form of Grothendieck's inequality (see [26, Section 10]). THEOREM 16. If (~k) is a sequence of unit vectors in L 1 (of a measure space (S'2, .T, #)) which satisfy an upper-2-estimate (in L1) then there is a non-negative function ~ for which f ~bd/z-- 1 and for which the sequence (~k/qb) satisfies an upper-2-estimate in L2 of the probability space whose measure is dpdlz. The details of the reduction are left to the reader. The proof of Theorem 15 follows. PROOF. Consider the functions of the form oo

fe -- Z 6jCj~ltj 1

for all possible choices of sign e L1. Moreover, the map

-

(6j).

Each of these sums converges in L2 and hence in

~f~ is continuous from the compact space {-1, 1}o~into L1. The functions fe certainly need not be bounded: but the functions ge given by g~ -- tan- 1 fe are bounded. The aim will be to show that for an appropriate choice of e, and every k,

I ac~, (or being a constant depending only upon M).


191

Define s : R --+ R by

fo x tan -1 t dt.

s ( x ) --

s is Lipschitz, with constant re/2. This ensures continuity of the function S : L 1 ~ R given by S(f) = Es(f).

There is, therefore, a choice of signs s = (8j) for which S ( f s ) is maximum. By altering the signs of the 7tj, it may be assumed that the maximum occurs for e:(1,1,1

. . . . ).

Use f to denote the function f(1,1,1 ....) integer k, take

A

-- ~

--

CjI[rj and g to denote tan -1 f . For a fixed

E

cj ~ j - ck 7tk -- f - 2Ck ftk.

j~k Then, by the maximality of the choice of signs,

(lO)

0 aij for all i and j. Nazarov's Theorem and Bang's Lemma show, respectively, that if X is either L1 or L2, and (xi) is a sequence of unit vectors in X, satisfying an upper-2-estimate, then for any square-summable sequence (wi), there is a functional 4~ in X* for which I~b(xi)l > tOi for every i. The obvious question is "For which X is this true?" A natural guess would be "Spaces of cotype 2". The adventurous might like to conjecture that a still more general statement is true; a statement that also includes a version of Theorem 11 (up to some constant).


193

References [1] K.M. Ball, Cube slicing in R n , Proc. Amer. Math. Soc. 97 (1986), 465-473. [2] K.M. Ball, Volumes of sections of cubes and related problems, Israel Seminar on G.A.EA., 1987-1988, Lecture Notes in Math. 1376, Lindenstrauss and Milman, eds, Springer-Verlag (1989), 251-260. [3] K.M. Ball, The plank problem for symmetric bodies, Invent. Math. 104 (1991), 535-543. [4] K.M. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), 351-359. [5] K.M. Ball, A lower bound for the optimal density of lattice packings, Duke Math. J. 68 (1992), 217-221. [6] T. Bang, A solution ofthe "Plank problem", Proc. Amer. Math. Soc. 2 (1951), 990-993. [7] I. Barany and Z. Ftiredi, Computing the volume is difficult, Discrete Comput. Geom. 2 (1987), 319-326. n C.R. Acad. Scis. Paris 321 (1995), 865-868. [8] E Barthe, Mesures unimodales et sections des boules B p, [9] F. Barthe, Indgalitds de Brascamp-Lieb et convexitd, C. R. Acad. Scis. Paris 324 (1997), 885-888. [10] H.J. Brascamp and E.H. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions, Adv. Math. 20 (1976), 151-173. [11] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417. [12] J. Bourgain, J. Lindenstrauss and V. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141. [13] H. Busemann and C. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94. [14] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (1988), 479-504. [ 15] K. de Leeuw, J.P. Kahane and Y. Katznelson, Sur les coefficients de Fourier des fonctions continues, C. R. Acad. Sci. Paris 285 (1977), 1001-1003. [16] R.J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. 140 (1994), 435-447. [ 17] R.J. Gardner, Geometric Tomography, Cambridge University Press, New York (1995). [ 18] R.J. Gardner, A. Koldobsky and Th. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. 149 (1999), 691-703 [19] A. Giannopoulos, A note on a problem of H. Busemann and C.M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239-244. [20] A. Giannopoulos, Private communication. [21] A.A. Giannopoulos and V.D. Milman, Euclidean structure in finite dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707-779. [22] A. Giannopoulos and M. Papadimitrakis, Isotropic surface area measures, Mathematika, to appear. [23] E. Gluskin, Extremal properties of rectangular parallelepipeds and their applications to the geometry of Banach spaces, Mat. Sb. (N. S.) 136 (1988), 85-95. [24] D. Hensley, Slicing the cube in R n and probability, Proc. Amer. Math. Soc. 73 (1979), 95-100. [25] D. Hensley, Slicing convex bodies-bounds for slice area in terms of the body's covariance, Proc. Amer. Math. Soc. 79 (1980), 619-625. [26] W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1-84. [27] M. Kanter, Unimodality and dominance for symmetric random vectors, Trans. Amer. Math. Soc. 229 (1977), 65-85. [28] A. Koldobsky, Intersection bodies, positive definite distributions and the Busemann-Petty problem, Amer. J. Math. 120 (1998), 827-840 [29] D. Larman and C.A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164-175. [30] E.H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179-208. [31] E Lust-Piquard, On the coefficient problem: a version of the Kahane-Katznelson-DeLeeuw Theorem for spaces of matrices, J. Funct. Anal. 149 (1997), 352-376. [32] R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), 309-323.

194

K. Ball

[33] M. Meyer and A. Pajor, Sections ofthe unit ball of~np, J. Funct. Anal. 80 (1988), 109-123. [34] E Nazarov, The Bang solution of the coefficient problem, Algebra i Analiz 9 (1997), 272-287. [35] C. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824-828. [36] G. Schechtman, Th. Schlumprecht and J. Zinn, On the Gaussian measure of the intersection of symmetric convex sets, Ann. Probab. (1998), 346-357. [37] M. Talagrand, Regularity of Gaussian processes, Acta Math. 159 (1987), 99-149. [38] J.D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543-553. [39] G. Zhang, A positive solution to the Busemann-Petty problem in R 4, Ann. Math. (1999).

CHAPTER

5

A p-sets in Analysis: Results, Problems and Related Aspects Jean Bourgain School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA E-mail: bourgain @math. ias. edu

Contents 1. 2. 3. 4. 5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence and construction of A p - s e t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Kolmogorov rearrangement problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lattice points on spheres; Quantum limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Restriction of the Fourier transform to spheres and the Hausdorff dimension of Besicovitch sets 6.1. Further comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. On the distribution of Dirichlet sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF T H E G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 195

....

197 199 204 208 213 215 222 223 230


A p-sets in analysis: Results, problems and related aspects

197

1. Introduction

A subset S of the integers Z is called a Ap-set (p ~> 1) provided for some constant C and all scalar sequences (an)n~S we have the inequality

n~S

LP(T)~< C (n~Sia~ 12)1/2

E an e inx

L P (T)

E

a n e inx

nES

2,

a n e inx

if p < 2,

(1.1)

(1.2)

L 1(T)

nES

where qr refers to the circle group. The smallest constant C satisfying (1.1), respectively (1.2), will be naturally called the Ap-constant K p ( S ) of the set S. This notion may obviously be extended to the setting of an arbitrary compact Abelian group. The main purpose of this paper is to bring together a number of items directly related to this concept. Those play an often important role in diverse mathematical areas, including harmonic analysis, of course, but also PDE's and analytic number theory. Without intention to survey these topics in any detail, we will mention some of the basic facts and elaborate a bit more on the related problems relevant to this expos6. Some of these generated in fact active research areas. Thus, we will try to change somewhat its general perception as a perhaps interesting, but certainly narrow subject. One of the major papers on "thin sets" in harmonic analysis, in particular on Ap-sets, is that of W. Rudin [38]. It raises in particular the natural question on the existence of A p sets that are not Aq for any q > p. We will discuss related results and problems in the next section. Probabilistic methods turn out to play an important role here. There have been also some byproducts of the methods developed for the purpose of this research, leading for instance to progress on Kolmogorov's rearrangement problem. On the other hand, there are several instances where one is led to estimate Ap-constants for specific finite sets of integers or real numbers. We mention two of them. (i) A basic tool in the theory of dispersive PDE's are Strichartz' type inequalities. For instance, the Cauchy problem for the nonlinear Schr6dinger equation (NLS) depends essentially on the following inequality for the linear Schr6dinger group (eitA)t~R 2(d + 2)

(1.3)

where A denotes the usual Laplacian. Considering the corresponding problem on bounded domains, for instance with periodic boundary conditions, the expression (eitAr ( x ) -

f ~"(se)ei<xse+t~2)

dse

(1.4)

is replaced by an exponential sum of the form

E an e i(nx+lnl2t)

nEZ d

(1.5)

J. Bourgain

198

considered as a function on the (d + 1)-torus q[,d+1. In order to obtain an analogue of (1.3), the issue becomes clearly to evaluate

Kp({(n, It/] 2) E ~d+l ] ]r/] ~ N / ) .

(1.6)

This question is largely unresolved, especially in higher dimensions. (ii) One of the major themes in analytic number theory is to evaluate moments of the ~'-function and more generally Dirichlet series and sums of the form

E annit' n,,~N

(1.7)

where t is restricted to some interval [0, T]. The problem of bounding expressions

fo T

P

ann it dt n,~,N

E

(1.8)

in various ranges of N and T has been extensively studied. Estimates for (1.6) with p an even integer play an important role in zero-density estimates of the ~'-function and L-functions (cf. [32]). On the other hand, several most significant conjectures (known as Montgomery's conjectures) remain essentially unresolved. Some of these conjectures, if true, would also have interesting implications in geometric measure theory, such as maximal Hausdorff dimension of any Besicovitch set in IKd, presently only known in dimension d = 2. Recall that a Besicovitch set in •d is a measurable subset ,4 of ]Kd containing a line or line segment in every direction. A well-known construction in the plane (going back to Besicovitch) shows that such sets may be of zero measure. But it seems reasonable to conjecture that they always have maximal dimension, thus dim,/[ = d .

(1.9)

This problem is also of primary importance in Fourier Analysis related to restriction theory of the Fourier transform to surfaces. What we have in mind here are improvements of the well-known Tomas-Stein theorem for the sphere S d-l, do"

I1~11 L 2(~+,~ d-1 (~d)

~< c

d#

L2(Sd-l)

(1.1o)

when one makes the stronger assumption of a measure # on S d-1 with bounded density dlz/da rather than assuming d#/da E LZ(a). As mentioned, there is an amount of recent research and results on the subject and part of it will be described briefly here. We made up the reference list strictly for the purpose of this short expos6 and even as such there are surely serious omissions. It should certainly not be viewed as a measure for the significance of the work of the various contributors to the subject.


199

2. Existence and construction of Ap-sets Consider first the case p > 2. The methods involved to produce Ap-sets are either of combinatorial (when p is an even integer) or probabilistic nature. THEOREM 2.1. For every p > 2, there exists a subset S ~ Z which is a A p-set and not a

Aq-set for any q > p. Denote

S N - - S A [0, N]. One has

Kp(S)[SN[ 1/2 >/

einX

~]SNI N-1/p,

(2.1)

hence

[SNI ~ Kp(S)2N 2/p.

(2.2)

Thus the intersection of a Ap-set S with [0, N] (hence also any arithmetic progression of length N) has cardinality at most C N 2/p. PROPOSITION 2.2. Fix p > 2. Then for any integer N > O, there exists a subset S C

{ 1 . . . . . N} satisfying I S [ - [N2/p],

(2.3)

Kp(S) < Cp.

(2.4)

In order to derive Theorem 2.1 from Proposition 2.2, recall the Littlewood-Paley theorem (2.5) for 1 1 2/(-1~ 2. Our understanding of Ap-sets for p < 2 is much less satisfactory. The following result is due to Bachelis and Ebenstein [3] based on a theorem of Rosenthal [37] as main ingredient. THEOREM 2.7. Let S C Z. Then {p E ]1, 2 [ I S is a Ap-set}

is an open interval. Thus, Theorem 2.7 exhibits a different behaviour from the p > 2 case. Remark that for p > 2, the interval {p E ]2, c~[I S is a Ap-set } may or may not contain its endpoint. On the other hand, the only known examples of Ap-sets for p < 2 are Aq-sets for some q > 2. It may be verified that for a random set SN C {1 . . . . . N} of size 6(N)N with density 6(N) satisfying lim log N-+cx~

N~ log 6 ( N )

-1

--

~

(2.32)

one gets for all p > 1 lim N--+~

Kp(SN)

-

cx~.

(2.33)

In fact, for p < 2, probabilistic techniques alone do not seem to suffice to produce nontrivial examples.

3. Further comments (1) Theorems 2.1 and 2.7 from the previous section remain valid if q[' is replaced by any (infinite) compact Abelian group.


205

(2) B2-sets are sometimes also called Sidon sets. Thus a set S C Z is a Sidon sequence if all the sums a + b, a, b E S are distinct. Remark that the name "Sidon set" is also attached to another related but different concept that will be discussed later. As mentioned, Sidon (= B2) sets are A4-sets. Although it is easy (as explained earlier) to construct an infinite A4-set S satisfying

I S N I - IS A [1,

N] I >

N 1/2

(3.1)

for all N

the situation for Sidon sequences is different. Erd6s showed that if S is an infinite Sidon set, then lim ISNl~/log N N

~/N

~< 1.

(3.2)

In the other direction, the best known result remained for a long time a construction of Ajtai, Komlos and Szemeredi who produced a Sidon sequence satisfying the density property lim ISNI >0. N (N log N) 1/3

(3.3)

This was only recently improved by I. Ruzsa [39], who constructed an example of a Sidon sequence S for which

ISNI > N •176

9 / - ~ / 2 - 1.

(3.4)

The reader is referred to the paper [39] for further references. (3) The other meaning of "Sidon set" is that of a subset S C Z such that for some constant C, the inequality

~la~l ~ C nES

Z an e inx

(3.5)

nES

holds for all sequences {a, In E AS}. Equivalently, any functionAf E C(T) for which the support of the Fourier transform f is contained in S, thus supp f C S, satisfies f -- A (T) ( = the space of absolutely convergent Fourier series). There has been an extensive study of the structure of those sets over the past years. Answering a problem raised by Rudin in [38], Pisier showed that S C Z is a Sidon set r S is a Ap-set for all p < cx~ and sup

Kp(S)

p>2 v/P

< c~.

In particular, finite unions of Sidon sets are still Sidon sets (Drury's theorem).

(3.6)

J. Bourgain

206

Call a subset S C Z "independent" if any relation elnl

nt-e2n2-+-...nt-ehnh--O

(es -- 0,1, - 1 )

(3.7)

between distinct elements of S is necessarily trivial, i.e. (3.7) =r

e= = 0

(1 ~< s ~< h).

(3.8)

Independent sets are Sidon sets and the interpolating measures are produced by Rieszproducts H(1 +

bncosnx)

( - 1 ~< bn

>.4 and p ~ 2(d+l) d - 3

"

For d ~> 3, no estimate of the form (5.5) for some p > 2 seems to be known 9 Next, we discuss the notion of a quantum limit. These are the weak*-limits of sequences of m e a s u r e s ~ j on ,-~d of the form d#j dx

- Igoj12,

(5.7)

where the qgj are L2-normalized eigenfunctions of eigenvalue ~,j with limj_+oc ~,j -----OO. For details on the following results, the reader should consult [28]. PROPOSITION 5.3. The density of any quantum limit v on 7~2 is a trigonometric polynomial whose Fourier transform is supported by at most 2 circles centered at the origin. PROPOSITION 5.4. For 3 0, define G p (S) as the graph with S A Z d as vertex set and connecting 2 points provided they are at distance < p. PROPOSITION 5.6 (Connes [16]). There is a function gr(R), l i m n ~ gr(R) --cx~, such that if S is an r-dimensional sphere in R d of radius R, then for O < p < gr(R), the connected components of Gp(S) are contained in one of the ( r - 1)-spheres I21 . . . . . f2m formed by intersection of S with a subspace of R d of dimension r. Proposition 5.6 as well as Proposition 5.1 have been motivated by problems on uniqueness of trigonometric series. More precisely, the uniqueness question for spherical summation: Let )-~ czJ a~ e ix~ be a trigonometric series such that

Z

a~ e ix~ R ~ >

0

pointwise.

(5.9)

I~I~ 2 is a more recent result due to the author. The starting point is the following fact (due to Connes) and which is a consequence of Proposition 5.6. PROPOSITION 5.7. Let qgj be a sequence of eigenfunctions of the Laplacian on 7fd such that l i m j ~ q g j ( x ) - 0 pointwise for x in a nonempty open subset of qFcl (we do not assume any normalization). Then lim IIgoj112 -

j--+ cx:~

o.

(5.10)

The reader interested in details and references on uniqueness problems for summation of trigonometric series may consult [ 1,2] which is a recent survey on the subject.

6. Restriction of the Fourier transform to spheres and the Hausdorff dimension of Besicovitch sets The first result we mention is the Tomas-Stein theorem (see [45,40]), which plays an important role in harmonic analysis.

216

J. Bourgain

PROPOSITION 6.1. Let S d-l, o" be the unit sphere with its surface measure or. Let lZ M (S) be a measure such that /z 1

for k r k'.

(6.2)

Thus

K ~ 4.

(6.26) A

PROPOSITION 6.5. There is the following distributional inequality, assuming suppq~ C B(0, N), ll4'll2 ~< 1 mes[(x, t) ~ Td+l I ]eitZx~] > X] N d/4.

The proof of Proposition 6.4 is based on simple arithmetic considerations. Proposition 6.5 is related to Proposition 5.2. Both are an application of the circle method technique. Applications of Propositions 6.4 and 6.5 to the Cauchy problem (6.17) for NLS with periodic boundary conditions, thus 4~ ~ HS (Td), are discussed in [11 ]. See also [23]. Next, coming back to Propositions 6.1 and 6.2, we discuss Stein's restriction conjecture. The conjecture is that A

IIf IxllLr(o)~ CIIflILP(Rd)

(6.29)

provided p
1. r p

(6.30)

Thus Proposition 6.2 corresponds to the case r - 2, p ~< 2(d + 1)/(d + 3). Observe that if one disposes of the inequality A

(6.31)

Ilf ]SilL'(o-) ~< Cllfllp, then also the following one holds A

I]flsllLr(,~)~Cllfllp

if r < p .

(6.32)

J. Bourgain

220

Indeed, (6.31) corresponds to boundedness of the map

T" L p (IRd) --~ L ' (o-) " f ~ f ' l s

(6.33)

hence

T* " L cx~(o-)____~ L pt (IRd). Thus, invoking the theory of summing operators, T* is rt-summing, for any r < p. This means that there is a probability measure v on S d- 1 satisfying (6.34)

IIT*gollp, ~ Crllqglltr'(dv).

But because T is rotational invariant, a standard averaging argument permits us to replace dv by do-. Hence we get (6.32). Consequently, Stein's conjecture essentially amounts to the inequality ~IIf IsIItlfo~ ~< Cpllfllp

2d for p < d + 1"

(6.35)

(This argument is only applicable for the sphere.) What is known? For d = 2, the restriction conjecture is correct and due to Stein. For d ~> 3, the problem is open and much harder. The truth of inequality (6.35) has deep consequences in geometric measure theory, which we explain next. DEFINITION 6.6. A measurable subset .,4 of R d is called a Besicovitch set provided .,4 contains a line or line segment in every direction. An old construction (see [18]), going back to Besicovitch, permits us to produce a measure-zero Besicovitch set in the plane (d -- 2). Then, straightforward product techniques show that there exist Besicovitch sets of measure-zero in any dimension d ~> 2. The following conjecture is verified for d = 2 but open for d ~> 3. It can be shown to be a formal consequence of (6.35) (see [9]). CONJECTURE 6.7. A Besicovitch set in R d has always full Hausdorff dimension. Thus dim,,4 = d.

(6.36)

REMARK. As we will see in the next section, there is another problem on the distribution of Dirichlet sums, known as Montgomery's conjecture, that implies Conjecture 6.7 equally well. It should be mentioned that the role of Besicovitch sets in harmonic analysis became first apparent from C. Fefferman's work on the ball multiplier. He proved the following: For d ~> 2, the map f --+ I

ff(~)e ix~ d~ I~ 3) was obtained in [9]. More precisely, for d = 3, (6.35) was established for p < 58/43. The first step consists in obtaining certain "nontrivial" geometric information related to Conjecture 6.7. We introduce the following maximal function f~*, for given 6 > 0 (which we refer to as "Kakeya maximal function"). Let f be a function on R d. Define for ~ E S d - l

f~*(~)

-

supWi Ifo ~

(6.42)

which is still the best result for Hausdorff dimension at the time of this writing. Very recently, the author showed that for a Besicovitch set A in dimension R d, one has 13 12 d i m A >> ~-~ d -+- 2--5 - e

(6.43)

which improves on (6.41) for large d. The argument is mainly based on T. Gowers' proof (contained in the paper [24]) of the Balog-Szemeredi theorem mentioned in Section 3. Some further progress in this direction, appears in the recent papers [30,29]. About conjecture (6.35) on restrictions, the best result in d = 3 at this point is due to Tao, Vargas and Vega (see [44] and forthcoming paper). PROPOSITION 6.1 1. Inequality (6.35) holdswith d - 3for p < 26/19.

6.1. Further comments (1) Related to the restriction conjecture is another problem known as the Bochner-Riesz summation conjecture in harmonic analysis. We first introduce the Bochner-Riesz multiplier m~, 0 ~< )v ~< (d - 1)/2,

m)~(~)

[ (1 -l~12) )~

/0

if I~1 ~

1,

(6.44)

ifl~l > 1.

Thus

cos Ix l

~(x)

(6.45)

ix ld-@+~

CONJECTURE 6.12. mx is a bounded Fouriermultiplier on LP(•d), p 7~ 2, i.e.

f f"(~)m x (~) e ix'$ d~

Cp,)~llfllp

(6.46)

p

if and only if (6.47)

)v>0

and 2d d-k- 1 +2)~

2d

n . 15

(6.50)

The result has been subsequently subject to further small improvements. (2) The reader may wish to consult [ 12] for a survey on oscillatory integral operators and related aspects in geometric measure theory, including in particular the topics discussed here. Most of these results have been improved later however. (3) For a recent survey on results and problems related to Besicovitch sets, see [47].

7. On the distribution of Dirichlet sums

A Dirichlet series is an expression of the form

•

ann it ,

(7.1)

n--1 where the coefficients {an } are apriori arbitrary (thus not necessarily of arithmetic nature). Our purpose here is to mention some results and problems (known as Montgomery's conjectures) on the distribution of Dirichlet sums

2N ~

ann it.

(7.2)

n=N

The most relevant case here is the situation where

2~)1/2 max

lanl

N. Then, with high probability, the following holds

Z IF(0)] 2 < C[N + (log T)Jf]]

(7.31)

06.f A

for any trigonometric polynomial F, supp F C I, of the form (7.30) and any subset ,T o f t whose elements are at least T1-separated, i.e.

1

IO - O'l > -~ for 0 ~ O' in ~ . PROOF (Sketch). Generate again the random set I as I = Ioj = {n c [0, T] n ZiOn(CO) - 1},

(7.32)

J. Bourgain

228

where the (~n) are independent 0-1 valued random variables of the required mean

f ~n(CO)dCO-- 8 = ~-. N

(7.33)

(7.31) is clearly equivalent to

C1N

Z IF(O)[ 2 < Oc~'N[IF[>C2 I~T]

(7.34)

for some constants C1, C2. Choosing C2 appropriately one may moreover assume in this statement that N I~1 < ~ . logT

(7.35)

Thus we need to estimate

f sup[o~7~a"~',.>O a nondecreasing sequence of sub-a-algebras of .T. A sequence (dn)n>~O of (Bochner) integrable B-valued functions is a martingale difference sequence relative to the filtration (3Cn)n>>Oif for all n >~ 0, the function dn is fin-measurable and E(dn+l ].Tn) - - 0 where the latter condition is simply that

fa

dn+l dP =O

forallA6f'n.

(1)

The Banach space B is UMD (unconditional for martingale differences) if tip(B) is finite for some p E (1, ec); equivalently, for all p 6 (1, oc). Here tip(B) denotes the least/3 [1, oc] such that

~-~dk k=0

p

(2)

k=0

for all B-valued martingale difference sequences d, all sequences e of numbers in {1, - 1}, and all nonnegative integers n. In this definition, the filtration (Un)n~>0 must vary, and the probability space must also vary unless it is nonatomic. By (1), if (an)n>~Ois a sequence of scalars, then (andn)n>~O is also a martingale difference sequence. So if B is UMD and the

Martingales and singular integrals in Banach spaces

237

series ~ - - 0 akdk converges in the Lebesgue-Bochner space L~ ($2, ~ , P), then the series converges unconditionally. That the finiteness of tip(B) for some p E (1, cx~) implies its finiteness for all p E (1, cx~) is due to Pisier; see Maurey [ 109]. It is also an immediate consequence of Theorem 1 below. Let fn Y~=0dk and Y~=0ekdk. The martingale g is the transform of the martingale f by e and this • 1-transform satisfies --

gn

--

[Igllp ~ ~p(B)llfllp.

(3)

Here IIf IIp - supn~>0 [Ifn IIp. Maximal functions will also play a role. The maximal function f * of f is defined on S2 by f*(co) = SUPn~>0 IIf~(co)ll. The inequality (3) holds with the same constant tip(B) if, in the definition of g, the number ek is replaced by an U(k- 1)v0measurable function that maps S-2 into the interval [ - 1 , 1]. If H is a real or complex Hilbert space, then tip(H) = p* - 1 for all p E (1, cx~) where p* = p x/q and 1/p + 1/q = 1; see Section 5. This includes R and C. Clearly, tip(B) >/ tip(R). Therefore, for all Banach spaces B,

tip(B)/> p * - 1.

(4)

Other results also follow from the scalar case such as tip (g P) = p* - 1 (integrate term-byterm). Indeed, flp(g~) - tip(B), which implies that if B is a UMD space, then so is ~ for all p E (1, cx~), and similarly for L~(0, 1). The spaces el, ~ , LI(0, 1), and L ~ ( 0 , 1) are not UMD. In fact, UMD spaces are reflexive, even superreflexive; see Remark 4 in Maurey [ 109] and the proof of this in Aldous [ 1]. However, this property does not characterize UMD spaces: there are superreflexive spaces that are not UMD [123]. Since every superreflexive space can be given an equivalent uniformly convex norm [71 ] and, by (2), every Banach space that is isomorphic to a UMD space is also UMD, the geometric property of uniform convexity does not imply UMD. As can be seen above, the classical reflexive Lebesgue spaces are UMD. Other examples include the reflexive Orlicz spaces [87], the reflexive trace-class spaces [87,20], and the reflexive noncommutative L p (M, r)-spaces associated with a v o n Neumann algebra M possessing a faithful, normal, semifinite trace r [ 12]. Is there a geometric property that these and other UMD spaces share that no other Banach space has? A real or complex Banach space B is ~'-convex if there is a biconvex function ~" : B x B R such that ~"(0, 0) > 0 and ~'(x, y) ~< [Ix + y [I if IIx II = Ily II = 1,

(5)

(A function ~" is biconvex on B • B if both ~'(x, .) and ~'(., y) are convex on B for all x, y E B.) The ~'-convexity property characterizes UMD spaces [30,36]. THEOREM 1. A Banach space B is UMD if and only if it is ~-convex. This is proved in [30] with a slightly different but equivalent notion of ~'-convexity. The equivalence is proved in [36].

238

D.L. Burkholder

A biconvex function ( that satisfies (5) must also satisfy ((0, 0) ~< 1" if IIx II = 1, then ( ( x , x ) V (x, - x ) ~> 0. But, by the definition of or, E V (Zoo) ~< 0 for all Z 6 Z(0, 0). This implies that U(0, 0) ~< 0 so U(0, 0) = 0. Let u(x, y) = ~ - U(x, y). If IIx - Yll >~ 2, then u(x, y) 2/u0(0, 0). This completes the proof that fi(B) = 2/u0(0, 0). REMARK 5. If ~" is any biconvex function satisfying (5) and g is a +l-transform of a B-valued martingale f such that P(g* 7> 1 ) = 1, then 211fllJ ~ C(0, 0)

(18)

and the inequality is sharp if ~" = ~'e. The main step in proving (18) is the application of the biconvex version of (12) and (15) to U = ~'. That (18) is sharp follows from a more general version of Lemma 3 in which both the starting point and the ending point of the zigzag martingale Z associated with f and g are taken into account; see Lemma 5.1 in [36]. LEMMA 6. Let 1 a)~ or IId~+l(O))ll > 26;.}.


243

1)n be the indicator function of the set {/z < n ~< v A a }, the empty set if n -- 0, and a set in f n - 1 if n >~ 1 (recall that Ildn+l I[ is .Tn-measurable). Therefore, Fn = Y ~ = 0 vkdk and Gn -- ~ = 0 ek vkdk define martingales F and G such that, by (9),

Let

P ( g * > fl ~. , f * 2, strictly concave. Therefore, 45 is increasing on [0, to] and decreasing on [to, 1] thus proving (5) and completing the proof that a UMD space is (-convex. 71

REMARK 8.

From (22) it follows that (B(0, 0) /> 2/[1 + tiP(B)]. A different method (see Section 6 in [36]) gives (a(0, 0) ~> 1~tip(B). I f B is a Hilbert space and p -- 2, then equality holds in both.

3. Singular integral operators and UMD Singular integral operators arise in many areas of mathematics, for example, in complex analysis, harmonic analysis, partial differential equations, and quasiconformal mappings. See Stein [134,136] and the many references therein for some of the history and much of the theory. Here we study an old problem of interest in many applications: How do such operators behave in a Banach space setting? Inequality (3) gives some information about the possible size of a + 1-transform g of a B-valued martingale f in comparison with the size of f . For example, if tip(B) is infinite, then Ilgllp can be exceedingly large even if ]lf l i p is small. Similar results hold for conjugates of B-valued harmonic functions. Let u and v be B-valued functions harmonic in the open unit disk of the complex plane with v(0) -- 0. Suppose that the Cauchy-Riemann equations, Ux = Vy and Uy = - V x , are satisfied. Let

Ilullpp =

sup

O 0, and the restriction of S-2 to the unit sphere is integrable. Let IIS2 II1 denote the integral of IS2l with respect to surface measure on the unit sphere. Then the limit f T f ( x ) -- l i m / f(x - y)K(y)dy e$O d[y I>e exists almost everywhere if f 6 L~ (R n) for some p satisfying 1 ~< p < ec. If 1 < p < ec, then (25)

IITfllz, ~ loep(B)ll~lll Ilfllp. THEOREM 9. A Banach space B is an H T space if and only if it is UMD.

That a UMD space is HT is due to Burkholder and McConnell; see Burkholder [33] where the proof, designed to be as accessible as possible, is based on the Rademacher functions rather than on the It6 stochastic calculus, the most natural setting. The question is also raised of whether an HT space is UMD. In response, Bourgain [18] proved that, indeed, HT implies UMD. PROOF THAT A U M D

SPACEIS HT.

C~p(B) ~< flZ(B),

This follows from the inequality

1 < p < ec.

(26)

It is enough to show that

f0 rr ]v( ei0 ) I1" dO ~< fi2p (B) f0 2rr ]lu(e i~

II~ dO

(27)

in the special case that the function u is harmonic and v is its conjugate on the entire complex plane with the ranges of both functions included in some finite-dimensional subspace of B. As usual, v(0) = 0. Let Z = (Zt)t>>o be a complex Brownian motion starting at zero with X and Y the real and imaginary parts of Z. The inequality (27) is equivalent to

D.L. Burkholder

246

where r (co) -- inf{t ) 0" IZt (co)l = 1} since Zr is uniformly distributed on the unit circle 9 Because u is harmonic, It6's formula gives

u ( Z r ) -- u ( 0 ) +

f0 Tux(Zt) dXt + uy(Zt) dYt

with a similar formula for v(Zr). The inequality (28) can now be proved with the use of the decoupling method. Let Z ~be another complex Brownian motion starting at zero. Assume as we can that Z and Z ~ are defined on the same probability space and that Z and Z f are independent. Then

II (z )ll

fo r vx(Zt) dXt -+-vy(Zt) dYt

=

/~p(B)

p

fo +v~(Zt)dX~ + vy(Zt)dY t

(29) p

Inequality (29) is the analogue of the decoupling lemma in Burkholder [33] and follows directly from the discussion in Garling [76]. A different proof is given in Burkholder [36]. By the Cauchy-Riemann equations and the fact that (r, X, Y, Y~, - X ~) and (r, X, Y, X ~, Y~) have the same distribution, the norm on the right of (29) is equal to

~0z Ux(Z,) dX~ + .y(Z,) dY/

fo r ux(Zt) dY t + uy(Zt) d(-X~) p

p

Here the norm on the right is less than or equal to

/7

tip(B) u(O) +

f0

ux(Zt) dXt + uy(Zt) dYt

-

and completes the proof of (26). Hence, a UMD space is HT. To prove Bourgain's result that an HT space is UMD, we shall use some of the properties of the periodic Hilbert transform H. Let f " R --+ B be defined by

f (0) -- E cveiV~ 1)

where the spectrum of f , the set {v 6 Z" cv ~ 0}, is finite 9Here, if B is a complex Banach space, then cv 6 B, but if B is a real Banach space, then cv = 89 (a~ - ibm) where av = a - v 6 B and by - - b - v 6 B. So for real Banach spaces, f can also be described as the trigonometric polynomial given by 1 ~ao + E v~>l

(av cos vO + by sin vO).


247

The periodic Hilbert transform is defined for such f by

H f (O) -- - i Z ( s g n v) cve ivO v#o Note that u(re i~ -- Y~v cvrlvleiv~ and v(re i~ = - i y~'v#0(sgn v) cvrlvle iv~ define Bvalued harmonic functions u and v on C that satisfy the C a u c h y - R i e m a n n equations, v(0) - 0 , u(e i0) - f(O), and v(e i0) - Hf(O). Just as for the classical case in which B - R, the integral in the definition of Ilu IIp is nondecreasing as a function of r. Therefore, by the continuity of u and v on the closed unit disk and the definition of Otp (B),

(30)

IIH fllp o on [0, 1) such that F0 = Go -- 0, and for n ~> 1, /7

F, = ~ ~ok(ro. . . . . k--1

rk-1)rk,

Gn - Z k=l

ek~Ok(ro . . . . . rk-1 )rk,

D.L. Burkholder

248

where qg~" {17 - 1 }~ --+ B and ek 6 { 1, - 1 } for all k ~> 1. Then, to prove (31), it will suffice to show that (32)

2 (B) ll Fn llp [IGnllp ~ Olp

The restriction to martingales satisfying F0 = Go - 0 has no effect on tip(B); see, for example, Section 2 of Burkholder [34]. The restriction to dyadic martingales of the above form also has no effect as can be seen from the proof of tip(B) - t o (B) that is given in R e m a r k 7. Let 7r be the even function on R with !k (0) - 1 if cos 0 ~> 0 and ~p(0) - - 1 if cos 0 < 0. Then

F F• F F 7/"

7r

(pk(~(to) . . . . . ~ ( t k - 1 ) ) ~ ( t k )

k=l

dto

dtn

2rr

2Jr

n

iionll

.

.

.

.

7/"

7/"

ek~ok(~(to) . . . . . ~ ( t k - 1 ) ) ~ ( t k )

k=l

dto

dtn

2Jr

2Jr

since (rk)0~ q - 1. So the existence of a concave majorant u of v satisfying (39) implies that fl ~> p* - 1. Therefore, tip(R)/> p* - 1 by L e m m a 12. The argument giving q - 1 as a lower bound can of course be avoided by using duality: tip (R) = flq ( R ) . A natural question arises: Is this lower bound p* - 1 also an upper bound? Equivalently, does there exist a concave majorant u of v satisfying (39) if fl = p* - 1 in the definition of v ? For example, if p = 2, then the identity function has the desired properties. If p > 2, t h e n / 3 -- p - 1 and equality must hold throughout (40) for the desired function u. So u(xo) = 0 and L = R -- u ( 1 ) / ( 1 - x0). This implies that u must be of the form u(x) = ax - b on the interval [x0, 1]. Therefore, by (39), u(x) = a x p - 1 - bx p on the interval [1, l/x0] and L = R implies that b = a(1 - 2 / p ) . Both u and v vanish at x0 and in order for u to be concave and majorize v, it is necessary that u' ( x 0 + ) = v' (x0),

which yields a l p 2 ( 1 - 1/p)p-1. Thus, u is uniquely determined on the interval [x0, l/x0]. Because v is concave on the interval ( - ~ , x 0 ] and also on the interval [ l / x 0 , c~), the desired u can be defined on the complement of Ix0, l/x0] by u(x) = v(x) or in any other way such that the result is a concave majorant of v on R satisfying (39). Therefore, tip(R) ~< p* - 1 for all p ~> 2. The case 1 < p < 2 has a similar proof, as well as being an immediate consequence of duality. So /~p(R) = p* - 1

(41)

and this completes the proof of T h e o r e m 11. REMARK 13. If u is defined by u = U (., 1) where, for all x, y E R,

U(x, y) = ap {

x+y 2

-(p*-

1)

x-y 2

I{

x+y 2

x-y +

2

p-1


253

with Otp = p(1 - 1/p*) p - | , then u is also a concave majorant of v on R satisfying (39) for all p 6 (1, ~ ) . If p > 2, for example, then this function satisfies u ( x ) = ax - b for x E [x0, 1] just as uniqueness on this interval demands.

5. Differential subordination 5.1. A sharp martingale inequality As in Section 2, let B be a real or complex Banach space, f a B-valued martingale, and d its difference sequence" fn - Y~=0 dk for all nonnegative integers n. In this section, we relax the condition on the B-valued martingale g. As before, f and g are martingales relative to a common filtration (U,)n~>0 but here the only additional condition on the difference sequence e of g is that lien (co)II ~< IId~(~)I[

for all co E S2.

(42)

The martingale g is differentially subordinate to f . The condition of Section 2 that en = sndn, where sn E {1,--1}, is a special case in which tip(B) plays a key role. The corresponding quantity in the differentially subordinate case will be denoted by yp (B). So if (42)holds and 1 ?' II f IIp. The following theorem [30,38] compares the behavior of tip(B) and yp(B). THEOREM 14. Let 1 < p < oc. (i) The extended real number yp(B) is finite if and only if B is isomorphic to a Hilbert space. (ii) I f H is a Hilbert space, then /3p (H) = yp (H) = p* - 1.

(44)

For example, if r E (1,2) U (2, ec) and p E (1, ec), then ~p(~r) is finite but yp(gr) is infinite. By (i), this is of course true for every UMD Banach space that is not isomorphic to a Hilbert space. PROOF OF (i). The finiteness of yp(B) for some p E (1, cx~) implies that yp(B) is finite for all p E (1, ec); see Section 5 in [30]. So in the proof of (i) we can assume that p = 2. The "if" part then follows from the easy part of (ii): y2(H) = 1. For the "only if" part, let (rn)n~o be the sequence of Rademacher functions on [0, 1), (an)n~>0 a sequence in B, and b E B with Ilbll - 1. Define martingale difference sequences in L~(0, 1) by dn - anrn and

D.L. Burkholder

254

en = [[an[[rnb. Note that the associated martingales f and g are differentially subordinate to each other. Therefore, the finiteness of yz(B) implies that for all n,

•

/7

Ila~ Ilrkb

akrk

k=O

2

k=O

~ l l a ~ Ilrk 2

=

k=0

[[akl] 2 k=0

So, by a result of Kwapiefi [ 100], B is isomorphic to a Hilbert space.

El

PROOF OF (ii). By (41) and the obvious inequality tip(R) ~ 2, then M"(O) = - O t p ( A + B + C),

(50)

where A -- p ( p B -- p(p

-

C -

-

p(p

1)(llhll 2 -Ilkll2)(llxll + IlYll)p-2, 2 ) [ l l k l l 2 - (y',k)2]llYll -~ (llxll + Ilyl]) p-l, 1)(p - 2)[(x', h ) + ( y ' , k)]211xll(llxll + Ilyll) p-3

so M"(0) ~< 0: A is nonnegative since Ilkll ~ Ilhll, B is nonnegative by the CauchySchwarz inequality, and C is also clearly nonnegative. If 1 < p ~< 2, a similar expression for ( p - 1)M"(0) is obtained from (50) by interchanging x and y, h and k, and then multiplying the right-side by - 1 . This can be seen from (p-

1)U(x, y ) - -oep(llx I I - (p - 1)IlYlI)(IlYll-+-IIx II) p-~,

which rests on the identity (p - 1)(p* - 1) = 1, valid for 1 0. We can and do assume that H0 is a proper subspace (otherwise enlarge H slightly). Let a 6 H be in the orthogonal complement of H0 with 0 < Ila II < 1 and write Fn = a + fn and Gn = a + gn. Then, by (42) and (48), U(Fn+l, Gn+l) ~ U(Fn, Gn) + (9(Fn, Gn),dn+l) + (1/~(Fn, Gn),en+l). Each of these four terms is integrable and, by (1), the last two have zero expectation. Therefore, EU(Fn+I, Gn+l) 3, then the best constant on the right side of (51) is p - 1. The proof given above has the advantage of yielding the best constant for the right side if p ~> 2 and the best constant on the left if 1 < p ~< 2; in both of these cases the constant is p - 1. The best constants in the other cases are not known. Inequality (51) does not carry over with finite constants to any Banach space that is not isomorphic to a Hilbert space. This is clear from a result of Kwapiefi [100]; see the proof of the first part of Theorem 14 above. The following theorem provides a martingale square function inequality in the setting of rearrangement invariant function spaces; see Lindenstrauss and Tzafriri [ 107] for background on such spaces. If p ~ [1, oo), then L p (0, 1) and many other Orlicz spaces are examples. Here B is a rearrangement invariant function space on (0, 1) and the upper Boyd index qB of B plays an important role. It is the least q 6 [ 1, oo] such that the norm of the dilation operator Ds on B satisfies

IIDsII ~< s llq

for all s E (0, 1),

where ( D s x ) ( t ) = x ( t / s ) if t 6 (0, s], ( D ~ x ) ( t ) = 0 if t E (s, 1), and the function x : ( 0 , 1) ~ R belongs to B. The following theorem is due to Johnson and Schechtman [96]; also see Antipa [4] and Hitczenko [90]. THEOREM 15. Let B be a rearrangement invariant function space on (0, 1) with finite upper Boyd index qB. There exist positive real numbers c and C depending only on qB such that if f is a real-valued martingale on (0, 1) with respect to Lebesgue measure on the er-algebra of Borel subsets of (0, 1), then

clls(f)

B

IIf*llB

CIIs(z)IIB.

(52)

Conversely, if either side of this two-sided inequality holds for all such martingales (the filtrations must also vary), then the upper Boyd index qB is finite.

258

D.L. Burkholder

This contains the Burkholder-Davis-Gundy inequality [43], which is the inequality

c•

0 Ilfn Ill)- Then f converges a.e. as can be seen as follows. Let )~ > 0 and define r:S-2 ~ [0, cx~] by r(co) = inf{n: Ilfn(co)ll > )~}. We shall see below that F = (frAn)n>~O converges a.e. This implies that f converges a.e. on the set {f* ~< )~} since f = F on this set. The complementary set satisfies P(f* > )~) ~< Ilflll/)~ by the Doob maximal inequality applied to the submartingale (llfnll)n>>O, so P(f* < ec) = 1 by the finiteness of Ilflll. Consequently, f converges a.e. The first step in the proof that F converges a.e. is to show that the expectation of F* is finite. Observe that F* ~< )~ on {r = ec} and F* = IIf~ II on {r < ec}. Therefore, E F* ~< )~ + limn--,oc fr B is fk_l-measurable and integrable. By (1), f is a martingale. Typically, P is the Haar measure on the infinite polydisk s = (0 D) ~176 and the value of fn at a point (e i~176 , e i01 . . . . ) in s is written as fl

f , (0) = x + ~

~0k(00 . . . . . Ok-1 )e i~ 9

k=l

THEOREM 18 ([68]). A complex Banach space B has the ARNP if and only if every L 1bounded analytic martingale converges almost everywhere. This theorem and the corresponding theorem for the RNP (Theorem 16) have quite different proofs. We omit Edgar's proof. The two theorems yield an obvious corollary, which is also a consequence of (57) and the vector-valued version of the E and M. Riesz theorem due to Ryan [132], as was observed by Bukhvalov [26]. COROLLARY 19. A complex Banach space with the RNP has the ARNP. But these two properties are not equivalent. The Banach space L~ (0, 1) has the ARNP (see the next section) but it is well known that it does not have the RNP (see Diestel and Uhl [56]). Further work related to the ARNP includes Dowling and Edgar [65], Garling [77], Ghoussoub and Maurey [80], Ghoussoub, Maurey, and Schachermayer [82], Ghoussoub, Lindenstrauss, and Maurey [81 ], and Bu [24].

D.L. Burkholder

264

8. Characterization of AUMD spaces 0 (B) the least ot 6 [1 cx~] such Here let B be a complex Banach space, 0 < p < oc, and Otp that 0 (B) IIf IIp Ilgllp ~ Ofp for all f and g where f is a B-valued analytic martingale (see Section 7) and g is a -+-l-transform of f (see Section 2). If the numbers E~ E {1 , - 1} in the definition of g are replaced by complex-valued U(~_l)v0-measurable functions v~ that have their values in the closed unit disk, then the best constant may change a little but its finiteness, or lack of it, will not be affected. There will be no change if the vk have their values in [ - 1 , 1]. The space B is an A UMD space if Ctp 0 (B) is finite for some p 6 (0 c~); equivalently, for all p ~ (0, 0o). This equivalence can be seen by using the inequality IIf*llp ~< el/pllfl]p, which holds for all p E (0, c~) and all analytic martingales [77], using the trivial inequality IIf IIp ~< IIf* IIp, and observing that the argument in the dyadic case (see Section 2 above) shows that if Ilg*llp ~< cp(B)llf*llp holds for some p E (0, cx~) and all f and g as above, then it holds for all p 6 (0, ~x~). Notice that if p E (1, oc), then Otp 0 (B) ~< tip(B) So a UMD space is an AUMD space. But the converse is not true: the space L ~ (0, 1) is AUMD but is not UMD because it is not reflexive. That L ~ (0, 1) is AUMD is an immediate consequence of the fact that C is UMD, hence is AUMD, and Theorem 9 of Garling [77]: if B is AUMD, then so is L~ (0, 1) for all p E [1, cx~). Note that Ilf* Ill ~ ellfll l, a special case of Garling's inequality for analytic martingales, implies that an AUMD space is an ARNP space" if B is AUMD and f is an L~-bounded analytic martingale, then f * is integrable so there is an integrable function foc such that fn = E (fee [ .T'n) as in (55), which implies that f converges almost everywhere. Therefore, by Edgar's theorem above, the space B is ARNE Using multipliers, Blower [ 16] proved the following characterization of AUMD. THEOREM 20. A complex Banach space B is AUMD if an only if there is a real number y such that

fo re

• n~O

mnan einO dO 0 and for every natural number m, there is an n > m and an operator T: G --+ En such that II T II II T -1 II ~< 1 + e. To see that we can assume this l property, let Cp = (Y~n @Gn)ep, where (Gn) is a listing of the (En) where each En appears ! ! infinitely many times in the sequence (Gn). Then clearly Cp is isometric to ( ~ E3Cp)ep. Note that C p is isometric to a subspace of C p' . Also, given any Ek and any m, the space (Y~in_ 1 ~)Ei)~pEDpEkis (1 -+- e)-isomorphic to one of the Ei's and we must have i > m just ! by dimension considerations. It follows that C p is isometric to a complemented subspace of Cp and hence by the Petczyfiski decomposition method Cp ~ C'p" An application of Proposition 1.5 shows that C p has a basis. The simplest way to do this is to observe that we can rearrange the En so that each Ezn G E2n+! has a basis (x~)~" l

~,~,~t'xn~kni)i:l)n~

with basis constant O. For later reference, we mention that because of the structure of compact sets in sums of Banach spaces, the approximation property passes through gp-sums. PROPOSITION 2.14. If (Xn) is a sequence of Banach spaces with the approximation property, then for every 1 /n. To get a space with a finite dimensional decomposition which fails to have a basis, we take X -- (y-~'GEn)e2. It follows that X is a complemented subspace of C2. We now have a space with a unconditional finite dimensional decomposition and it cannot have a basis taken from the En's. The main problem is that this space might have a basis coming from Proposition 1.4. To prevent this from happening, we choose the En's carefully as subspaces of s with Pn ,~ 2. The main point then is to show that we can pick the En's so that En @ s fails to have a good basis, and choose p~ so that any subspace of s of dimension n-1

~< Y]~i=l dim Ei is a good Hilbert space. Now, in order to build a good basis for E~, the span of {E1 . . . . . En- 1} is too small to be of help, while elements chosen from the span of En+l . . . . have the property that any set of dim En of these elements are spanning a good Hilbert space and therefore cannot help either. For (2) we have the same subspaces En, only now we need to glue them together more carefully. We now need to form the En's into a Banach space in such a way that we do not introduce new projections. We do this by overlapping the En's carefully and gluing them together in such a way that there are uniformly bounded finite rank operators passing through the overlaps but no good projections. This program is carried out by carefully embedding the E~ 's (with overlap) into C2. This is a very delicate operation and the complexity of Read's paper reflects this. In both cases above, the space X we construct is isomorphic to a complemented subspace of C2 and X has the bounded approximation property. Since C2 is reflexive, X has the commuting bounded approximation property and so by Proposition 6.8, X | C2 has a basis. That is, in both cases above, our space X is complemented in a space with a basis. E3 Proposition 6.5 does not hold for unconditional finite dimensional decompositions since the Kalton-Peck space [54] has a unconditional finite dimensional decomposition into two dimensional subspaces but fails to have a unconditional basis (or even be complemented in a space with a unconditional basis). However, a space X has a unconditional finite dimensional decomposition if and only if it has a unconditional finite dimensional expansion of the identity. Also, every Banach space with a unconditional finite dimensional decomposition embeds into a Banach space with a unconditional basis (see [66, Theorem 1.g.5, p. 51 ]). We ask

302

P.G. Casaba

PROBLEM 6.8. If X has a unconditional finite dimensional decomposition and X has GLLUST, must X have a unconditional basis? Casazza and Kalton [14] have given a partial positive answer to Problem 6.8: If X has GL-LUST and a unconditional finite dimensional decomposition X = ~ O En with SUPn dim En < ~ , then X has an unconditional basis (whose basis elements may be chosen from the En's). In particular, what the Kalton-Peck space [54] is missing for it to have a unconditional basis is GL-LUST. Another general method for constructing bases uses the universal spaces C p of Definition 1.10. This is an immediate consequence of the construction of Johnson (see Theorem 5.9). PROPOSITION 6.9. If X is separable and has commuting bounded approximation property, then X G Cp has a basis, for all 1 ~ p 0 so that for every finite dimensional subspace E of X there is another finite dimensional subspace F of X so that span (E U F) ~2 E G F and E • F has a basis with basis constant ~< K. That is, if every subspace of X has the local basis property that every finite dimensional subspace of X is well complemented in a finite dimensional subspace of X with a good basis. One consequence of Lusky's proof of Proposition 6.10 is that every/21-space has this property, i.e., every subspace of a s has the local basis property.

Approximation Properties

303

PROPOSITION 6.1 1. There is a constant C > 0 so that f o r every 1 1, let

kx(K,n)

-

sup

-

inf{rk T" T e L(X), IITII 80). It is unknown if the disk algebra has the uniform approximation property. If true, this would be even more useful and interesting in light of the fact that it is unknown if H ~ has the approximation property. Clearly, if H is a Hilbert space then k/4 (1, n) = n. Hence if X is isomorphic to a Hilbert space, then there is a constant K such that

k x ( K , n) -- n

for all n.

The converse of this is also true by the complemented subspaces theorem of Lindenstrauss and Tzafriri [64]. From our discussion concerning the weak Hilbert space T 2 in Section 1, we have that T 2 has the uniform projection approximation property with KT2 ( 8 , n ) - - 2n (or even K T z ( K , n ) -- (1 + e)n for any fixed e and K -- K(e)). That is, for any 2ndimensional subspace E C T 2, E C span{(ti)in_=l , F}, where F is an n-dimensional subspace of span(ti)i~=n+l. But now F is 8-complemented in T 2. Johnson and Pisier [49] showed that such linear behavior of kx actually characterizes weak Hilbert spaces. THEOREM 7.2. A Banach space X is a weak Hilbert space if and only if there are constants K, C such that

k x ( K , n) O. Then there is a finite dimensional subspace F of X containing E such that for every subspace Y of X containing E with dim Y / E -- k there is a linear operator T : Y --+ F with IITII ~< 1 + ~ and T x = x for every x ~ E. By refining Proposition 9.5 in the reflexive case, Lindenstrauss [61 ] shows THEOREM 9.6. Every reflexive Banach space has the separable complementation property with norm one projections. Actually, Lindenstrauss' method shows more. For a reflexive Banach space X and any separable subspaces Y C X and Z C X*, there is a projection P on X with separable range such that Y C P ( X ) , Z C P*(X*) and IIPII -- 1. We observed in Section 2 that if every

312

P.G. Casazza

separable subspace of X is contained in a separable subspace with the approximation property, then X has the approximation property. Johnson [42] proves the corresponding result for the bounded approximation property. This proof is based on ideas of Lindenstrauss [61 ] which are interesting in that it uses discontinuous, non-linear operators to construct a family of approximating bounded linear operators. THEOREM 9.7. Let X be a non-separable Banach space. Then X has the X-bounded approximation property (respectively the fez-property, the uniform approximation property, the uniform projection approximation property) if and only if every separable subspace of X is contained in a separable subspace with the ),-bounded approximation property (respectively the rc)~-property, the uniform approximation property, the uniform projection approximation p rope rty). PROOF. We first check the bounded approximation property. If X has the )~-bounded approximation property and Y is a separable subspace of X, let (En) be an increasing sequence of finite dimensional subspaces of Y whose union is dense in Y. Now we can construct by induction a sequence of finite rank operators (Tn) on X with the property that Tn+l IEnUTn(X) = I. Now letting Z be the closed linear subspace of X (which contains Y) generated by (Tn (X)) will do it. Conversely, assume that every separable subspace of X is contained in a separable subspace with the ~.-bounded approximation property. Let E be any finite dimensional subspace of X. By Proposition 9.5, there exists a separable subspace Y of X so that, given any finite dimensional subspace F of X with E C F, there exists an operator TF : F --+ Y with IITFll ~< 1 + 1/dim F and TFIE -- IE. By our hypotheses there is a separable subspace Z of X with Y C Z and Z has )~-bounded approximation property. Choose a finite rank operator L: Z --+ Z so that IILI[ ~< )~ and LIE = IE. Now consider the net SF : X --+ L Z of (non-linear and discontinuous) functions defined by

LTFx SFX --

0

x 6F, otherwise,

where the direction on the net is by inclusion on F. Since dim L Z < oc, a compactness argument yields a pointwise convergent subnet to say Lx for each x 6 X. Then, L is linear, llLll ~< )~, L has finite rank, and LIE = IE. The zrz-property is done similarly. For the uniform approximation property, the if part can be done as in the bounded approximation property case. For the only if part, choose a separable subspace Y of X and let (Yn) be dense in Sy with each Yn repeated countably many times in the sequence. Let (En) be the spans of all finite subsets of (yn). For every n, choose an operator Tn with IlTn II ~< K, rank Tn C(X)

> O.

Thus T determines an extension of the compact operators by C(X). Conversely any such extension determines, up to a compact perturbation, an essentially normal operator such that 7r(T) = z, where z is the identity function on X. A third viewpoint identifies this extension with the monomorphism that identifies C(X) with the subalgebra C* (t) of the Calkin algebra. They put a natural equivalence on extensions which corresponds to identifying operators which are unitarily equivalent up to a compact perturbation. The set of

K.R. Davidson and S.J. Szarek

324

equivalence classes Ext(X) becomes a semigroup under the operation of direct sum. Moreover this construction makes sense for any compact metric space X, not just subsets of the plane. They established that, in fact, Ext(X) is a group. Moreover it is a generalized homology theory paired in a natural way with topological K-theory. This opened the door to a new era in C*-algebras in which K-theory and topological methods came to play a central role. Even the existence of a zero element is non-trivial. In the case of a single operator or planar X, this reduces to the Weyl-von Neumann-Berg theorem [27] which shows that every normal operator is a small compact perturbation of a diagonalizable operator. This is easily generalized to arbitrary metric spaces, and it shows that any representation of C(X) on Hilbert space is close in an appropriate sense to a representation by diagonal operators. BDF show by elementary methods that any essentially normal operator T is unitarily equivalent to a small compact perturbation of T 9 D, where D is a diagonal normal operator with a ( D ) = ae(D) = ae(T). This leads us somewhat afield from the original problem. However the connection is made via quasidiagonality. A set of operators T is quasidiagonal if there is an increasing sequence Pk of finite rank projections converging strongly to I such that l i m n ~ II[T, P]II = 0 for every T E T. In particular, if T is quasidiagonal and e > 0, then there is a sequence Pk such that

T-- Z(Pk-- Pk-1)T(Pk-- Pk-1)= K k>/1

is compact and IlK II < e. Indeed, one just drops to an appropriate subsequence of the original one. Thus T - K has the form ~ke> 1 Tk where T~ act on finite dimensional spaces. Such operators are called block diagonal. Salinas [149] showed that the closure of the zero element in Ext(X) in the topology of pointwise-norm convergence, where we think of an extension as a monomorphism from C(X) into/2(7-{)//(7, is the set of all quasidiagonal extensions. In particular, unlike the planar case, the trivial element need not be closed. He used this to establish homotopy invariance of the Ext functor. In the case of a single operator, this shows that if T is essentially normal with zero index data, then it has a small compact perturbation which is block diagonal, say

T-K-T'-

ZeTk. k>~l

Moreover, it follows that [T', T'*] -- y~ke> 1[Tk, Tk*]. Therefore lim

k---+ (x)

II t:Tk, :T;J II

-

0

A positive solution to our original question would yield normal matrices Nk such that limk~ IIT~ - N~ II - 0. Hence N -- Y~ke>1 Nk is a normal operator such that T - N is compact. This was the motivation for the work of the first author [55]. If T -- y~,e k~>l Tk is essentially normal with ae(T) = X, then after a small compact perturbation, T may be replaced

Local operator theory, random matrices and Banach spaces

325

with T G D where D is any normal operator with eigenvalues in X. Thus each summand Tk can be replaced by Tk 9 D~, where Dk is diagonal. An absorption theorem is proven: THEOREM 1.3 (Davidson). I f T is any n • n matrix, then there are an n x n n o r m a l matrix N a n d an 2n • 2n n o r m a l matrix M such that

IIM - Z 9 Nil ~< 75[1[T, Z*]l[ 1/2. One difficulty in applying this to the BDF problem is that any holes in the spectrum of T are obliterated by summing with a normal with eigenvalues dense in the unit disk. In [29], Berg and the first author showed that similar operator-theoretic techniques can be used to show first that every essentially normal operator with zero index data is quasidiagonal. And then refinements of the absorption principle which control the spectrum were used to prove the planar version of BDF. This had the advantage of providing quantitative estimates for 'nice' spectra. But the correct positive answer to our question came again from a more abstract approach. Huaxin Lin [113] considered T = ~ke>~l Tk as an element of the von Neumann algebra 93l -- Ilk~>l 9)tnk. A positive solution is equivalent to constructing a normal operator N = Y'~ke>1Nk asymptotic to T, that is, verifying l i m k ~

[ITk -- Nk ] l - 0. This is a per-

turbation by the ideal 3 -- ~ k ~>1 93ln~ of elements J = ~ ek/> 1 Jk where limk~ oc I[Jk 11= 0. Thus T determines an element t = q ( T ) in the quotient algebra 9Jt/~. The problem is reduced to lifting each normal element of 93I/3 to a normal element of 93t. This was accomplished by a long tortuous argument. THEOREM 1.4 (Lin). Given e > O, is there a ~ > 0 so that w h e n e v e r A a n d B are n x n H e r m i t i a n matrices o f n o r m one, f o r any n >~ 1, a n d ]lAB - BAll < 6, then there are H e r m i t i a n matrices A1 a n d B1 w h i c h exactly c o m m u t e such that ]IA - All] < e a n d

IIB - B1 II < ~. A remarkable new proof of Lin's theorem is now available due to Friis and ROrdam [78]. They take the same approach, but base their argument only on two elementary facts about 93l/3. It has stable rank one, meaning that the invertible elements are dense. And every Hermitian element can be approximated by one with finite spectra. Both of these results can be obtained by using well-known facts about matrices on finite dimensional spaces. From this, a short argument cleverly using no more than the basic functional calculus yields the desired lifting of normal operators. Thus the original question has a positive solution. (See Loring's book [ 115] for a C*-algebraic view of this problem and its solution.) In a second paper [79], they show how the planar case of the Brown-Douglas-Fillmore Theorem follows. An important matrix question remains: PROBLEM 1.5. What is the dependence of 3 on e for almost commuting Hermitian matrices?


326

Optimal estimates for 3 should be 0(82) as in the absorption results. However the Lin and Friis-Rcrdam results yield only qualitative information, and do not provide information on the nature of the optimal 6 function. A related question deals with pairs of almost commuting pairs of unitary matrices. Voiculescu [173] constructed the pair Uej --ej+lmodn and Vej --e2zrij/nej for 1 ~< j ~< n. It is evident that [I[U, V]ll < 2rc/n. However he showed that this pair is not close to a commuting pair of unitaries. He speculated that both this example and the commuting triples example occur because of topological obstructions. Asymptotically, the spectrum approximates a torus (or a two-sphere for Hermitian triples), and thus there is a heuristic justification for this viewpoint. Loring [ 114] showed that this was indeed the case for this pair of unitaries by establishing a K-theoretic obstruction coming from the non-trivial homology of the two-sphere. Exel and Loring [74] obtained an elementary determinant obstruction that shows that this pair of unitaries is far from any commuting pair, unitary or not. 1.2. Unitary orbits of normal matrices Consider a normal matrix N in 9Xk. By the spectral theorem, there is an orthonormal basis which diagonalizes N. So N is unitarily equivalent to diag(~,l . . . . . )~k) where the diagonal entries form an enumeration of the spectrum of N including multiplicity. Two normal matrices are unitarily equivalent precisely when they have the same spectrum and the same multiplicity of each eigenvalue. A natural question from numerical analysis and from the approximation theory of operators asks how the spectrum can change under small perturbations. The answer one gets depends very much on what kind of perturbations are allowed. In particular, if the perturbed matrix is also normal, then one obtains a more satisfying answer than if the perturbation is arbitrary. Consider first the case of a Hermitian matrix A. Then the eigenvalues may be enumerated so that ,kl ~> ... ~> ~.k. If B is a small perturbation of A, say IIA - B 1[ < e, then the Hermitian matrix R = (B + B*)/2 also satisfies IIA - R[[ < e, and so we may assume that the perturbation B is Hermitian to begin with. In that case, write the eigenvalues as # l ~> " " ~> #k. Then a 1912 argument due to Weyl [183] gives: THEOREM 1.6 (Weyl). Let A and B be k x k Hermitian matrices with eigenvalues ordered as )~l ~ "" >1 )~k and # l ~ "" >~ #k, respectively. Then sup [#j - ,kj [ ~< [] A - B II. l~j~k

Moreover, f o r each A, this value is attained by some such B. Indeed, suppose for example that )~j ~ # j . Consider the spectral subspace for A corresponding to {)~1. . . . . ~j } and the spectral subspace of B for {#j . . . . . #k }. Since the dimensions of these spaces add up to more than k, they contain a common unit vector x. Then (Ax, x) lies in the convex hull of {~1 . . . . . ~,j }, whence is ~> )~j. Similarly (Bx, x) 0, is there a constant C (e) independent of n such that for every matrix T with [[T [[ ~< l, there exists a diagonal operator D and an invertible operator W such that

IT -

WDW -1

and

]]W[[ [[W-' [1 ~ C(e)?

A compactness argument shows that for each fixed dimension n, there is a constant C(e, n) which works.

1.4. Extensions of pure states and matrix paving In this section, we will discuss a well-known problem of Kadison and Singer [99], which asks whether every pure state on the algebra D of diagonal operators on ~2 with respect to the standard orthonormal basis extends uniquely to a (necessarily pure) state on s163 We note that 79 is a generic discrete maximal abelian subalgebra or masa. It is proved in [99] (see also [4]) that the answer to the uniqueness question is negative for non-discrete masas. The interest in the Kadison-Singer problem lies, in particular, in the fact that a positive solution would shed new light on the structure of pure states on s163 In fact it would go a long way towards a simple characterization of such states. Indeed, pure states on 79 ~ s _~


334

C (fiN) are just point evaluations at elements of fiN, the Stone-Cech compactification of N. The canonical extension of a state from 79 to s is obtained by composing the state with the conditional expectation E which takes each T in s to its diagonal part in 79. In other words, if L/is the ultrafilter associated to a pure state q) on 79, then the extension 7t to/~(/~2) is given by ~ ( T ) = ( p ( E ( T ) ) = lim(Ten,en). n~/g

This suggests the open question raised in [99, w whether every pure state on/~(~2) is an extension of a pure state on some discrete masa, or indeed a given masa. However, it has been shown in [6], cf. [4, w that the two questions are in fact equivalent. See also [5] for a related positive result for the Calkin algebra. On the other hand, every pure state on/~(~2) is of the form qg(T) = limncU (Txn, Xn) for some sequence (Xn) of unit vector in ~2 and some ultrafilter in N [ 186]. Moreover, if q9 is not a vector state, the weak limit w-limneU Xn is necessarily 0. Since weak-null sequences in ~2 can be refined to be asymptotically orthogonal, it follows that the difficulty in settling both this and the uniqueness question lies in the difference between limits over ultrafilters and regular limits. Thus, at the first sight, the Kadison-Singer problem seems to be a strictly infinitedimensional question. However, it was shown by Anderson [4] that it is equivalent to a finite dimensional question known as the paving problem. PROBLEM 1.18. Does there exist a positive integer k such that, for any n >~ 1 and any matrix A 6 9Jtn with zero diagonal, one can find diagonal projections P1, P2, 9.., Pk 6 9Jtn such that

(i)

EPi=I

and

PiAPi

(ii)

i=1

1

~ IIAII?

i=1

This is clearly the kind of question that fits into our framework. To clarify the connection we will sketch the argument. Observe first that Problem 1.1 8 is formally equivalent to the following: Given s > O, does there exist a positive integer k such that, for any matrix T in s one can find diagonal projections P1, P2, ..., Pk such that ~ 1 Pi - I and k

k

E( ))Pi

(ii') i=1

E

PiTPi - E(T)

~<slITII?

i=1

Indeed, one gets s in place of 1/2 in the condition (ii) via iteration, with k depending on s. Passing from finite matrices with a uniform estimate on k to infinite matrices considered as operators on ~2, one obtains the partition of N corresponding to the decomposition of the identity on ~2 into a sum of projections from finite partitions via a diagonal argument. Let q9 be an extension to s of a pure state on 79. We claim that qg(D1TD2) = q)(D1)qg(T)q)(D2)

for D1, D2 E 79, T E s163

(l)


335

We note that a priori ~p is multiplicative only on 79 as a point evaluation on C (fiN) _~ 79. We postpone the proof of (1) and show first how it allows to complete the argument showing that an affirmative answer to Problem 1.1 8 implies that q9(T) -- q9(E (T)) for T E/2 (~ 2 ). Let e > 0. Fix T ~ E(g2) with IIT II ~ ~ _ P i T P i - E ( T )

q)

Pi TPi

- ~o(E(T))

i=1

Since e > 0 was arbitrary, it follows that qg(T) = qg(E (T)), as required. It remains to prove (1). To this end, consider the GNS representation of (E(g2), ~p). There is a Hilbert space H, a norm one vector x 6 H and a ,-representation 7r of s on/2(7-/) such that for all T in s =

An elementary argument shows then that x must be an eigenvector for each 7r(D), D ~ 79 with an eigenvalue qg(D). Hence, for any T in E(g2),

~o(D1TD2) = (Tr(D1TD2)x, x) -- (Tr(D1)rc(T)Tr(D2)x, x) = (Tr(T)Tv(D2)x 7r(D1

x)--(Tr(T)(qo(D2)x) 9o(D1)x)

= qo(D2)qo(D1)(rc(T)x, x ) = qo(D2)qo(D1)qo(T). The converse, uniqueness implies paving [4, (3.6)], can be proved in a similar spirit. One way is to show first that paving is implied by the relative Dixmier property, which says that for T 6 E(s

79(T) := conv{U*TU: U ~ D, U unitary} M 79 ~- 0. This immediately implies that E (T) belongs to 79(T). This is done in very much the same way as the argument presented above. On the other hand, if E ( T ) is not in 79(T), then E(T) can be separated from 79(T) by a functional qg. Heuristically, in view of the balanced

336


nature of 79(T), q9 may be assumed to be positive and with more work, an extension of a pure state on D. By construction, q) is different form the canonical extension obtained by composing with the conditional expectation E. There has been a lot of work on Problem 1.18 in the intervening years, mostly yielding partial and related results. In particular, it was shown by Berman, Halpern, Kaftal and Weiss [30], cf. [43,98], that the answer is positive for matrices with nonnegative entries. A major progress was achieved by Bourgain and Tzafriri. Their work was motivated principally by applications to local structure of L p-spaces, see [98] in this collection for more details. First, in [42], they showed that for Problem 1.18, there exists a diagonal projection P with IIP A P II ~< 89IIA II and rank P ~ 6n, where 6 is a universal positive constant. A closely related result was obtained earlier by Kashin [ 102]. It then clearly follows by iteration that there is a decomposition of identity P1, P2 . . . . . Pk verifying the conditions (i) and (ii)of Problem 1.18 such that k = O(logn). Then in [44], they obtained by far the strongest results to date. Problem 1.18 is solved in the affirmative when the absolute values of entries of the matrix A are relatively small, specifically O(1/(logn) l+~) for some 7/> 0 (Theorem 2.3). Their solution also applies to the cases of Hankel and Laurent matrices with certain regularity properties But the major accomplishment is a saturation result that follows. As earlier, we identify s with the algebra 79. Similarly, the power set P(N) --= {0, 1}r~ is thought of as a subset of s In particular, a C N is associated with the sequence (aj) ~ s the indicator function of a , and with the diagonal projection P,, in 79. THEOREM 1.1 9 (Bourgain-Tzafriri). Given ~ > 0 and T ~ /2(~2), one can find a positive measure v supported on the w*-compact set

K - K(r, of total mass

Ilvll ~

c N: IIP (r - E(r))P

[I
1 f o r all j >~ 1, where C is a universal numerical constant.

For clarity, we state also the finite dimensional version of Theorem 1.19 from which the theorem easily follows by a diagonal argument. PROPOSITION 1.20. There is a universal constant c > 0 so that given ~ > O, n ~ N and a matrix A E 9J'(n with zero diagonal, there exist a finite sequence of nonnegative weights (ti) with E i ti = 1 and diagonal projections (Pi) such that

IleiAPi II ~ EIIAII f o r all i >~ 1

and

E

ti Pi ~ c~ 2 I. i

Let us point out first that the last condition in this proposition implies by trace evaluation that the rank of at least one of the Pi's is at least ce2n, thus recovering the result from [42]


337

mentioned above with the best possible dependence on ~. Note also that, except for that optimal dependence, it is enough to prove Proposition 1.20 for some ~ E (0, 1). We wish to emphasize that Theorem 1.19 is very close to implying the affirmative answer to the Kadison-Singer problem. Indeed, if we somewhat carelessly apply a (pure) state q9 to both sides of the assertion of Theorem 1.19, we "obtain" (2)

~O(fKadV ) -- fKcP(a)dv ~ 1.

Hence there is a a E K such that qg(a) = qg(Po) > 0. Thus qg(Po) = 1 because q9 is multiplicative on 79 and Po is idempotent. We now proceed as in the derivation of the original Kadison-Singer problem from the paving problem. As a 6 K, one has IIP~(T - E(T))P~II 0. Therefore (p(T) = (p(E(T)), whence qg(T) is determined by its diagonal part. The weak point of this "argument" lies in the fact that the integral fK a d r ( a ) makes only weak* sense, while in equality (2) we implicitly used weak convergence. The argument would work if the measure v was atomic, though. Still, Theorem 1.19 shows that there is an abundance of diagonal projections P~ verifying [[P~ (T - E(T))P~ 11~< ellT[[. That abundance just isn't formally strong enough to guarantee that the collection of such a ' s will intersect every ultrafilter. We show now a simple example to that effect. Let d E 1~ and let Id be the set of all words of length 2d in the alphabet {A, B } consisting of d A's and d B's. So n := #Id = (2d)!/(d!) 2. Next, for s = 1, 2 . . . . . 2d, let as be the set of those words in Id whose sth letter is A. Clearly, Z'd :-- {al, 0"2 . . . . . CrZd } provides a cover of Id. It is not a minimal cover, but d + 1 sets are required. Let r[t be the hereditary subset of the power set 79 (Id) generated by Zd, namely X~ := {a: a C as for some 1 ~< s ~< 2d}. Then every subcover, or partition, of Id consisting of elements of r ~ must have at least d + 1 -- O(logn) elements. On the other hand, it is easily seen that 1 ~-~,2~l Xo, -- 89Thus the set of projections {Po" a E X~} verifies the condition in the assertion of Proposition 1.20, but does not verify the condition of Problem 1.18 with k independent of n. An infinite example verifying the condition of Theorem 1.19, but not that of the infinite variant of Problem 1.18 is routinely obtained by identifying N with ~ d Id and setting Z'--{aCI~:

aAIdEr~foralld~l}.

Of course, this is just a combinatorial contraption. There is no a priori reason why r produced above would correspond to an actual operator T E s163 The methods of [42-44] are quite sophisticated. Without going into details, we mention that the first step in finding large subset a C { 1, 2 . . . . . n} for which IIP~ A P~ I[ is small involves a random procedure. The first approximation is a = {j: ~j = 1 }, where


338

~n is a sequence of independent Bernoulli selectors. These are random variables satisfying P(~j = 1) = 1 - P(~j = 0) - - ~ for j = 1,2 . . . . . n for some properly chosen E (0, 1). The norm of P~ A P~ is then the maximum of a random process, which is analyzed using decoupling inequalities (see, e.g., [106]), majorized by a more manageable maximum of a Gaussian process, and estimated via metric entropy and Dudley's majoration (see [112, (12.2)]). This works if the entries of the matrix A are rather small, such as the O ( 1 / ( l o g n ) l+u) bound mentioned earlier. Such random subsets also yield a partition of {1,2 . . . . . n }. In the general case, one only obtains an estimate on the norm of P~ A P~ as a map from s to ~ . In the context of Proposition 1.20, one has instead a weighted s The final step uses the Little Grothendieck Theorem [ 137]. Alternatively, some of the steps may be done by using the measure concentration phenomena (cf. Section 2.2) already employed in [103] or majorizing measures, cf. [165]. See [98] in this collection for more details on some of the above arguments and related issues. We conclude this section by commenting on the type of examples that need to be analyzed in hope of further progress. To be a potential counterexample, a matrix A -- (aij) ni j__ 1 (meaning a sequence of n • n matrices that together provide a counterexample) must have the following features: (i) ]lAB] must be much smaller than ]](]aij ]) ]], or otherwise we could apply the argument that works for nonnegative entries. (ii) ]aij] do not admit a (uniform) O ( 1 / ( l o g n ) l+u) bound. (iii) On the other hand, the substantial entries of A must be sufficiently abundant, or otherwise one could avoid them by the same combinatorial argument that works for nonnegative entries. (iv) The combinatorial structure of the substantial part of A must be quite rigid to distinguish between the conditions from Proposition 1.20 and Problem 1.18. One structure that comes to mind is related to adjacency matrices of Ramanujan graphs (see, e.g., [116]). Let B = (bij) be such a matrix corresponding to a d-regular graph on n vertices. Then []B ]] = d, and it is achieved on the eigenvector (1, 1 . . . . . 1), while all the remaining eigenvalues are bounded by 2~/d - 1. So the n • n matrix ( 89(bij - k ~ n ) / ~ / d - 1) is of norm at most 1, and appears to enjoy the features (i)-(iv), some of which are admittedly vague. The question would then be to determine whether matrices obtained this way from various constructions of Ramanujan graphs can be paved. It seems at the first sight that new techniques are required for any kind of answer. ~1, ~2 . . . . .

1.5. Hyper-reflexivity If r is an operator algebra contained in s then L a t A denotes the lattice of all of its invariant subspaces. Dually, given a collection 12 of subspaces, Alg/2 denotes the algebra of all operators leaving each element of 12 invariant. The algebra ,A is reflexive if Alg Lat,4 = A. There is a quantitative version of reflexivity which has proven to be a powerful tool when it is available. Notice that if PL is the projection onto an invariant subspace L of .,4, then for any operators T E s and A 6 .A,

II

II- II

A)P li

-

All.


339

Hence the inequality

fls

"-- sup

IIPL TPL LI

dist(T, A).

LE/Z

The algebra A is hyper-reflexive if there is a constant C such that dist(T, A) O, 79 (F ~ M + t) ~ 1 - ~ ( t ) < e x p ( - t 2 / 2 L 2 ) .

(4)

One has the same upper estimate e x p ( - t 2 / 2 L 2) if the median M is replaced by the expected value fR, F dyn.


347

For future reference we point out here that, for a convex function, its median with respect to a Gaussian measure does not exceed the expected value; see [71,107] or Corollary 1.7.3 in [76]. The relevance of Theorem 2.6 to random (Gaussian) matrices is based on the elementary and well-known fact, described in Section 1.2 of this article, that spectral parameters like singular values (respectively eigenvalues in the selfadjoint or unitary case) are Lipschitz functions with respect to the matrix elements; see [35,33] for related more general results. In particular, for each k c {1. . . . . n}, the k-th largest singular value sk(X) (respectively 17 the eigenvalue ,kk(X)) is Lipschitz with constant 1 if X (Xjk)j,k=| is considered as an - -

element of the Euclidean space R n2 (respectively the submanifold of R n2 corresponding to the selfadjoint matrices). If one insists on thinking of X as a matrix, this corresponds to considering the underlying Hilbert-Schmidt metric. Accordingly, in the context of applying Theorem 2.6, the Lipschitz constant of sk (G (n)) is 1/~/-n (because of the variances of the entries being l / n , which corresponds to the identification G = G (n) = 1/x/~ X). Respectively, for a Gaussian selfadjoint matrix A = A (n) verifying the hypotheses of Theorem 2.2, the Lipschitz constant of )~k(A(~)) is v/2/n. The additional 2 is a consequence of the same variable appearing twice, in the jkth and kj th position. The above comments, and hence the results below, carry essentially word for word to the following often considered variants, all Gaussian unless explicitly stated otherwise. (i) A variant of A = A (") in which variances of the diagonal entries are assumed to be 2In rather than l / n , called often the Gaussian orthogonal ensemble or GOE. This is in fact the same as v/2 times the real part of G (n) . Note for future reference that GOE can be represented as Y + A, where Y is a (diagonal) Gaussian random matrix independent of A, and so many results for GOE transfer formally to A. (ii) The complex non-selfadjoint case, all the entries being independent and of the form g + ig', where g, g' are independent real N(O, 1/2n) Gaussian random variables. (iii) The complex selfadjoint case: formally the same conditions as in Theorem 2.2 (except for the obvious modification in the symmetry condition), but the abovediagonal entries are as in (ii) while the diagonal entries are real N(0, 1/n)'s. Again, this is the real part of the matrix in (ii) times v/2, and is frequently referred to as the Gaussian unitary ensemble or GUE. (iv) Rectangular, real anti-symmetric or complex anti-selfadjoint matrices. (v) Orthogonal or unitary matrices distributed uniformly on SO(n), respectively U (n). It is also easily seen that in the first three cases above the Lipschitz constants are respectively 2 x / ~ , 1 / v / ~ and 1/v/-n. The anti-symmetric/anti-selfadjoint/rectangular cases are treated the same way, and there are equally useful results for orthogonal/unitary matrices cf. [124,86]. There appears to be no easily available exposition of the unitary case, even though all ingredients are available, cf. [176]. Still, a word of caution is needed. As noted in Section 1.1, eigenvalues are not very regular functions of general (non-normal) matrices. Combining the above remarks and Theorem 2.6 we get THEOREM 2.7. Given n E N, there exist positive scalars Sl, $2 Sn such that the singular values of the n x n real Gaussian random matrix G = G (n) satisfy .

79(Isk(G)- ski ~> t) < 2 e x p ( - n t 2 / 2 )

.

.

.

,

(5)

348


f o r all t ~ 0 and k -- 1, 2 . . . . . n. This holds, in particular, if sk's are the medians or the expected values of sk(G). Similar results hold f o r the eigenvalues of the real symmetric Gaussian random matrix A -- A (n) with - n t 2/4 in the exponent on the right side of (5). Moreover, the corresponding deterministic sequence ~.l, )~2 . . . . . )~n can be assumed to be symmetric, i.e., )~k = --)~n-k+l f o r k = 1, 2 . . . . , n. Likewise, related results hold f o r other Gaussian random matrices, in particular those described in (i)-(iii) above.

A shortcoming of the above result is that it doesn't see the possible relationships between the deterministic sequences sl, s2 . . . . . Sn (or)~1, ~.2 . . . . . )~n) for different n's. Still, it allows to formally deduce statements in the spirit of Theorem 2.3 from the corresponding Theorem 2.2 like results. Consider, as an illustration, the ensemble A = A(n). Once we know that there is a rough estimate on, say, EIIAII (e.g., of the type of Theorem 2.1), the analogue of (5) for A implies via an elementary calculation ETr(A v) - ~ x v d#(n)(x) ~ sin(F). Then (12)

1 - w/~ < E s m ( f ) < Ms,(F) < Esl ( F ) < 1 + and consequently, f o r any t > 0,

max{79(s, ( F ) >~ 1 + ~

+ t), V ( s m ( r )

,)1

~ t) < C e x p ( - c p n min{t 2, t2/d})

(~5)

where d is the degree of P, c e > 0 depends only on P and C is a universal constant. One necessarily has maxl 0 the price St is a normally distributed random variable. Bachelier's use of Brownian motion in the context of finance was thus prior to the use of this process in the context of physics by A. Einstein [21] and, independently, by M. Smoluchowski [50] some five years later. Having fixed the model, Bachelier now turned to the question of pricing an option (to be precise: a European call option); such an option is the right to buy one unit of the underlying stock at a fixed time 7" (the expiration time) and at a fixed price K (the strike price). This determines the value CT of the option at time T as a function of the (unknown) value ST of the stock at time T, namely CT = (ST - K ) + .

(1)

Indeed, a moment's reflection reveals that the option is worthless (at time T) if ST K: in the latter case the holder of the option will exercise the option to buy one unit of stock at a price K and can immediately resell it at the present market price ST to make a profit ST - K. (In practice, a cash settlement is often made.)

370

F. Delbaen and W. Schachermayer

Hence we know, conditionally on the random variable St, what the option will be worth at time t = T. But what we really want to know is what the option is worth today, i.e., at time t = 0. To determine this quantity Bachelier simply took the expected value, i.e., Co " - E [CT] -- E [(ST -- K)+]

(2)

which leads to a simple explicit formula for Co involving the normal distribution function. The approach of formula (2) has been used in actuarial mathematics for centuries and is based on a belief in the law of large numbers: if the stochastic model (St)o 0, there is a measure Q0 such that Q0 is equivalent to P, that ]IQ - QoIJ ~< e and that under Q0 the process S is a sigma-martingale. See [ 15] for details and for an example that shows that one cannot do better than a sigmamartingale. As indicated above, the central point of the proof is the fact that C is weak*-closed. This is done using the Krein-Smulian theorem, also called the Banach-Diedonn6 theorem. This theorem says that a convex set C in the dual X* of a Banach space X is ~ (X*, X) (i.e., weak*-closed) if and only if C n (nBx,) is ~(X*, X) closed for each n ~> 1. If X = L1 and X* = L ~ we can, using the characterization of relatively weakly compact sets in L1 as uniformly integrable subsets of L1, make this even more precise. A convex set C C L ~ is weak*-closed if and only if, for each sequence ( f , ) ~ > l in C that is uniformly bounded and converges in probability to a function f , we have that f E C. Since in our context the set C is a cone we have to show the following fact. CLAIM 3.5. Let (Hn)n>~j be a sequence of 1-admissible integrands, let (fn)n>~l be a sequence in L ~ ~ , ~) such that - 1 1 supt I(H n 9S)t] is in L2(]P). That this is possible follows from an upcrossing type lemma and the maximality of f . After an additional technical reduction, which we skip, the D o o b - M e y e r decomposition theorem now allows to decompose S into a martingale M and a predictable process of finite variation A, i.e., S - M + A. The D o o b Meyer decompositions of ( H n 9S) are then given by ( H " 9M) and ( H ~ 9A). In order to control the jumps of M and A we use the following generalization of an inequality of Stein.

382


PROPOSITION 3.8. Let (.Tn)n=0 ..... N be a discrete time filtration on the probability space (12, ~ , I?), let ( fn)n--1 ..... U be adapted to (.Tn)n=O ..... N, let gn be the predictable projection, i.e., gn = E [fn I f ' n - l ] then we have

EI(ZJgn[q)P/q]I/P~2E[(~]fnlq)p/q]1/p whenever 1 ~ p ~l t

> (E[fn I.T'n-1])n>~l

(17)

has norm less than or equal to 2.

The next step is to show that the convex hull of ( ( H n. M)~)n>>l and of ( ( H n. A)~)n>>l are both bounded (in a good sense). Since ( ( H n . S ) ~ ) n ~ l is bounded in L2(I?) both sets are at the same time bounded or unbounded. But if they are unbounded we are faced with the fact that ( H n 9A)t increases in a "linear" way whereas ( H n . M ) t , due to the orthogonality of its increments, grows only in a way related to the square root. This leads to a contradiction. Once the boundedness is proved, it is a straightforward track to find convex combinations K n of (Hn)n>~l that converge to a predictable process K such that (K 9S ) ~ = f . The latter technique is a combination of techniques of Memin (see [39]). The separation argument in the proof of the fundamental theorem can be exploited further. It yields to the following duality result (see [24,9]). THEOREM 3.9. A s s u m e that S is a locally bounded semi-martingale such that M e is not empty. Then f o r f ~ 0 we get sup E Q [ f ] : inf{ot I 3g ~ K with ot 4- g ~ f } : : or0.

(18)

Furthermore, if the quantities are finite, the infimum is a minimum and there is a maximal element g E K max with oto 4- g ~ f .

4. The continuous case

When the process S is continuous or, more generally, locally bounded, the assumption on S to be a semi-martingale turns out to be a necessary condition for the conclusion of the fundamental theorem of asset pricing to hold true. In fact we can prove the following result (see [ 11 ]), stated under a finite time horizon, say [0, 1]. THEOREM 4.1. I f the locally bounded process S = (St)o 0, and a sequence o f simple strategies Hn such that (Hn . S) >>.- e n , en --+ O, (Hn . S) 1 ~

f .

Applications to mathematicalfinance

383

We recall that a strategy H is simple, if there is a finite sequence of stopping times

0 0, fTT+ e hu2 d ( M , M)u = oc.

Technically the latter case is more difficult but it has a simple solution: there is a very special kind of arbitrage, namely there is H supported by ~T, 1~ and such that (H 9S) ~> 0 i.e., the outcome at every time is nonnegative, and iP[(H 9S)T+e > 0] > 0 for all e > 0. dt + d Wt where W is Brownian motion An example of this nature is given by dSt - -~

defined in its natural filtered space. A strategy of the form above is given by Kt -1/(~/71og(t-1)) and then we stop the process ( K . S) either when it reaches a level 1 or when it hits 0 for the first time after 0. The iterated logarithm theorem implies that, immediately after 0 the process (K 9S) is strictly positive! The next result is closely related to these arguments. It was shown in [12] and, independently and under slightly stronger hypothesis, in [37]. THEOREM 4.2. I f the continuous martingale S satisfies (NA) then there is a measure Q absolutely continuous with respect to iP and such that S is a local martingale under Q. The support o f the measure Q can be chosen to be equal to {L1 > 0} where Lt -- e x p ( hu dMu - I h2 d ( M , M)u) stopped when h2 d ( M , M)u hits oc, or what is the same: when L hits O.

fo

fo

fo

Even when S satisfies (NA) and fo h] d(M, M}u < ec a.s., the process L need not be a martingale, i.e., it can happen that E[L1] < 1. This means that the measure Q is not necessarily given by dS = L1 diP. See [47] and [ 16]. We also have the following: PROPOSITION 4.3. I f S is a continuous semi-martingale decomposed as dS = d M + h d ( M , M ) then the set KI = {(H. S ) ~ ] H is 1-admissible} is bounded in Lo(IP) if and only if f 0 h2, d ( M , M)u < cx) a.s.

The 3-dimensional Bessel process shows that this does not rule out arbitrage (compare

[13])~ 5. Changes of num~raire and a related Banach space Throughout this section we suppose that the process S modeling the price of d stocks is a d-dimensional semi-martingale that is locally bounded. We also suppose that the process

Applications to mathematical finance

385

S admits a local martingale measure. We repeat that the set of equivalent local martingale measures is denoted by M e, the notation M being reserved for the closed convex set (in fact the closure of M e) of absolutely continuous local martingale measures for S. Ka will denote the cone of outcomes of (a-)admissible strategies, K - - U a Ka. As seen before, maximal elements play an important role in the proof of the fundamental theorem. However, more can be said. If V = 1 + (H 9S) is a strictly positive process satisfying V~ > 0 a.s., then we might use V as a new money unit (say $ instead of CHF or Euro). The market 1 s ) . Economic interpretation is now described by the (d + 1)-dimensional process X - (V, leads us to conjecture that S satisfies (NA) if and only if X satisfies(NA): It does not matter whether you do the bookkeeping in $, CHF or in Euro! However since the admissible strategies may change, we have to be more careful. THEOREM 5.1. If S satisfies (NFLVR) then the following assertions are equivalent for a process V = 1 + (H . S), where H is admissible and V ~ > 0 a.s.: 1 s (1) X -- (V, V ) satisfies (NA), (2) there is an equivalent local martingale Q measure f o r the process S, such that V is a uniformly integrable Q-martingale, i.e., E q [ V ~ ] = 1 or equivalently E q [ ( H 9 S)~] =0, (3) (H . S ) ~ is maximal in K1 (and hence also in K). It is not true that in this case V is a uniformly integrable martingale for each element R in the set of equivalent local martingale measures M e, see [47] or [16] for this surprising fact. The above theorem also yields another proof of the theorems of Jacka [31 ] and AnselStricker [ 1]. THEOREM 5.2. If S satisfies (NFLVR) and f is a positive random variable, then the following conditions are equivalent: (a) f = ~ + (H . S ) ~ with (H . S ) ~ maximal in K, (b) there is Q E 1 ~ e such that E q [ f ] = sup{E R[f][R E M} < cx~. The Bishop-Phelps theorem now immediately implies THEOREM 5.3. If S satisfies (NFLVR) and continuous local martingale measures are already equivalent, then M is reduced to a singleton. PROOF. As the set of absolutely continuous local martingale measures M is a bounded, closed and convex subset of L1, the set { f I f attains its supremum on M} is a norm dense subset of L ~ . The set of all elements of the form c~ + ( H - S ) ~ , where the process ( H . S) is bounded is therefore dense in L ~ and because it is closed (this follows essentially from claim 3.5) we have that it equals L ~ . This implies that all elements of L ~ are constant on M, hence M can only be a one-point-set. [-1 Another application of Banach space theory is given by James' theorem on weakly compact sets. We state the result in its negative form, see [9] for details.

386


THEOREM 5.4. Suppose S is continuous, satisfies (NFLVR) and suppose that all martingales with respect to (.Yt)o ~o(xo) - ~llx - xoll for all x 6 C. Consider in the Banach space X 9 R the subsets: C1 - {(x, t) E C @ R; t/> ~o(x)}, C2 - - { ( x , t) E X ~) R; t ~< ~p(xo) -

EIIx - xoll ].

They are both closed and convex and the interior of C2 is non empty and disjoint from C1. Using the H a h n - B a n a c h theorem, we can find G = (y*, c~) 6 X* E3 R = (X G R)* such that (y*, c~) r (0, 0), and m E R such that: (1) if x ~ C and t >~ qg(x), then y * ( x ) + at >~m; (2) if x 6 X and t O, X > 0 and assume that u E X satisfies qg(u) < infx q9 + e. Then there exist q E Q and v~ ~ X such that: (i) qg(v~) < infx ~o + e.

(ii) d(u, v~) < X. (iii) For all x E X, qg(x) + 2eX-Zq(x) >~qg(v~) + 2eX-Zq(v~). In view of the three perturbed minimization principles discussed above, it is natural to inquire about the classes of functions on a complete metric space X that are suitable to be perturbation spaces for a minimization principle. In the next section, we shall isolate a fairly general condition on a class A of continuous functions on X that makes it eligible to be a perturbation space. We shall then say that A is an admissible class. We will see that the perturbations used in Ekeland's principle as well as the one of Borwein-Preiss, readily satisfy our condition. The same will hold for various spaces of smooth functions defined on a suitable Banach space. However, the proofs of the facts that, on certain Banach spaces, the spaces of linear functionals or cones of plurisubharmonic functions are admissible classes, are more involved. The rest of the paper contains various applications of these principles to Banach space theory, potential theory, non-smooth analysis, non-linear analysis, the calculus of variations as well as to the theory of viscosity solutions for Hamilton-Jacobi equations.

2. Dentability and perturbed minimization principles 2.1. A general perturbed minimization principle Let (X, d) be a metric space and let (A, 3) be a metric space of real valued functions defined on X. For any subset F of X, we shall denote by ,AF the class of functions in A that are bounded above on F. For f E AF, and t > 0, we denote by S(F, f, t) the following slice of F

S(F, f, t) = {x E F" f (x) > s u p f ( F ) - t}. DEFINITION 2.1. The space (X, d) is said to be uniformly A-dentable if for every nonempty set F C X, every f E AF, and every e > 0, there exists g 6 ~AF such that 6(f, g) ~ K sup{If(x) - g(x)l; x e X} f o r all f, g ~ A. (ii) (A, 6) is a completemetric space. (iii) The product space (X, d) is uniformly A-dentable. Then, f o r any lower semi-continuous function q):X --+ R U {+cx~} that is bounded below with D(~o) 7~ 0, the set {g E A; q9 - g attains a strong minimum on X}

is a dense G a subset G of,4. We shall then say that (,4, 6) is an admissible class of perturbations for the space (X, d). PROOF. We claim that the set /.gn = {g c A ; 3Xn e X with ( ~ o - g ) ( x n ) < inf{(~o - g)(x): d ( x , x n ) >~ I / n } } is an open dense subset of A. Indeed, ~n is open because of assumption (i). To see that/gn is dense, let g E A and e > 0. We need to find h E A, 6(h, g) < e, and Xn E X such that

(~o - h)(xn) < inf{(~o - h)(x); d ( x , x n ) >~ 1/n}.

(*)

To do that, note that the functional (g, - 1 ) in A is bounded above on the epigraph F of q) in ~', hence for any e' > 0, there exists a non-empty slice S - S(/t, F, t) of F with diameter less than e ' and determined by a function h = (h, - 1 ) ~ A verifying 6(h, g) < d. Take e ~ < min{e, I / n } and pick any (Xn,)~n) E S. For any x E X such that d ( x , x n ) 1/n, we have that (x, qg(x)) ~ F \ S, so that h (x) - q9(x) ~~ q)(Xn), we obtain that the function h verifies (,). Since A is a complete metric space, G - On>~l L/n is a dense G~-subset of A, by the Baire category theorem. We claim now that if g ~ G, then q) - g attains a strong minimum on X. Indeed for each n ~> 1, there exists Xn E X such that

(~o - g)(xn) < inf{(cp - g)(x); d ( x , x n ) >f 1/n}.

Perturbed minimization principles and applications

399

We necessarily have that d (Xp, Xn) < 1/n for each p > n. Indeed, if not then by the definition ofxn, we have (qg-g)(xp) > (q)-g)(xn). On the other hand, we willhaved(xn,Xp) >I 1/n >~ 1/p which gives, by the definition of Xp, that (q9 - g)(xn) > (q9 - g)(xp). This is clearly a contradiction. Thus (Xn)n is a Cauchy sequence converging to some x ~ 6 X and we claim that x ~ is a strong m i n i m u m for 99 - g. Indeed, since 99 is lower semi-continuous,

(~p - g ) ( x ~ ) 0, consider a point (x0, so) 6 F such that h(xo) - so > suPF/~ -- r 2. Let k be the functional in .A1 defined by/r -- (h - r d (, x0), - 1) and let S be the slice of F given by

s - { (s, ~) e F; /,(s, ~) > ~(xo, so) - r2}. It is easy to see that the d - d i a m e t e r of S is less than r, which means that X is .A1-uniformly dentable. A similar proof works for .A2. Note that Ekeland's result would then follow from T h e o r e m 2.2 and the triangular inequality. Note also that we could have used the space .A - Lip(X) of Lipschitz functions on X as an admissible space. We shall now investigate the possibility of having other classes of functions as perturbation spaces. For simplicity, we shall only deal, in the sequel, with the case where X is a Banach space.

400

R. Deville and N. Ghoussoub

A function b ' X --+ R is said to be a bump function on X if it has a bounded and nonempty support. PROPOSITION 2.1. Let ,A be a Banach space o f continuous functions on a Banach space

X satisfying the following properties: (i) For each g 6 A, IlgllA t> Ilgll~ = sup{lg(x)]; x E x}. (ii) A is translation invariant, i.e., if g E A and x ~ X, then rxg : X --+ R given by rxg(t) = g(x + t) is in A and II~xgllA = IlgllA. (iii) A is dilation invariant, i.e., if g ~ A and ot ~ R then g~ : X --+ R given by g~ (t) = g(ott) is in A. (iv) There exists a bump function b in A. Then A is an admissible space o f perturbations f o r the space X. PROOF. According to (ii) and (iv), we can find a bump function b in A such that b(0) r 0. Using (iii) and replacing b(x) by otl b(oQx) with suitable coefficients C~l, ot2 E R, we can assume that b(0) > 0, Ilbll~ < e and b(x) - 0 whenever Ilxll ~> e. Let now ~ - (g, - 1) E A be bounded above on a closed subset F of X and let (x0, so) F be such that g(xo) - so > sup F ~ - b(0) and consider the function h(x) -- b(x - xo) and k = (g + h , - 1 ) . By (ii), h 6 A and Ilhll.a - IlbllA < e, which implies that Ilg - ~:IIA < e. On the other hand, consider the following slice of F,

S--

{

]

(x, s) 6 F; k(x, s) > sup ~ . F

It is non-empty since it contains (xo, so). On the other hand, if (x, s) 6 F and IIx - xoll ~ e, then b(x - x0) = 0 and (x, s) cannot belong to S. It follows that the d-diameter of S is less than 2e. Consequently, A is an admissible family of perturbations. E] COROLLARY 2.1 (Localization). Assume A is a Banach space o f bounded continuous functions on X satisfying conditions (i)-(iv) above. Then, f o r some constant a > O, depending only on X and A, the following holds: I f 99 : X --+ R U { + ~ } is lower semi-continuous and bounded below with D(99) ~: 0 and if Yo ~ X satisfies qg(y0) < infx q9 + ae 2 f o r some e ~ (0, 1), then there exist g ~ A and xo ~ X such that (i) Ilx0 - y0ll ~< e, (ii) Ilgll,a ~< e, (iii) q9 + g attains its minimum at xo. PROOF. We can clearly assume that there exists a bump function b in A with b(O) - 1, 0 ~ Ilbll~ - 1. Let a -- 1 / 4 M and suppose that e and Y0 are given. Define the function


401

We have qS(yo) < infx ~o - a~ 2 and qS(y) >~ infx ~o whenever Ily - yoll ~> e. From Proposition 2.1 and Theorem 2.1, we can find xo E X and k E A such that []kl[A ~< min{e/2, ae2/2} and q5 4- k attains its minimum at xo. The above conditions imply that ]Ix0 y0ll < e. Thus, the function g e ,A defined by g(x) -- - 2 a e Z b ( x--xo ----E- ) + k(x) satisfies claims (i), (ii) and (iii) of the corollary. E] REMARK 2.1. Again, we can recover Ekeland's minimization principle on Banach spaces from the (easily verifiable) fact that the space ~41 of all bounded Lipschitz functions on X equipped with the norm [

[If[lA, = sup{ If(x)l; x E X} + sup{ /

If(x) IIx -

f(y)l

; x=/=y}

yll

satisfies the conditions of Proposition 2.1 and hence it is an admissible space of perturbations. Note that, as an additional bonus, we get that the perturbation is also small in the uniform norm as well as in the Lipschitz norm. To derive an analogue of the Borwein-Preiss Theorem, we can consider the space .42 of all bounded Lipschitz functions f on X that also verify the following second order condition

I[fl[

_ supJ If(x + 2h) - 2 f ( x + h) + f ( x ) [ 9x, h E X } / h2

<ec.

The space .,2[2 equipped with the norm IIf ]IA2 = IIf [IA~ + Ilf II is also an admissible space of perturbations. Clearly, the above norms will correspond to the C 1 and C2-norms whenever the functions are differentiable. But since X is in general infinite dimensional, we have to deal with two types of difficulties: firstly, the appearance of various different and generally nonequivalent types of differentiability and secondly, the problem of admissibility of these spaces of differentiable functions which requires extra assumptions on the Banach spaces involved. We deal with some of these problems in Section 3.

2.3. Martingales and admissible cones of perturbations In this section, we briefly sketch the relationship between admissible cones of perturbations and the theory of balayage associated to such a cone. Let (s I7, P) be a probability space and let (17n)n be an increasing sequence of suba-fields of I7 (i.e., r n C 17n+1). Recall that a sequence (Fn)n of real-valued integrable random variables on s is said to be a martingale (respectively a submartingale) with respect to ( r n ) n if for each n E N, (i) Fn is 17n-measurable, and (ii) fA Fn dP -- fa Fn+l dP (respectively fa Fn dP 1. We call II II a p-norm for 0 < p ~< 1, if in addition it is p-subadditive, that is (iv) Ilxl +x211 p ~< Ilxt IIp + IIx211p f o r x l , x 2 E X. The Aoki-Rolewicz theorem asserts that every quasi-norm is equivalent to a p - n o r m for some p (0 < p ~< 1). A complete quasi-normed vector space X will be called a quasiBanach space. If the quasi-norm on X is also p-subadditive, we say that X is a p-Banach


403

space. For the basics about non-locally convex vector spaces, we refer the reader to the book [57]. Let A = {z 6 C, Izl < 1 } be the open unit disc. Denote by 0A or T the unit circle {z C; Iz[ = 1} and by z~ = A U 0A. DEFINITION 2.4. Let g : X --+ [ - c o , + c o ) be an upper semi-continuous function. It is said to be plurisubharmonic on X if for every x, y 6 X we have

g(x) ~

f0 zrg(x + ei~

dO 2re

We denote by PSH(X) the class of all such functions. LIPp(X) will denote the set of functions g on X satisfying for some K > 0, Ig(x) - g(Y)l ~< Kllx - y[I p for all x, y in X. We shall write PSHp(X) for PSH(X) A LIPp(X) and we shall equip it with the following norm Ilgllp -- max{Ig(O)l, sup{Ig(x) - g(y)l/llx - YlIP; x, y 6 X, x - r y}}. Here is the Plurisubharmonic perturbed minimization principle [47]. See also [45]. THEOREM 2.3 (Ghoussoub-Maurey). Let X be a p-Banach space for some p (0 < p 0 for each n 6 N. Modulo the appropriate hypothesis, one can show in quite a general setting that A is an admissible space of perturbations on any strict A~-subset C of K. An interesting aspect of this approach is that the perturbed minimization principle also holds not just for lower semi-continuous functions but also for strict A~-functions: i.e., those whose epigraph in K x R is a strict A~-set. See examples below. Conversely, if one starts with a general complete metric space C and a Banach space A of continuous functions on C that forms an admissible vector space of perturbations, then one can find a compactification K of C in which C sits as a strict A~-set. Again, this program is developed in its full generality in the upcoming monograph [48] and we shall only state here the linear c a s e - studied extensively in [46] - where C is a closed convex subset of a Banach space X and A is the dual space X*. The case where A is the space of weakly continuous harmonic functions was considered in [49]. In the linear case, C is assumed to be a subset of a dual space Y* (usually the double dual X** of the Banach space X containing C. A-convexity coincides then with the usual convexity, meaning that A-convex compact sets are exactly the convex weak* compact subsets of Y*. In this case, A~-sets are called w*-H~-sets (i.e., their complement is a countable union of convex weak* compact sets) since A-convex compact sets are - in this setting - just intersections of hyperplanes determined by linear functionals in Y. Here is the main result of the linear theory [46]. THEOREM 2.4 (Ghoussoub-Maurey). Let K be a w*-compact convex set in some dual Banach space Y* and let C be any strict w*-H~ -subset of K. For any strict w*-H~-function llt on C, we have: (1) The set {y ~ Y; lp + y attains its maximum on C} is dense in Y. (2) If K is w*-metrizable, then the set {y ~ Y; ~ + y exposes C from below } is a dense G~ in Y: i.e., (Y, el Hi) is an admissible space of perturbations for (C, w*). (3) If C is norm separable then the set {y ~ Y; ~p + y strongly exposes C from below} is a dense G~ in Y: i.e., (Y, ]1 I]) is an admissible space of perturbations for (C, I] I])-


405

Moreover, if in (2) and (3), ~p is assumed to be concave then the minimum is (uniquely!) attained on an extreme point o f C.

Here are a few basic examples of strict w*-H~-sets and functions. PROPOSITION 2.2. Let K be a weak*-compact convex subset o f a dual Banach space Y*. Then (1) A countable intersection o f open half-spaces in K determined by functionals in Y is a strict w*-H~-set in K. (2) I f f is a functional in Y** that is in the first Baire class f o r the weak*-topology, then it is a strict w*-H~-function on K and any half-space determined by f in K is a strict w*-H~-subset o f K. (3) If 99 is convex, w*-lower semi-continuous and norm-Lipschitz on K, then lp = -cp is a strict w*-H~-function on K. (4) The supremum o f a bounded sequence o f strict w*-H~-function on K is also a strict w*-H~ -function. (5) I f C is a strict w*-H~-set in K and if K is weak*-metrizable then any relatively weak*-closed subset D o f C is a strict w*-H~-set in K and any weak*-lower semicontinuous function on C is a strict w*-H~-function. (6) I f C is a norm separable strict w*-H~-set in K, then any norm closed subset o f C is a strict w*-H~-set in K and any norm-lower semi-continuous function on C is a strict w*-H~-function. In particular, if K is norm separable, then any norm-lower semi-continuous function on K is a strict w*-H~-function.

Finally, we have the following compactification theorem [46]. THEOREM 2.3 (Ghoussoub-Maurey). Let C be a closed convex bounded subset o f a separable Banach space X. The following are then equivalent: (i) C has the R a d o n - N i k o d y m property. (ii) (X*, II II) is an admissible space o f perturbations f o r (C, II II). (iii) There exist a separable subspace Y o f X* and an isometric embedding T : X --+ Y* such that T (C) is a strict w*-H~ in Y*.

3. Perturbed minimization and differentiability Let again q9 be a function defined on a Banach space X with values in R U { + ~ }. In principle, we are only concerned here with the concepts of Frgchet and G~teaux-differentiability. However, since the arguments in both cases are similar, we shall avoid repetition, by working with the notion of differentiability associated with any bornology on X. Recall that a bornology fl is just a class of bounded subsets of X such that the topology r/~ on X* that corresponds to the uniform convergence of linear functionals in X* on the sets of/3 defines a locally convex topology on X*. Note that if/3 is the class of all bounded subsets (respectively all singletons) of X, then r/~ coincides with the norm (respectively weak*) topology on X*.

406


DEFINITION 3.1. Say that q9 is ~-differentiable at xo ~ D ( f ) with/3-derivative qg'(xo) =

p 6 X* if for any A 6 t3, lim t -1 [qg(x0 + th) - ~p(xo) - (p, th)] - 0 t--+0

uniformly for h E A. When 13 is the class of all bounded subsets (respectively all singletons) of X, the 13differentiability coincides with the classical notion of Fr~chet-differentiability (respectively GCtteaux-differentiab ility). We shall denote by cl~(x) the space of all real valued, bounded Lipschitz and /3differentiable functions g on X such that g~:(X, II II) ~ (X*, ~ ) is continuous, endowed with the norm IIg II~ = sup { Ig (x)l; x E x } + sup { IIgt (x)ll; x E x ] = IIg II~ + IIg' II~ . We shall use the notation c l ( x ) (respectively c l ( x ) ) when we are dealing with Fr6chet (respectively G~teaux) differentiability. Analogously, we can define the spaces C~ (X) (respectively C~ (X)) equipped with the C2-norm. 3.1. Smooth minimization principles Two immediate applications of Theorem 2.1 and Proposition 2.1 are the following [24]: THEOREM 3.1 (First order smooth minimization principle). Suppose X is a Banach space on which there exists a C1-Frechet smooth (respectively, G~teaux-differentiable) and Lipschitz continuous bump function. Then C~ (X) (respectively C1F(X)) is an admissible space of perturbations: i.e., for each e > 0 and f o r each lower semi continuous and bounded below function ~p: X --+ R U {+cx~ } such that q9 is not identically equal to +c~, there exists a C 1 (respectively, GCtteaux differentiable), Lipschitz continuous function p such that: (1) IIPlI~ = sup{Ip(x)l; x E X} < E, (2) IIP'll~ = sup{llP'(x)llx*; x E X} < e, and (3) q9 + p has a strong minimum at some point xo E X. THEOREM 3.2 (Second order smooth minimization principle). Suppose X is a Banach space on which there exists a C2-Frechet smooth bump function b with Lipschitz derivative. Then C2 (X) is an admissible space of perturbations: i.e., f o r each e > 0 and f o r each lower semi continuous and bounded below function qg : X --+ R U {+cx~} such that 99 is not identically equal to +cx~, there exists a C 2-function p on X with pt Lipschitz continuous such that: (1) IIPlI~ = sup{Ip(x)l; x ~ X} < e, (2) IIP'll~ = sup{llP'(x)llx*; x E X} < e, (3) IIP"II~ = sup{llP"(x)llB(x); x ~ X} < e, and (4) q9 + p has a strong minimum at some point xo E X.


407

REMARK 3.1. It is worth comparing the above with the result of Borwein and Preiss (Theorem 1.3). Indeed, if X is a Hilbert space or more generally an LP space (1 < p < ec), then the perturbation q in Theorem 1.3 can be taken to be of the form q ( x ) = 89 - wll 2 for some w (usually not equal to re). This quadratic perturbations is more explicit than the above, however it lacks the boundedness properties of the perturbations in Theorem 3.2.

3.2. Smooth norms and smooth bumps The existence of a smooth bump function is a key hypothesis in the smooth minimization principles stated above. From now on, we shall use the following notation: (H1) There exists a C 1-smooth, Lipschitz continuous bump function b: X ~ R. (H1G) There exists a Gfiteaux-differentiable, Lipschitz continuous bump function b : X - - + R.. (H2) There exists a C2-smooth bump function b : X ~ R with Lipschitz derivative. We begin this section by pointing out that these assumptions are necessary for the validity of the smooth minimization principles. PROPOSITION 3.1. Let qg: X --+ R be a (Lipschitz continuous)function on X satisfying 99(x) > Of o r all x E X and l i m l l x l l ~ qg(x) = 0. (1) I f there exists a C 1-smooth, Lipschitz continuous function g such that 99 - g has a global minimum attained at some point xo, then X satisfies (HI). (2) I f there exists a Gfiteaux-differentiable, Lipschitz continuous function g such that 99 - g has a global minimum attained at some point xo, then X satisfies (H1G). (3) I f there exists a C2-smooth function g with Lipschitz derivative such that 99 - g has a global minimum attained at some point xo, then X satisfies (H2). Our next result, due to Leduc [59], asserts that the existence of a smooth bump function can be characterized by the existence of a smooth function looking like a norm, but not convex in general. PROPOSITION 3.2. Let X be a Banach space. The following assertions are equivalent: (i) There exists on X a Lipschitz continuous function d : X --+ R +, which is C l-smooth on X\{0}, and a constant K > O, such that [Ixll ~< d ( x ) 0. The function d : X ~ R defined by d(0) - 0 and d ( x ) - ( f + ~ b ( t x ) dt) -1 for x ~ 0 satisfies the required properties. Conversely, let r :R --+ R be a continuously differentiable function such that r = 0 on (-cx~, 1] U [3, +cx~) and r(2) ~ 0. The function b : X --+ R defined by b(x) = r ( d ( x ) ) for x E X is a continuously Fr6chet differentiable bump on X. E] An immediate application of the above proposition (and a similar proof in the case of Gfiteaux-differentiability) is that the existence of a smooth norm implies the existence of a


408

smooth bump function. That is if X has an equivalent Fr6chet-differentiable norm (respectively, G~teaux-differentiable norm), then X satisfies (HI) (respectively, (HI G)). A consequence of the above result is that every reflexive space and every space with separable dual satisfy (HI), (because these spaces admit an equivalent Frechet-differentiable norm). Finally, we note that Haydon [52,54,53] constructed an example of a Banach space which satisfies (HI) (this space even admits a C ~ - s m o o t h Lipschitz continuous bump function), but has no equivalent Fr6chet-differentiable norm, thus showing that the converse is not true in general. On the other hand, for separable Banach spaces, a converse is available: THEOREM 3.3. Let X be a separable Banach space. The following conditions are then equivalent: (1) X satisfies (H 1). (2) There exists on X an equivalent Frdchet-differentiable norm. (3) The dual X* of X is separable. PROOF. (3) implies (2) is a classical result of Kadec and Klee. Since X* is separable, we let N* (x *) --

Ilx*II+ ~

2-n dist(x *, En ),

n

where (En)n is an increasing sequence of finite dimensional subspaces such that X* = Un En. It is easy to see that N* is weak*-locally uniformly convex: that is, a sequence (Xn*)n on the N*-unit sphere of X* norm-converges to x*, whenever it weak*-converges to x* and N* (x*) - 1. The predual norm N on X is then Frdchet differentiable on X \ {0} and its derivative is (norm to norm)-continuous from X \ {0} to X*. For more details, we refer the reader to the monograph [25]. The proof that (1) implies (3) uses the smooth minimization principle, so we shall provide a sketch. Let b be a C l-smooth Lipschitz continuous bump function on X. Denote U -- {x E X; b(x) ~ 0} and define f ' U --+ R by f ( x ) = 1/b(x). We claim that f is C 1smooth on U and that {ft (x); x E U} is norm dense in X*. Indeed, fix p E X* and e > 0. The function q)" X --+ R U {+oo} defined by qg(x) = - p ( x ) + f ( x ) if x E U and qg(x) -- cx~ otherwise, is lower semi continuous on X. By the first order smooth minimization principle, there exists a C 1-function g on X such that IIg~Iloc < e and q9 + g has a minimum at some point x0 E U. We have f ' ( x o ) - p + g'(xo), hence I[f'(x0) - Pll < e which proves the claim. On the other hand, since f t is continuous and U is separable, {f~ (x); x 6 U} is separable. Together with the claim, this implies that X* is separable. D REMARK 3.2. As noted above, if X is a Banach space which admits an equivalent G~teaux-differentiable norm, then there exists on X a Lipschitzian GSteaux differentiable bump function b such that b t is norm to weak* continuous. In particular, X satisfies (H1G). The class of Banach spaces satisfying (H1 G) is very large: Banach spaces which are weakly compactly generated (in particular, all reflexive Banach spaces and all separable Banach spaces) admit an equivalent G~teaux-differentiable norm. Therefore, they satisfy (H1G). On the other hand we shall see in Section 3.3 that goc does not satisfy (H1G).


409

We conclude this section by showing that the existence of a smooth bump function implies the existence of other bump functions enjoying special properties. PROPOSITION 3.3. Let X be a Banach space on which there exists a C 1-smooth Lipschitz continuous bump function b, then: (1) There exists a C 1-smooth Lipschitz continuous bump function bl such that b~ ~ 0 and bl (x) = 1 = maxx bl f o r x in a neighbourhood o f O. (2) There exists a C 1-smooth Lipschitz continuous bump function b2 which attains its strong maximum at O. (3) I f moreover X is separable and infinite dimensional, then there exists a C l - s m o o t h bump function b3 such that b~3(X) -- X*. PROOF. (1) By translation and replacing if necessary b by - b , we can assume that b(0) > 0. Let ~0: R --+ R be a C l - s m o o t h Lipschitz continuous function such that ~p(t) = 0 if t ~ b(0)/2. The function bl = ~o o b has the required properties. (2) Apply the smooth minimization principle to a constant function to get a function g which attains a strong minimum at some point x0. Replacing b by b(. - x0), we can assume that x0 = 0. Let a > f (0) such that IIx II < 1 whenever f (x) < a. Let ~ : R --+ R be a Cl-smooth Lipschitz continuous function such that gr(t) = 0 if t/> a and 7r' (t) < 0 if t < a. The function b2 = ~ o b has the required properties. (3) The construction of b3, which is given in [2], requires a series of functions of the form x --+ cnbl (anx - Xn).Yn, where an, Cn > O, Xn C X and Yn E X* are suitably chosen. Observe here that under the assumptions of (3), there exists on X an equivalent norm I1.11 which is Frechet-differentiable on X\{0}. If b : X --+ R is of the form b(x) = qg(llx II), where q) is a C l bump function on R with support in (0, + e c ) , then, according to James theorem, the set {a.b'(x); a > O, x ~ X} coincides with X* if and only if X is reflexive. Thus the bump function b3 cannot be of this form in general. D For higher order smoothness, the situation is more delicate. Let us mention here that for 2 ~< p < + ~ , the norm of the LP spaces is C2-smooth and its derivative is Lipschitzian on the unit sphere. Therefore these spaces satisfy (H2) and the second order minimization principle can be applied to these spaces. On the other hand, if X satisfies (H2), then X is superreflexive of type 2 (see [25]). If one does not require a Lipschitz condition on the derivative, the situation is different: for instance, there exists on spaces of continuous functions on countable compact, an equivalent C ~ - s m o o t h norm [53], although these spaces are not reflexive. Actually one can even construct an equivalent analytic norm on these spaces. Finally, in spaces not containing co, the existence of a C ~ - s m o o t h bump function is equivalent to the existence of a polynomial P on X such that P (0) = 0 and P (x) >~ 1 whenever Ilx II = 1.

3.3. Generic differentiability o f convex functions We show in this section how the smooth minimization principle is an efficient tool to establish generic differentiability of convex functions. The following is a theorem of Ekeland and Lebourg [32,33,59].

410


THEOREM 3.4. A s s u m e X satisfies (HI), then it is an A s p l u n d space, i.e., every convex and continuous function ~p " X --+ R is differentiable on a dense set. PROOF. Fix xo E X and e > O. Set F ( x ) --

1 - q ) ( x ) + b2(m(x_xo) ) +cxz

if b ( M ( x - xo)) 7~ O, otherwise.

The function F is lower semi-continuous, and bounded below for M large enough. By the smooth minimization principle, there exists g" X --+ R, C 1-smooth, such that F + g attains its m i n i m u m at some point Xl where b ( M ( x l - xo)) ~ O. If M > 0 is large enough, then Ilx 1 - x0 II < e. Moreover, if we let

f (x) -- b 2 ( M ( x _ xo)) + g ( x ) ,

the function q) - f has a local m a x i m u m at x l, so q9 is superdifferentiable at the point X l and hence at each point of a dense subset D of X. Since q9 is convex and continuous, q) is subdifferentiable at every point (by the H a h n - B a n a c h theorem). Therefore, q) is differentiable at every point of D. D An immediate application of T h e o r e m 3.4 is: COROLLARY 3.1. A s s u m e that the norm o f X satisfies f o r every x ~ X,

lim sup h--,0

Ilx + h II + IIx - h II - 21Ix II

> 0.

(,)

Ilhll

Then X does not satisfy (HI) (i.e., there exists no bump b" X --+ R which is C l - s m o o t h and Lipschitz continuous).

PROOF. Indeed, a norm satisfying (,) is an example of a nowhere differentiable convex continuous function. This corollary allows to deduce that a certain number of spaces do not satisfy assumption (HI): indeed, the usual norm on the space C([0, 1]) satisfies

lim sup h~0

IIx + hll + IIx - h l l - 211xll

= 2

Ilhll

for every x 6 C ([0, 1]). Therefore C ([0, 1]) does not satisfy (HI). The same remark applies to the space g l (N) and to the space L 1([0, 1]). The fact that C ([0, 1]), g 1(N) and L 1([0, 1]) do not satisfy (H 1) also follows from Proposition 3.3, since these spaces are separable with non separable duals. D


411

PROBLEM 1. Is the converse of Theorem 3.4 true? The answer is yes for separable spaces, since the class of separable Asplund spaces and the class of separable Banach spaces satisfying (H 1) both coincide with the class of spaces with separable dual. Thus, any counterexample should be a non separable Banach space. Possible candidates for a counterexample is C ( K ) , where K is the compact space constructed by Kunen, or the long James space J(col). Here is the "Gfiteaux" counterpart of Theorem 3.4 which can be proved in a similar way. THEOREM 3.5. A s s u m e that X satisfies (H1G). Then every convex and continuous function ~p" X ---> R is G~teaux-differentiable on a dense set. Let X -- s (N) and ~0"X ---> R be the convex continuous function defined by ~o(x) -limsup ]xn]. Since for every h 6 X, ]]hl]- 1 and for every x 6 X, we have lim sup

~o(x + th) + ~p(x - th) - 2~0(x)

t-~O

= 2,

t

the function ~0 is nowhere Gfiteaux-differentiable. Therefore there exists no Lipschitz continuous bump function b" X --+ R which is Gfiteaux-differentiable at every point of s REMARK 3.3. Let X be a Banach space and let A1 be the Banach space of all bounded Lipschitz functions on X (equipped with the norm given in Remark 2.1). Recently, Bachir [4] proved that the supremum norm on the Banach space ~41 is generically Fr6chetdifferentiable on ,AI. His proof relies on the general perturbed minimization principle and on a duality result involving a new notion of conjugacy. THEOREM 3.6. A s s u m e that there exists a bump b" X --+ R which is differentiable at every point, Lipschitz continuous and such that, f o r some p ~ (1,2], sup lim sup x6X

b ( x + h) + b ( x - h) - 2b(x)

< +ec.

Ilhll p

h-+O

Then, f o r every convex and continuous function ~p" X ~ o f X such that f o r every x ~ D, ~p is differentiable and

lim sup h~O

IIx -4- hll-4-IIx - h l l -

211xll

R, there exists a dense subset G

< +o~.

IIhlIP

A consequence of this result is that on s 1 ~< q < 2, there is no bump function which is differentiable at every point, Lipschitz continuous and such that, for some p E (q, 2], sup lim sup x~X

h--+O

b ( x + h) + b ( x - h) - 2b(x)

< +oc.

]]h]lP

The following theorem follows from Theorem 3.6 with p -- 1.


412

THEOREM 3.7. Let X be a Banach space satisfying (H2) and let qg: X --+ R be a convex

continuous function. Then the set of points xo ~ X such that q9 is differentiable at xo and such that, for h small enough, Iqg(x0 + h) - qg(x0) - qg'(x0).hl ~< Cllhl] 2, is dense. In other words, a concave continuous function defined on a space satisfying (H2) has a non empty proximal subgradient on a dense set. For a study of proximal subgradients in Hilbert spaces, we refer to [ 15], and to [ 16] for a more extensive study with application to optimal control. PROBLEM 2. Does Theorem 3.7 remain true if the assumption "X is a Banach space satisfying (H2)" is replaced by the weaker hypothesis "there exists on X a C2-smooth bump function"? In particular, since there exists a C ~ - s m o o t h bump function on c0(N), is it true that a concave continuous function defined on co (N) has a non-empty proximal subgradient on a dense set? We conclude this section with an important open problem. Recall that Alexandroff's theorem states that a convex continuous function in R n has second order expansions almost everywhere, i.e., for almost every x ~ R n, there exists (p, Q) ~ R n • S(n) such that f (x + h) = f (x) + (p, h) + Q(h, h) + o(llhl12). PROBLEM 3. Does Alexandroff's theorem extend to an infinite dimensional setting? More precisely, if H is a separable Hilbert space and if f : H ~ R is a convex continuous function, does there exist (a dense set of) points x 6 H such that there exists (p, Q) 6 H x S(H) satisfying f (x + th) = f (x) + t.(p, h) + t2.Q(h, h) + o(t 2) for every h 6 H. Note that the convex continuous function f ( x ) = I[x+l[ 2 on s shows that we cannot hope to have f (x + h) = f (x) + (p, h) + Q(h, h) + o(llhll2), and the function defined by the same formula on t~2(F), with F uncountable, shows that it is necessary to assume that H is separable.

3.4. The existence of separating polynomials Let X be a real Banach space, let p >t 1 be a real number and let f be a real valued function defined on X. We say that f has a Taylor expansion of order p at the point x E X, if there is a polynomial P of degree at most n, where n = [p] is the integer part of p, verifying

[f(x + h ) -

f(x)-

P(h)[-o(IihliP).

We say that f is TP-smooth if it has a Taylor expansion of order p at every point. Note that if f is m-times Fr6chet differentiable on X, then from Taylor's theorem we have that f is T P-smooth for 1 ~~ 1 for all x in the unit sphere of X. It is known (and easy to prove) that if X admits a separating polynomial, then X admits a C ~ - s m o o t h bump function.

413


Recall that the norm II. II on a Banach space X has modulus o f convexity o f p o w e r type p if there exists a constant C > 0 such that for each e E [0, 2], 6(e) "-- inf{ 1 -

x+y 2

9 x, y ~ x

Ilxll ~ 1; Ilyll ~ 1; IIx

- y ll/> e} ~

cE p

The following result is due to Deville et al. [30]. We include its proof since it uses the linear minimization principle of Bourgain-Stegall. THEOREM 3.8. Let p >~ 1 be a real number and let X be a Banach space. Assume: (1) There exists on X a T P-smooth bump function. (2) The norm o f X is uniformly convex with modulus o f convexity o f p o w e r type p. There exists then on X a separating polynomial o f degree ~ 1 be a real number which is not an even integer. Then there is no T P-smooth bump function on L p as long as it is infinite dimensional. PROOF OF COROLLARY 3.3. Assume there exists a TP-smooth bump on Lp. Since the modulus of convexity of the norm in L p is of power type p, Theorem 3.8 shows that there is a separating polynomial P on L p of degree less or equal than p. The polynomial Q defined by Q (x) = P (x) + P ( - x ) is separating, even and of degree less or equal than the degree of P. Since p is not an even integer, the degree of Q is strictly less than p. This contradicts the fact that there exists no separating polynomial of degree < p on LP. [3

For the proof of Theorem 3.8, we shall use the following elementary lemmas from Fabian et al. [37]" LEMMA 3.1. Let 6(e) be the modulus o f convexity o f the norm I1.11of X. Let x , h ~ X and f ~ X* such that f ( x ) - ]]xl[ - ]if I ] - 1, f ( h ) - - 0 a n d e O, there exist xl, x2 E X, Pl E D - u l (Xl) and P2 e D - u 2 ( x 2 ) such that: (1) Ilxl -x211.(1 + IlPl II + liP211) < e, (2) IIel + P2 II < e, and (3) Ul(Xl) + u2(x2) < limo~o inf{ul(y) + u2(z); IlY - zll < rl} + ~. PROOF (Sketch). Let d be the Lipschitz continuous function given by Proposition 3.2. For each a > 0, define Wa :X • X --+ R by Wa(X, y) = Ul (X) + ue(y) + ad2(x - y). Wa is a lower semi continuous and bounded below function on the Banach space X x X and B(x, y) = b ( x ) b ( y ) is a Lipschitz continuous, cm-smooth bump function on X • X.


417

A c c o r d i n g to the s m o o t h m i n i m i z a t i o n principle, there exists a Lipschitz continuous, C 1s m o o t h function g" X x X --+ R such that: (a) wa 4- g has a strong m i n i m u m at s o m e point (Xa, ya) 9 X x X , (b) Ilgll~ - - s u p { l g ( x , Y)I; (x, y) 9 X x X} < l / a , (c) sup{ll ~Og (X , y)][ ~ " (X , y ) 9 X x X} < 1 / a and 0g (X, y)[] ~ ; (x, y) 9 X x X} < 1 / a (d) sup{ II~S

[]

Og D e n o t e Xl -- Xa, X2 -- Ya, Pl = - a ( d 2 ( x - Y))' -~(Xa, Ya) and p2 - a ( d 2 ( x - y ) ) ' o__~g ay (Xa, Ya). For a large enough, properties (1), (2) and (3) are satisfied. COROLLARY 4.1. Let X be a B a n a c h space satisfying ( H I ) a n d let ~p : X --+ R be a lower s e m i - c o n t i n u o u s a n d b o u n d e d below f u n c t i o n such that D(qg) ~ 0. Fix e, )~ > O, then, f o r every xo 9 X satisfying qg(xo) < infx q9 + eL, there exist Xl 9 X a n d Pl 9 D - q g ( X l ) such that IlPl Ilg* < ~ a n d []Xl - xoll
0. Define h ( x ) - ~.o(llx xoll - e) and let 0 < a < e)~o - (qg(xo) - infx qg). C h o o s e eo 9 (0, e) in such a way that, if h ( x ) < qg(xo) - e)~o + a - infx qg, then Ilx - xoll < eo. A p p l y i n g T h e o r e m 4.1 with U l - q9 and u2 = h, there exists Xl, x2 9 X and there exist p l 9 D - q g ( X l ) , P2 9 D - h ( x 2 ) such that: (a) Ilxl - x2 II < e - eo, (b) IIP~ + P2 II < Z - Zo, and (c) qg(Xl) + h(x2) < l i m o ~ o inf{qg(x) + h ( y ) ; IIx - yll < r/} + a. Since h is Lipschitz c o n t i n u o u s with Lipschitz constant )~o, we have liP211 ~< )~o. B y (b), we obtain that IlPl II < ~. Since q9 is lower s e m i - c o n t i n u o u s a n d h is Lipschitz continuous, we have

lim inf{qg(x) 4- h ( y ) ; 0----~0

IIx - yll < ~} -

inf {qg(x) 4- h ( x ) ; x 9 X}

e}. L is a closed subset of [0, 1] and the set F0 = {p 6 C([0, 1], X); PIE = )tiE} is a closed subset of the Banach space C ([0, 1], X) (endowed with its usual supremum norm) that is included in F . Finally, define I :L --+ R by I (p) -- max {(r + s

t E [0, 1]\L },

where g : X --+ R is given by the formula g(x) = max{0

; 62 -

e.dist(x, F) }.


425

The functional I is continuous, bounded below and defined on the closed subset of a Banach space. According to Ekeland's minimization principle, there exists )to 6 F0 such that: (i) I ( y o ) 0 such that the set F -- F, -- {x ~ X; Ilqg'(x)lls, ~ ~ and ~o(x) -- c} is a closed subset of X that separates a and b. By the preceding theorem, there exists x 6 F such that ~0~(x) = 0, which is a contradiction. M REMARK 5.1. There exist also non-smooth versions of the mountain pass theorem. A first result in this direction has been obtained by Chang [12]. More recently, Corvellec, de Giovanni and Marzzochi [17] and Fang [39] introduced the notion of weak slope which allows them to develop critical point theory for lower semi continuous functions.

As in minimization problems, it is possible to give second order information in the mountain pass theorem. This is done in [41,42] under some conditions of uniform continuity of the first and second order derivatives of 99. These conditions do not seem to be severe in applications. THEOREM 5.2 (Fang-Ghoussoub). Let X be a B a n a c h space a n d let qg : X --+ R be a C 2functional. Let a, b ~ X, F, c a n d F be as in Theorem 5.1. Suppose that q9t a n d r are H61der continuous on a n e i g h b o u r h o o d o f {r = c}. Then there exists a sequence (Xn) C X such that:


426

(1) limqg(Xn) = c; (2) limllqg'(xn)]l = 0 ; (3) limdist(xn, F) = O; 1 2 for every h in a subspace En of codimension at most 1. (4) ~o"(xn)(h, h) >1-~llhll

6. Applications to Hamilton-Jacobi equations in Banach spaces General first order Hamilton-Jacobi equations are of the form

H (x, u(x), Du(x)) = 0 in the stationary case, and of the form

H (t, x, u t, x), OuSt, x)) = o in the evolution case. They arise in optimal control theory and in differential games. We refer to [ 19] and [20] for a detailed discussion of these equations, and, for instance, to [6] for an introduction to the theory of Hamilton-Jacobi equations. The problem is to deal with existence and uniqueness of a global "solution" of these equations, under suitable hypothesis on the Hamiltonian H (monotonicity of H in the variable u, continuity properties of H, and in some cases, convexity properties of H in the variable Du(x)). In this section, we shall concentrate on a couple of examples that will illustrate the power of the smooth minimization principles described previously.

6.1. The maximum principle for stationary first order Hamilton-Jacobi equations Let I2 be a bounded open subset of R n and H" S2 x R n ~ R be an arbitrary continuous function ( H is called an Hamiltonian). Consider the following Hamilton-Jacobi equation: u(x) + H(x, Du(x)) = 0 u (x) -- 0

for all x 6 S2, for all x 6 OS-2.

(H J1)

We first examine the maximum principle for classical solutions. Let u ~ C ( ~ ) N C 1 (if2) be a (classical) subsolution: i.e., u(x) + H(x, Du(x)) / H(xl, P l ) - H(x2, Pl) -+- H(x2, P l ) - H(x2, P2) - e.

(1)

Since Ilxl - x 2 l l ( l l p l II + 1) ~ e, we obtain that H(xl, Pm) - H(x2, P l ) ~ 0 as e --+ 0. Since IIp~ - p2 II ~< e, we get that H (x2, Pl) - H (x2, P2) ~ 0 as e --+ 0. Plugging this information in (1), we get: inf(v - u) ~> 0.

[]

x

REMARK 6.1. It is important to have comparison theorems for semi-continuous functions rather than for uniformly continuous solutions for several reasons: This need appears in the existence results via Perron's method where discontinuous viscosity subsolutions are used in the proof as well as in optimal control with boundary conditions where the value function can be discontinuous.

6.2. The maximum principle for parabolic Hamilton-Jacobi equations Our aim here is to prove uniqueness of a b o u n d e d uniformly continuous viscosity solution u : R + • X --+ R of the following evolution equation:

ut + H(x, Ux) -- O, u(O, x) - uo(x), where u o : X ~ tinuous.

(HJ2)

R is the initial condition which is assumed bounded and uniformly con-

429


DEFINITION 6.2. Say that a function u : R + x X --+ R is a viscosity subsolution of (HJ2) if u is upper semi continuous and, for every (t, x) 6 X and every (a, p) E D + u ( t , x):

u(O, x) Y > Y0, where J is an injection into a suitable space Y0. Similarly, the surjective hull 9As"~(X, Y) consists of all operators T E ~ ( X , Y) that become a member of 9,1 by lifting the domain: X0 Q X T > Y, where Q is a surjection defined on a suitable space X0. Due to the extension and lifting properties, in the definitions above we may take Y0 - loc (I[) and X0 -- l l (]I) with an appropriate index set. It is clear that 9Areg, ~inj and 9As"~ are ideals and that ~[reg C ~inj. m

J. Diestel et al.

446

An ideal is called regular if 9,1 = p.~lreg, injective if 9,1 - - ~[inj, and surjective if 9,1 = ~sur. Every dual ideal is regular. Moreover, if 9,1 is injective, then 9,1dual is surjective; similarly, if 9.1 is surjective, then p_~dual is injective. However, in neither case can one conclude the converse. By Enflo's [46] negative solution of the approximation problem, ~ is properly contained in .~, but it is easy to see that ~ inj __ -~sur __ ~.. Obviously, f13sur - ~ . Moreover, Weis [189] proved that ~5 s"r - ~inj_ ~ [ 1 . Injectivity and surjectivity of 9,1 imply that A, the associated class of Banach spaces, is stable w h e n passing to subspaces and quotients, respectively. In case of symmetry, X E A and X* 6 A are equivalent. 1.20.

Properties of ideals considered so far are listed in Table 1.

Table 1 Ideal

Symmetric

Regular

Injective

Surjective

Factorization property

Idempotent

yes yes no yes no no no no no no

yes yes yes yes yes yes yes no yes yes

no yes yes yes no yes yes no yes yes

no yes no yes no yes no yes yes no

no no no yes ??? yes no ??? yes no

yes yes no yes ??? yes no ??? yes no

J~ ~9" ~gt ~3~ ~; ~I1 ~5c0

1.21. Let H denote an infinite dimensional Hilbert space and (. I') its inner product. With e 6 H we associate the functional e* "x --+ (xle). Every operator T E ~ ( H ) admits a Schmidt representation oo

T--

men|

n,

(1.21a)

n=l

where (en) and (fn) are suitable orthonormal sequences and t = (rn) is a null sequence. We m a y arrange by simple manipulations that r l / > 1:2 ~> . . . >/0, and then the coefficients rn are uniquely determined. Indeed, (1.21 a) implies that Ten = rn fn and T* fn = rnen where, deviating from our usual notation, T* stands for the Hilbert space adjoint of T. r nen, and rn is the nth eigenvalue of the positive operator ~/T--;T. We Hence T* Ten 2 refer to Sn (T) := rn as the nth s-number (singular number) of T. Italics indicate that lp and co are viewed as Banach spaces in the usual sense. However, in order to stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. These are permutation invariant (rearrangement invariant) -

-

Operator ideals

447

ideals a in the ring of bounded scalar sequences. That is, in addition to the ideal property we require that (rk) E a implies (rrr(k)) E a for every permutation Jr of N. Obviously, there is no other proper and closed sequence ideal in lec than co. Thus we have a counterpart of Calkin's theorem, which asserts that t3(H) contains one and only one proper and closed operator ideal: m

~(H)--.~(H)--93(H) ....

.

Consequently, proper ideals a and 9.1(H) are contained in co and . ~ ( H ) , respectively. Given any proper sequence ideal a, we consider the collection of all operators T E .~(H) having a Schmidt representation with t E a. This process yields a proper ideal in t2(H). Conversely, with every proper ideal 9.1(H) in s we may associate the set of all sequences t = (rn) such that the operator defined by (1.21a) belongs to 9.1(H). Clearly, this property does not depend on the special choice of the underlying orthonormal sequences. Thus we get a one-to-one correspondence: a +-~ 9.1(H). In particular, c0 +-~ .~(H). In this way, ideal theory in ~ ( H ) is reduced to the theory of permutation invariant ideals in [oc. This may be considered as a significant simplification, since a non-commutative situation is turned into a commutative one. Unfortunately, sequence ideals exist in abundance. For our purposes, the ideals [p and co are most important. The above procedure associates with [p the Schatten-von Neumann ideal ~ p(H). Given any Schmidt representation of r r (~p(H), put tip(T) := Iltllp II and note that this quantity is well-defined. It takes some doing to show that tip defines a norm when 1 a2(T)/> . . . / > 0. Occasionally, we will write an(T" X --+ Y) instead of an(T).

Operator ideals

453

3.2. Precise computation of approximation numbers is rare; however, for some diagonal operators Dt(~k) -- (r~k) this can be achieved. Let 1 ~ 0 such that [trace(ST) I ~< cllSI] for S 6 ~(Y, X). The 1-integral operators form a Banach ideal 31 with the ideal norm ll (T) := infc. Although the inequality [trace(ST)l ~< c[ISII looks much simpler than the inequality Itrace(ASoBT)[ /0 and a (regular) Borel probability # on B x , such that

IlZxll ~ 0 for which such a measure can be found. In order to make the support of/z small, we may pass to any weak* compact subset W of B x , such that Ilx[I - SUpx,cW [(x,x*)l. For example, if X is C ( K ) , then the set of all Dirac measures 6~" f ~ f ( ~ ) with ~ E K has this property. For 1 ~Y

> loc(Br*) (5.3b)

C(Bx,)

I

> L p ( B x , , Ix)

Indeed, let A take x to (x, .), viewed as a continuous function on B x , , while J sends y to ((y, y*)), viewed as a bounded family on Br,. Moreover, consider the canonical map I " C ( B x , ) --+ L p ( B x , , lz). Then inequality (5.3a) means that []JTxl[ L > Y**, where L is a suitable Banach lattice. Passing from L to L**, we may even suppose that the lattice is order complete. The class of these operators, denoted by s is a maximal Banach ideal with respect to the ideal norm Lust(T) := inf IIBII IIA II. As usual, the infimum ranges over all possible decompositions as described above. This theory goes back to Gordon and Lewis [56]. Solving a problem raised by Grothendieck [60, p. 72], they proved that 1-summing operators need not factor through L1. That its, ~31 ~ s A consequence of this result will be discussed in 12.10. On the other hand, we have gtl o s C ~1. 6.7. Banach spaces belonging to Lust are said to have local unconditional structure in the sense of Gordon and Lewis. This term is justified by the fact that, from the local point of view, those spaces behave just like Banach spaces with an unconditional basis. The most prominent examples of spaces X ~ Lust are the disk algebra [124, p. 25] and the Schattenvon Neumann classes ~5p(12) with p r 2; see [56] and [98].

7. Grothendieck's theorem Standard references: [32, Section 14], [35], [135, Chapter 8], [148, Chapter 5]. 7.1. Henceforth, Lp denotes any member of Lp. In particular, we may think of a classical space Lp(M, #). By defnition, the C(K)'s are included in t_~. Let 9,1 and ~3 be maximal quasi-Banach ideals. Then for all Lp and Lq, we have P,.l(Lp, Lq) c_ ~ ( L p , Lq) if and only if P,.[(lp, lq) c ~ ( l p , lq). This, in turn, occurs pren cisely when fl(T) 1 only

7.2.

depends on ot and ft.

Operator ideals

467

By far, the most striking result of the theory of operator ideals is Grothendieck's theorem, which he obtained as a corollary of his thdorkme fondamental de la thdorie mdtrique de produits tensoriels [60, p. 59]. It can be stated in various ways. The usual 7.3.

form is

t2(L1, L2) = gj31(L1, L2). We prefer, however, to work with inequalities, since this point of view yields additional information about the involved constants: (G1)

~ l ( T ' l 7 --~ l~) (~k) takes l l into 12. By Grothendieck's theorem, this map is 1-summing. Hence ( ~ ) e ll. Conversely, every sequence ( ~ ) E l l defines an element x -- ~ - 1 ~kx~ in 11. So all unconditional normalized bases of l l are equivalent to the standard basis.

8. Concrete operators 8.1. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal. For example, every continuous kernel K defined on the unit square induces a nuclear integral operator g(t) ~ fd K (s, t)g(t) dt from C[0, 1] to itself, but may fail to be nuclear as an operator from L2[0, 1] to itself. We are going to present a small but tasty selection of concrete operators, which make the subject of ideal theory more attractive. 8.2. The simplest examples are identity maps I'lnp --+ lq. These operators belong to t/ every quasi-Banach ideal 9.1 and o t ( I ' l p/7 ~ lq) is well-defined for the underlying ideal quasi-norm ct. The asymptotic behaviour of these quantities as n --+ oo offers valuable information, in particular, when we compare different quasi-Banach ideals. Here are two typical examples:

Vl

.

(I " lp --+ l

q) = {" 1-1/p+l/q 11

rtp(I " lq, ---->lq) ~ (n logn) 1/q

i f l f 9 g(s) -- ~

mf+

f (s - t)g(t) dt

are especially important examples of integral operators 9If f 6 C (qI'), then it follows from T f 9L l (7~) r f> C (7~) I > L1 (T) that Tf E 9tl (L1 (T)) C ~2(L1 ('IF)). For f E Cz(T) we can do better. Approximation by trigonometric polynomials gives En ( f ) ~< c n -z, which in turn implies aen(Tf" LI(T) -+ C(T)) ~ c n - A . Hence Tf E 9132 o 9.lp,c~(Ll(T)) with 1 / p - - )~. Finally, employing the theory of eigenvalue distributions [90, pp. 213-214], [137, p. 310], we arrive at a well-known result of Bernstein [ 11 ]" if f E C z (~) and 1/2 < ~. < 1, then f has an absolutely and uniformly convergent Fourier series.

8.7. Let 0" D -+ D be a analytic function on the open unit disk D. Then the composition operator

Co " f (w ) --> f o O(z) "- f (d#(z)) maps every Hardy space Hp (D) into itself. Shapiro and Taylor [ 180] were the first to find criteria that guarantee that a composition operator belongs to a given ideal. For instance, if (1 -14~1) - 1 has integrable boundary values and 1 ~< p < oo, then C 4) ~ O43p(Hp(D)). In the case 2 ~< p < oo and only in this case the condition above is also necessary. We know that Ce maps Hp (D) into H ~ (ID) if and only if ~b(II)) is contained in a disk r D with radius r less than 1. An equivalent property is the compactness of C~ : H ~ (D) --+ H ~ (D). Moreover, under this assumption, C4):Hp(D ) ~ Hp(D) is nuclear and even a member of 9.1o. Using Nevanlinna's counting function

N4)(w)'--

Z

l~

~1

for w --# 0 (0)

ZEr (tO)

Shapiro [ 178] answered the question of how much r has to compress ID into itself in order to insure that C~ compresses bounded subsets of Hp (I3) into relatively compact ones: Cr ~ ~(Hp(]D)) r

lim Nr Iwl~ 1 log

1

Iwl

=0.

Operator ideals

471

We stress that the right-hand condition is independent of 1 ~< p < ~ . In the case of H2 (D), this criterion was extended by Luecking and Zhu [107]: dA(w)

Ccb E ~r(H2(D)) ~ fD[Ncb(t~ r/2

(1 - Iwl2) 2

log ~

<ex~

if0~ 0 such that inequality (9.1a) holds for p -- 2 and fixed n. Then

pr~ n) (T) >Ilxll for all x ~ M,,. In the setting of spaces, X r FIT means that X contains the l~ 'S uniformly. Spaces in FIT are also referred to as B-convex; see also 11.5. On the other hand, we conclude from 1.18 that an operator T E s Y) does not belong to O lt if and only if there exist a constant c >t 0 and a sequence of elements x t, x2 . . . . ~ X such that oo

k=l

OQ

~< ~-~ I~kl ~ 0 such that

_ ilyllP)1/p

cllx II for all x 6 X and y 6 Y.

Y) is called uniformly q-convex if there exists a

x++x_

T x + - T x - II 0 there exists 6 > 0 such that IlY+ Txll + I l Y - Txll

- 1 ~ EIIxll

whenever IlYll- 1 and Ilxll ~ ~.

An operator T 6 ~ ( X , Y) is called uniformly convex if for every e > 0 there exists 6 > 0 such that

Tx+2

ii~<e

whenever

Ilxi II -

1 and

2 x++x ii> ~ 1 - 3 .

10.7. The operators just defined form one-sided ideals that are related to the Haar type and cotype ideals in the following way: 9 T ~ 9A~Ep(X, Y) if and only if Y admits a renorming such that T becomes uniformly p-smooth.

Operator ideals

479

9 T E PA~q(X, Y) if and only if X admits a renorming such that T becomes uniformly q-convex. These criteria remain true in the limiting cases as well. 10.8.

The following examples should be compared with (9.4a) and (9.4b).

Lp E ATp if 1 ~< p ~< 2

and

Lq EAT2 if2~ 0 such that

( io"

11 Z Txk c o s kt k=l

' )"' ( i0 • dt

~< c

rr

1 --

2

1/2

xk sin kt

k=l

for n e N and Xl . . . . . Xn e X. Again the smallest constant provides an equivalent ideal norm. The same ideal can be obtained by working, instead of the periodic Hilbert transform, with the Hilbert transform on the real line, some discrete versions thereof, or the related Riesz projections. If 1 < p < ex~, then L p 6 HT, but l l, l ~ ~ H'I'. The remarkable Burkholder-Bourgain theorem [20,14] asserts that UMD = HT. We do 9

not know whether this result extends to operators: JI93tX) -- ,9'E. 11.5.

Khintchine's inequality implies that the Rademacherprojection

~" f (t) w-> Z

(fo'

f (t)rk(t) d t ) rk(s)

k=l

maps L p[O, 1) into itself whenever 1 < p < ex~. An operator T e ~ ( X , Y) is compatible with the Rademacherprojection if there exists a continuous extension [7~, T]. The injective, surjective, symmetric and maximal Banach ideal 9~t~13 so obtained does not depend on the choice of p. All ideal norms prtp(T) := [117~, T]:[Lp[O, 1), X] --+ [Lp[0, 1), Y][[ are equivalent, but non-normalized for p # 2. A local characterization reads as follows: an operator T e s Y) belongs to 9~r if and only if there exists a constant c ~> 0 such that

(s0 s

k=l

)

T fn(t)rk(t) dt rk(s)

as

~< c

IIf~ (t) llp dt

for n 6 N and all X-valued Haar polynomials fn; see 10.1. Then the smallest admissible constant coincides with the ideal norm prrp defined above.

Operator ideals

483

Let us return to the relation r_ __p ual C 91ffp, stated in 9.4. Though we do not have equality, there is something like a reverse: 91~dual o 9 1 ~ C 91(~'.p. Moreover, 91gt C 91~. In the setting of spaces, we even have FIP = FIT; a formula which is usually phrased by saying that K-convexity and B-convexity are the same; see [ 145] and 9.9. 11.6. Next, we look for a vector-valued version of the Riesz representation theorem. Given a compact Hausdorff space K and a Banach space X, we want to represent operators A : C ( K ) --+ X by integration against X-valued measures. There are a few problems in the general case, but weak compactness of A is characterized by the existence of a regular X-valued Borel measure m on K such that

A f -- s

f (~)dm(~)

for all f 9 C(K).

Moreover,

A 9 gt 1(C(K), X) ~ m has a finite variation Iml, A 9 011 (C(K), X) ~ m has a derivative with respect to [ml. The latter condition means that there is a function a 9 [L 1(K, #), X] such that

m(B) = fB a(~) dlml(~)

for all Borel sets B in K.

An operator T 9 t2(X, Y) is said to have the Radon-Nikodym property if A 9 ~ 1 (C[0, 1], X) implies TA 9 011 (C[0, 1], Y). The same holds then with any C(K) instead of C[0, 1]. A large variety of equivalent properties is known; see [37, pp. 217-219]. For example, T f T carries absolutely continuous functions f :[0, 1] --+ X to functions g : [ 0 , 1] >X >Y that are differentiable almost everywhere. In this case, we have b

g(b) - g(a) --

fa

g' (t) dt

for0 ~< a 0 there exists T 6 ~ ( X ) such that mix T x II ~< e for x E K. 9 The natural map from X* ~ X into X* ~ e X or s is one-to-one. 9 The natural map from Y* ~ X into Y* ~ e X or s X) is one-to-one for all Y E k. 9 vl (T) -- v~ (T) for all T ~ ~ ( X ) . 9 Vl (T) -- v~ (T) for all T E ~(Y, X) and all Y 6 L. 9 The functional T ~-~ trace(T), defined for finite rank operators, admits a continuous extension to 9 t 1(X). 9 ~(Y, X) - .~(Y, X) for all Y ~ L. It seems to be open whether we may add: ~ ( X ) - ~ ( X ) . A Banach space X has the metric approximation property if, in the first condition of the above list, T 6 ~ ( X ) can be chosen such that IJTJl ~< 1. Now we are able to describe precisely the relationship between t l, vl and v 1o .

489

Operator ideals

t l (T) = P l (T) for all T 9 ~(Y, X) and all Y 9 k if X is reflexive. 9 Vl (T) - v~ (T) for all T 9 ~(Y, X) and all Y 9 k if and only if X has the approximation property. 9 t l (T) -- v~ (T) for all T 9 ~(Y, X) and all Y 9 k if and only if X has the metric approximation property. Approximation properties were introduced by Grothendieck in 1955. He also discovered various reformulations. For a long time many experts believed that all Banach spaces would have this property. So it occurred as a big surprise when Enflo [46] constructed a counterexample. Later, Szankowski [183] showed that even ~(12) fails this property. 9

12.9. In general, the natural map from X | Y to X | Y is neither one-to-one nor onto. Grothendieck conjectured that isomorphy only occurs when at least one of the spaces is finite dimensional [59, Chapter II, p. 136], [60, p. 74]. But this is not the case! Pisier [146] constructed an infinite dimensional space X such that X | X - X | X, algebraically and topologically. In addition, he arranged X and X* to have Rademacher cotype 2. Translating this result in the language of ideals yields ~ ( X * , X) -- 9tl (X*, X), and John [78] observed that .~(X, X * ) - 9tl (X, X*). Moreover, ~ ( X ) - 9tl (X). Loosely speaking, X admits only very few operators. It is still open if .~(X) -- 9tl (X) can be true for some infinite dimensional Banach space X. 12.10. Taking Grothendieck's point of view, the following considerations are carried out for operators acting between finite dimensional spaces. Given ideal norms ot and /3, the symbol ot -< 13 means that r ~< c f l ( T ) for all T 9 t2(E, F), where the constant c >~ 1 does not depend on E 9 F and F 9 F. If ot -1~

To> F, n~>rank(T)},

To> 11n

B

> F, n ~> rank(T) }.

Taking the operator norm as a starting point, Grothendieck observed that there exists a minimal set of 14 n a t u r a l ideal n o r m s , which is stable with respect to ot ~ ot dual . . . . .

490 ot ~

J. Diestel et al. ot lift provided that we identify uniformly equivalent members. Thanks to the relations

(oldual)SUr__ (olinj)dual,

oldual) dual -- Ol,

(oladj)ext-- (olinj) adj,

(oladj) adj = ~,

(olSUr)sUr--otSUr '

(olinj) inj = otinj,

(~dual)lift

(Olext)dual,

(oladj)lift--- (olsur) adj, (Olext)ext-.-Olext '

(~lift)lift--ollift '

it suffices to check the invariance under a ~-+ Otdual, Oll----> Otadj and a ~ ot inj. In the following diagram the arrows point from the smaller ideal norms to the larger ones, in the sense of the preodering - O, and for every non-negative measurable function g on

(0, cx~), we have g(u) du

t -~-1 dt

e]]F]]p,q) + Z

i--n+l

n+N

n+N

g ( e / N ) ) + ~ i=n+l

m([hi] > e[]F]]p,q)

i--n+l

~i

i=n+l

(since e II F II p,q >/ gi)

0 and set m = [nr The theorem is equivalent to the statement (which we shall prove) that the spaces s contain uniformly complemented subspaces that Set v = ~ i =mnl i - 1 / P e i and, for 1 ~< j 0

The associate space (L4~)I is the collection of measurable functions g for which the Orlicz norm

IIg II~ = sup

If0

Ifgl dx: IIf I1~ ~ 1

/

(31)

is finite.

THEOREM 8. L V, = (L4)' and

Ilgllr ~ Ilgll~ ~ 211gll~ for all g ~ L~v. PROOF. To prove the right-hand inequality, suppose that Ilfll~ ~ 1 and Ilgll~ ~ 1 (so that M ~ ( l f l ) ~< 1 and Mo(lgl) ~< 1). Applying Young's inequality, with s = If(x)l and

Special Banach lattices and their applications

513

t = Ig(x)l, and then integrating yields

f

oo Ifgl dx O : i n f qb()~)tp > 0 } . k,t/> 1 q~()~t)

(33) (34)

Note that ot~ ~ 0 :

sup ~b()~t) < ec ] 9 0 0. It is easily checked that A0 is a separable complete metric linear space with metric d ( f , g) =

sup m(E)=l

I f - gl dx. 1 + I f -- gl

The topology induced by this metric is the topology of convergence in measure on (0, cx~). Let L0 denote the closed subspace of A0 consisting of all measurable functions supported on [0, 1]. Recall that a quasi-norm II 9II on a vector space x is a p - n o r m (0 0 such that m{ITxl > tllxll} < c(t) for all x ~ X, where c(t) --+ 0 as t --+ oo. Observe that if (ei) are independent Bernoulli random variables (defined on a probability space (I2, r , P)) taking the values 4-1 with equal probability, then by symmetry and convexity P (I ~-~in= 1 ai eil >~max lai I) ~> 1/2 for all real scalars (ai). Let (xi) be a sequence in X. Then

m maxlTxil > t

I]xill p i=1

oo

E

siTxi

> t

i=1

2E(m(E

Ilxi

IIp

i=1

oo

i=1

siTxi

>t

6iXi i--1

517


Given a > 0, select to such that 2c(t0) < e. Then

m m a x l r x i l > to

Ilxill p

(40)

< e.

i=1

Now choose a disjoint sequence (An) of sets of positive measure and a corresponding sequence of unit vectors (Xn) which are maximal with respect to the conditions An {ITxnl >/to/m(An) l/p} for each n >/1. Then maxn>~l IT(m(An)l/Pxn)[ >/to on Un~>l An and 0 0, and observe that

m

(5)

anFn ) t =

n=l

5

n=l

dF(t/lan)--

5

4)(lanl/t)

n=l

for all t > 0. It will follow that the Orlicz sequence space ( ~ , II 9I10~)is a p-convex quasiBanach space isomorphic to [Xn], and then from the non-locally convex version of Theorem 12 (see [75]) that [Xn] contains a subspace isomorphic to ~r for some r ~> p. D

Notes.

For a proof of Theorem 12 see the texts [97] and [98] (as well as [96]). The set of p's such that g p embeds isomorphically into the Orlicz function space Lq~(0, 1) is determined in [63]. The corresponding problem for the function space L~(0, ec) is solved in [112] and [63]. A characterization of the Orlicz sequence spaces g0 which embed isomorphically into g~ is given in [95], where the problem of complemented g~p'S is also considered. An example is given there of a reflexive Orlicz sequence space which does not contain any gp as a complemented subspace (see also [97] and [98]). See [61-63] (and [60] for examples of Orlicz function spaces L~(0, 1) without any complemented gp) for results on complemented s in Orlicz function spaces. Kalton [75] extended Theorem 12 to the non-locally convex and the non-locally bounded cases and showed that the theory for complemented subspaces is significantly different in the non-locally convex setting. The proof given here of the Nikishin Factorization Theorem is modelled on the proofs in in [76] and [136]. The assumption that X be a p-Banach space can be relaxed (see [76]) to Rademacher type p (0 < p ~< 2). The Lorentz spaces Lp, l(#) and Lp,cc(#) also arise in the factorization theory of absolutely summing operators: Pisier [ 119] (see also the text [40]) proved that a (p, 1)-summing operator from a C(K) space into a Banach space admits a factorization (analogous to the Pietsch Factorization Theorem) through an Lp,1 (/z) space. See [44] for a recent application of this result, where it is used to show that cotype characterizes the H61der-continuity properties of the indefinite Pettis integral.


519

See the text [57] for the theory of stable Banach spaces and for the proof of the following refinement of Aldous' theorem: a closed subspace X of L1 contains almost isometric copies of gp, where p is the infimum of the type interval for X. Garling [55] proved that L 4,(0, 1) is a stable Banach space if 4~ satisfies A ~ . His arguments readily extend to cover the quasi-Banach case of closed subspaces of L p, for 0 1) are stable. It follows that every closed subspace of these spaces contains almost isometric copies of ~r for some r. See [55] for a representation theorem for the space of 'types' on L~, and [9] for types on Orlicz and Lorentz sequence spaces. See [8] for results on the stability of quotient spaces and of interpolation spaces. Proposition 3 and Theorem 14 are adapted from [49].

6. Some probabilistic applications One area of analysis in which Orlicz and Lorentz norms arise frequently is probability theory. Indeed, a useful way of studying the integrability of a random series is to determine its closed linear span in an appropriate function space. In this section we present two important results of this type. Khintchine's inequality states that a convergent Bernoulli series has finite moments of all orders, and that the closed linear span in Lp (0 < p < oe) of a Bernoulli sequence is isomorphic to g2. The Orlicz spaces L~q, where ~q(t) - - e ta -- 1, may be used to study the exponential integrability properties of a Bernoulli series. The following result of Rodin and Semenov [ 130] identifies the closed linear span of a Bernoulli sequence in L~q (q > 2) with the Lorentz sequence space g~p,~, where 1/p + 1/q -- 1. THEOREM 15. Let (en) be a sequence of independent Bernoulli random variables defined on a probability space (S-2, r , P). Then, for each q > 2, there exist constants C1, C2 such that

1 - - II(a,)llp,~ C1

C21l(an)llp,~

~-~ an En

(41)

~q

n=l

for all scalars (an), where 1 / p + 1/q -- 1. PROOF. First we prove a fundamental estimate, known as Hoeffding's inequality [67], for the distribution function of a convergent Bernoulli series: L

P

anen > t

~ t

0, where Cq depends only on q. First, we may assume that (lan I) is a decreasing sequence. Secondly, it suffices to prove (43) for all t > 2q. Let m be the largest integer such that t > 2qm 1/q + 1. Then

•

m

(3o

m

(3O

an 6n

Z

n--1

n=l

n=m+l

n=l

an 8n

n=m+l

o(3

2qm 1/q + 1) P

anen > t

qml/q n=m+l

~< 2exp

2}~n~

(44)

"

But o 2) [131]. First let us introduce the Orlicz spaces Yp =- L49p (0, cx3) (0 < p < ~ ) , where

dpp(t)

- / t2

I tp

for 0 2 there is a short elementary proof (essentially the proof given by Rosenthal [131 ]), which we present. Making the substitutions gn = f 2 and q = p / 2 (so that q > 1), (45) will follow from

~gi i=1

~< max 2 q Z i--1

Ilgi II1,2

I[gi IIq i=1

9

(46)

522

S.J. Dilworth

(The reverse inequality (with 2 replaced by 1) does not require independence but just follows from the positivity of the gi's and an application of H61der's inequality.) To prove (46) observe that ~< 2 q-1 ( f i gq -+- f i ( j ~ / i--1 i=1

( f i ) q gi i=1

gj )q-1 gi ) 9

So by independence of the gi's, we have

~< 2 q-1 ( f i Ilgi IIq + f i ( f i e gi i=1 q i=1 i--1 ~ 2q-1

("

~ I[gi IIq + i=l

Ilgilll j~/ gj -' 9 " qq-1 Ilgi Ill

i=1

t)

1 i:, I11_,t

(by H61der's inequality and the positivity of the gi's) ~< 2 q max

Ilgi IIq,

i=1

Ilgi II1

figi i=1

i=1

q-l), q

which yields (46). The proof in the range 0 < p < 2 requires more elaborate arguments for which the reader should consult [72]. [~ Using the fact (due to Marcinkiewicz and Zygmund [105]) that the norm in L p of a sum of independent mean zero random variables is equivalent to the norm of the square function, we can reformulate the previous theorem. COROLLARY 4. Let (Xn) be a sequence of independent mean zero random variables. Then, f o r 1 2) (first proved to exhibit new complemented subspaces of L p) in some sense characterizes L p. See [14] for a precise formulation of this fact and for further results on the linear span of independent random variables in rearrangement-invariant function spaces. See [71] and the text [114] for martingale inequalities in rearrangement-invariant spaces. The Orlicz space Yp is isomorphic to L p in the range 1 < p < cx~ [70] (and so L p has two representations as a rearrangement-invariant function space on (0, ec)). The 'other' scale of sums and intersections of Lp spaces, Mp = Lp(O, cx~)71L2(0, oo) (respectively Lp(O, oo) -k- L2(0, ~ ) ) for 0 < p < 2 (respectively 2 < p < ec), is closely related to (though isomorphically distinct from) the L p scale. Several Banach space properties of these spaces and of the L2(s spaces are obtained in [41,42], e.g., that there is no isomorphic embedding of L2(~p) into g2(gp) for p r 2. For the structure (e.g., containment of gr(gs) subspaces) of the 'mixed norm' spaces Lp(Lq) see the papers of Raynaud [ 122,123,127].

S.J. Dilworth

524

7. Embedding

Lw,q into Lq

In his seminal work [7] Banach discussed the question of the 'linear dimension' of the classical Banach spaces, that is, the question of the existence (or non-existence) of an isometric or an isomorphic embedding from one space into another. A famous question left open in [7] for many decades concerned the existence of an isomorphic embedding from ~q into Lp for 1 ~< p < q < 2. Surprisingly, the solution to this problem turned out to be probabilistic in nature: using stable processes Bretagnolle, Dacunha-Castelle and Krivine [16] proved that it is even possible to embed Lq isometrically into L p in the range 0 < p < q ~< 2 (cf. [82] for a self-contained proof). In this section we shall consider the question of embedding Lorentz spaces into Lq. It is appropriate to study this question not just for the L p,q spaces but for the wider class of Lw,q spaces, which will now be introduced. Let I denote the interval (0, 1) or the interval (0, cx~), and let w(t) be a positive decreasing weight function defined on I for which fd w(t) d t = 1 and f o w(t) dt = cx~ (the latter condition only for I = (0, cxz)). For 0 < q < c~, the Lorentz function space Lw,q(I) is the space of equivalence classes of real-valued measurable functions f on I for which the following norm (or q-norm if q < 1)

Ilf llw,q -

(f,

f * (t)q w(t) dt

)"q q

1

is finite. For 0 < q ~ 1, the condition that w should be decreasing guarantees (and, in fact, is equivalent to) the triangle inequality in Lw,q [ 101 ]. Given the existence of the stable embeddings of L p into Lq, a natural question to consider is the possibility of an isometric embedding of Lw,q (I) into Lq. It turns out that such embeddings exist for only a very restrictive class of weights [46]. THEOREM 17. Let 0 < q < oo and let I be either (0, 1) or (0, c~). There exists an isometric embedding of Lw,q (I) into Lq if and only if one of the following conditions is satisfied: (a) w(t) -- 1, in which case Lw,q (I) -- Lq (I) with equality of norms; (b) I = (0, 1), 0 < q ~< 1, and w(t) is a decreasing linear weight; that is, there exists ot E [0, 2] such that

w(t) =_ w~(t) = 1 + - -t~t. 2 The case q > 1 is easy: the space Lq is smooth, but Lw,q is not, unless w(t) ---- 1. For 0 < q < 1, the proof of Theorem 17 is quite technical as it uses the theory of Fourier transforms of distributions. For q = 1, however, a short geometrical argument can be given for the main step of the proof, which concerns finite-dimensional Lorentz sequence spaces. For a l >~ ... >~an >~0 (not all zero) and q > 0, the expression

II

.....

II -- (a' Cx )" +""

+

'/"


525

is the norm (or q-norm if q < 1) of an n-dimensional Lorentz space denoted ~n,q. For an infinite decreasing sequence w -- (an) of positive weights, for which ~-~n~=l an -- co, the Lorentz sequence space g w,q is defined similarly. PROPOSITION 4. The s p a c e g.n,1 (n ~ 2) is i s o m e t r i c to a s u b s p a c e o f L1 i f a n d only i f al - a2 -- a2 - a3 . . . . .

an-1 - an.

(47)

PROOF. First we prove by induction that ~n,1 is isometric to a subspace of L1 only if the weights ak satisfy (47). Let E be a Lorentz space of dimension n + 1 with weights a 1. . . . . an+ 1. If E is isometric to a subspace of L 1 then (47) holds by hypothesis, and so it suffices to prove that an-1 - an = an - an+l to complete the induction. In particular, we may assume that a n - l , an and an+l are not all equal. Let B* denote the unit ball of E*. Since the unit ball of E is a polytope, and since E is isometric to a subspace of L1, it follows (see, e.g., [12]) that B* is a z o n o t o p e (that is, a Minkowski sum of line segments). By [12, Theorem 3.3], all of the two-dimensional faces of B* are centrally symmetric. It is easily seen that the extreme points of B* are all the sign-changed permutations of the vector a = (al, a2 . . . . . an+l). In particular, one of the two-dimensional faces of B* has as its vertices all of the vectors obtained by permuting the last three coordinates of a. If an-t = an or if an = an+l, then this face is triangular, which contradicts the central symmetry requirement. If an-1 > an > an+l, then the face is h e x a g o n a l , and the symmetry condition forces an-1 - an = an - an+l as required. Thus (47) is a necessary condition. To show that (47) is also sufficient, one can check that if (47) is satisfied then B* has four kinds of two-dimensional faces: two classes of quadrilateral faces, one class of octagonal faces, and one class of hexagonal faces like the face described above. The first three kinds of faces are automatically centrally symmetric without any condition on the weights, while (47) guarantees that the hexagonal faces are also centrally symmetric. So, by [12, Theorem 3.3] once again, if (47) is satisfied, then B* is a zonotope, and hence E is isometric to a subspace of L1. D The question of embedding Lw,q isomorphically into Lq is answered by the following theorem of Schtitt [132]. THEOREM 18. L e t 1 1). Allen [4] obtained a simple description of the dual space for regular weight sequences. See [34] for proofs of the Kadets-Klee property and local uniform convexity of gw,p. The symmetric basic sequences in gw,p are characterized in [6]. Necessary and sufficient conditions are given for g w,p to have exactly two nonequivalent symmetric bases and an example is given of a subspace of a Lorentz sequence space with a symmetric basis which is not itself isomorphic to a Lorentz sequence space. Symmetric basic sequences in the dual of gw,p are considered in [33]. Bretagnolle and Dacunha-Castelle [ 17] proved that Lr is isomorphic to a subspace of L 1 if and only if 4) is equivalent to a 2-concave convex 4)-function (recall that 7t is 2-concave if g r ( ~ ) is concave), which is the Orlicz space analogue of Theorem 18. See [133] for an explicit embedding of a given 2-concave finite-dimensional Orlicz sequence space into L 1. The proof utilizes some combinatorial and probabilistic inequalities [84,85] which have also been used to compute projection constants for finite-dimensional Orlicz and Lorentz spaces [84]. Hermindez and Ruiz [65] proved that, for 0 < p < q < cx~, every Orlicz space Lr (0, cx)) whose Boyd index reciprocals lie strictly between p and q is lattice-isomorphic to a sublattice of Lp -+-Lq (see also [52]). Raynaud [123] proved that every p-convex and q-concave Orlicz space L~(0, oc) is lattice-isomorphic to a sublattice of Lp(Lq). See [128] for results on embedding symmetric spaces into L1 (gq), and [64] and [124,125] for further embedding results. A special case of a theorem of Kalton [76] asserts that the Banach space gp(gq) embeds into L0 if and only if 1 ~< p ~< q ~< 2. Briefly, let us mention a way in which Orlicz and Lorentz spaces have been integrated into a single class of spaces. Let 4) be a convex 4)-function and let w be a decreasing weight function. The Orlicz-Lorentz space A~,w(I) is the Banach space of measurable functions f on I for which

IIf II = inf, )~ > O:

f0

$(f*(t)/~.)w(t) dt

J

< e~.

See [81] for uniform convexity, [80] for extreme point structure (see also [29] for the uniqueness of the Choquet integral representation in Lw,1 (0, oc)), [79] for containment of co and g ~ , and [68] for local uniform convexity and the Kadets-Klee property (see also [ 134] for Lw, 1) of Orlicz-Lorentz spaces. See [ 109] and [ 123] for concavity properties and Boyd indices for Orlicz-Lorentz spaces.

References [1] D.J. Aldous, Subspaces of L 1 via random measures, Trans. Amer. Math. Soc. 267 (1981), 445-463.

528

S.J. Dilworth

[2] J. Alexopoulos, De La Vallde Poussin's theorem and weakly compact sets in Orlicz spaces, Quaestiones Math. 17 (1994), 231-248. [3] J. Alexopoulos, On subspaces ofnon-reflexive Orlicz spaces, Quaestiones Math. 21 (1998), 161-175. [4] G.D. Allen, Duals of Lorentz spaces, Pacific J. Math. 77 (1978), 287-291. [5] Z. Altshuler, Uniform convexity in Lorentz sequence spaces, Israel J. Math. 20 (1975), 260-274. [6] Z. Altshuler, P.G. Casazza and B.L. Lin, On symmetric basic sequences in Lorentz sequence spaces, Israel J. Math. 15 (1973), 144-155. [7] S. Banach, Th~orie des Operations Lin~aires, Warsaw (1932). [8] J. Bastero and Y. Raynaud, Quotients and interpolation spaces of stable Banach spaces, Studia Math. 93 (1989), 223-239. [9] J. Bastero and Y. Raynaud, Representing types in Orlicz and Lorentz sequence spaces, Math. Proc. Cambridge Philos. Soc. 107 (1990), 525-538. [10] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York (1988). [11] J. Bergh and J. L6fstr6m, Interpolation Spaces, Springer-Verlag, New York (1976). [12] E.D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-345. [13] D.W. Boyd, Indices for the Orlicz spaces, Pacific J. Math. 38 (1971), 315-323. [14] M.Sh. Braverman, Independent Random Variables in Rearrangement Invariant Spaces, Cambridge University Press, Cambridge (1994). [15] M. Braverman, Independent random variables in Lorentz spaces, Bull. London Math. Soc. 28 (1996), 79-87. [16] J. Bretagnolle, D. Dacunha-Castelle and J.L. Krivine, Lois stables et espaces L p, Ann. Inst. H. Poincarr, Probab. Statist. 2 (1966), 231-259. [17] J. Bretagnolle and D. Dacunha-Castelle, Applications de l'~tude de certaines formes lindaires al~atoires au plongement d'espaces de Banach dans les espaces LP, Ann. Sci. l~cole Norm. Sup. 2 (1969), 437-480. [18] D.L. Burkholder, B.J. Davis and R.E Gundy, Integral inequalities for convex functions of operators on martingales, Proc. Sixth Berkeley Symp. Math. Statist. Probab., Vol. 2, Univ. of California Press, Berkeley, CA (1972), 223-240. [ 19] N.L. Carothers, Rearrangement invariant subspaces of Lorentz function spaces, Israel J. Math. 40 (1981), 217-228. [20] N.L. Carothers, Rearrangement invariant subspaces of Lorentz function spaces, II, Rocky Mountain J. Math. 17 (1987), 607-616. [21] N.L. Carothers, Lorentz Function Spaces, Seminar Notes, Bowling Green State University (1993) (unpublished). [22] N.L. Carothers and S.J. Dilworth, Geometry of Lorentz spaces via interpolation, University of Texas Functional Analysis Seminar Longhorn Notes 1985-86, Univ. Texas, Austin (1987), 107-134. [23] N.L. Carothers and S.J. Dilworth, Inequalities for sums of independent random variables, Proc. Amer. Math. Soc. 104 (1988), 221-226. [24] N.L. Carothers and S.J. Dilworth, Subspaces of Lp,q, Proc. Amer. Math. Soc. 104 (1988), 537-545. [25] N.L. Carothers and S.J. Dilworth, Equidistributed random variables in Lp,q, J. Funct. Anal. 84 (1989), 146-159. [26] N.L. Carothers and S.J. Dilworth, Some Banach space embeddings of classical function spaces, Bull. Austral. Math. Soc. 43 (1991), 73-77. [27] N.L. Carothers, S.J. Dilworth and C.J. Lennard, On a localization of the UKK property and the fixed point property in L w, 1, Interaction Between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math. 175, Dekker, New York (1996), 111-124. [28] N.L. Carothers, S.J. Dilworth, C.J. Lennard and D.A. Trautman, A fixed-point property for the Lorentz space Lp,1 (#), Indiana Univ. Math. J. 40 (1991), 345-352. [29] N.L. Carothers, S.J. Dilworth and D.A. Trautman, On the geometry of the unit spheres of the Lorentz spaces Lw,1, Glasgow Math. J. 34 (1992), 21-25. H ot

//

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[127] Y. Raynaud, A note on symmetric basic sequences in L p ( L q ) , Math. Proc. Cambridge Philos. Soc. 112 (1992), 183-194. [128] Y. Raynaud and C. Schtitt, Some results on symmetric subspaces of L 1, Studia Math. 89 (1988), 27-35. [129] S. Reisner, A factorization theorem in Banach lattices and its application to Lorentz spaces, Ann. Inst. Fourier (Grenoble) 31 (1981), 239-255. [130] V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1 (1975), 207-222. [ 131 ] H.E Rosenthal, On the subspaces of L p (p > 2) spanned by sequences of independent random variables, Israel J. Math. 8 (1970), 273-303. [132] C. Schtitt, Lorentz spaces that are isomorphic to subspaces of L | , Trans. Amer. Math. Soc. 314 (1989), 583-595. [133] C. Schiitt, On the embedding of 2-concave Orlicz spaces into L l, Studia Math. 113 (1995), 73-80. [134] A.A. Sedaev, The H-property in symmetric spaces, Teor. Funkcii Funkcional. Anal. i Prilozen. 11 (1970), 67-80 (Russian). [135] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton ( 1971). [ 136] E Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press, Cambridge (1991).

CHAPTER

13

Some Aspects of the Invariant Subspace Problem

R Enflo and V. Lomonosov Department of Mathematics, Kent State University, Kent, OH 44242, USA E-mail: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Invariant subspaces of algebras containing compact operators . . . . . . . . . . . . . . . . . . . . . . . . 2. Generalizations of Burnside's theorem in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Generalizations of Burnside's theorem in Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

535 535 538 543

Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Extremal vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Definitions and some basic properties of extremal vectors . . . . . . . . . . . . . . . . . . . . . . .

543 545 546

4.2. Invariant subspaces and a two sequences theorem

...........................

547

5. Operators without invariant subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.0. Introduction and history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Outline of the original considerations and construction . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Remarks on some other constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. More examples of transitive operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

549 549 550 555 556

5.4. A transitive operator on el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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H A N D B O O K OF THE G E O M E T R Y OF B A N A C H SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 533


Some aspects of the invariant subspace problem

535

Introduction In this survey we present some circles of ideas and results which we feel constitute progress on the theory of invariant subspaces. We are not trying to give a complete survey of the theory as it stands today. There are many important techniques and directions - especially for operators on Hilbert s p a c e - that will not be discussed here. For some of these we refer the reader to [1,6] where also more references can be found. Behind the techniques of the first four sections in this survey is a common general idea: to find on the boundary of a convex set, points which are hard to move into the set by operators from a given s e t - and the connection between such points and invariant subspaces. In Section 1, this idea is carried out in the context of locally convex spaces. There we give results on invariant subspaces of algebras containing a compact operator. In Section 2 we consider first general Banach spaces. We present a generalization of the classical Burnside theorem for operator algebras. We also give connections to invariant subspaces of the so called two sequences theorem. So far this two sequences theorem has been proved only for Hilbert space. It should be an interesting problem to decide for which Banach spaces it is true. We discuss this further in Section 2. In Section 3 we present strengthenings of these results to operator algebras on Hilbert space and an invariant subspace theorem for essentially self-adjoint operators on real Hilbert space. The techniques in Section 3, developed by A. Simoni6 [37] - use the same underlying ideas as those in the first two chapters. However Simonic develops them in an impressive way that, for now, seems to work only in Hilbert space. In Section 4 we present techniques of extremal vectors and their connection to invariant subspaces. This techniques have, so far, been developed only in Hilbert space. However, they are, in essence, Banach space techniques, and also here, it should be interesting to find out for which Banach spaces they work. The techniques present a more constructive way to find the points which are hard to move. This leads, in particular to a strengthening of the two sequences theorem in Section 2 for some special cases. In Section 5 we present counterexamples to the invariant subspace problem. We also present results where counterexamples are used to support or disprove other conjectures.

1. Invariant subspaces of algebras containing compact operators Let E be a complex locally convex space, L(E) the algebra of continuous linear operators on E and R a sub-algebra of L(E). R is said to be transitive if E does not contain nontrivial closed subspaces that are invariant under each operator in R. A subspace M in E is said to be hyper-invariant for a subset S in L(E) if it is invariant under each operator which commutes with each operator from S. In 1935 von Neumann proved that each compact operator acting on a Hilbert space is non-transitive. In 1954 N. Aronszajn and K.T. Smith [4] generalized this statement to Banach spaces. In 1966 A.R. Bernstein and A. Robinson [10], using non-standard analysis, proved that a polynomially compact operator on a Hilbert space is non-transitive. E Halmos [21] converted their proof into one that uses only classical concepts. In 1968 W. Arveson and J. Feldman [5] proved that an algebra of operators on a Hilbert space with

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one quasi-nilpotent generator is non-transitive if it contains a compact operator. In 1973 C. Pearcy and N. Salinas [27] proved Corollary 2.2 of this paper for the case of an algebra of operators with one generator. In 1973 the second author gave a new approach to invariant subspace theorems [23]. The most general result proved by this method was the following THEOREM 1.1 ([22]). Let R be a sub-algebra of L(E). The algebra R is dense in L ( E ) with respect to the weak operator topology if and only if it is transitive and its closure in this topology contains a nonzero compact operator. PROOF OF THEOREM 1.1. LEMMA 1.2. The algebra R is transitive if and only if for each nonzero vector x E E its orbit Rx = UAER A x is dense in E and if and only if for each nonzero functional f E E* its orbit UAER A* f is dense in E* in the w*-topology. Proof of the lemma is obvious. LEMMA 1.3. If a transitive algebra R E L ( E ) contains a nonzero compact operator K, then this algebra contains a compact operator with a nonzero fixed vector. PROOF. From the definition of compact operator it follows that there exists an open convex neighborhood of zero U in E such that the closure K (U) of the set K (U) is compact. Let us pick a vector y E E such that K y ~ O. Put V -- U + y. It is clear that the vector y can be chosen in such a way that the origin is not contained in the set K ( V ) , that is ( - K y ) q~ K ( U ) . From Lemma 1.2 it follows that for any element x ~ K ( V ) there exists an operator A 6 R such that A x ~ V. Therefore the pre-images A - I ( v ) of the open set V generate a covering of the compact K (V). We can choose from that covering a finite sub-covering {Ui }, i = 1 . . . . . n. By construction there exist operators A 1. . . . . An in R such that Ai (Ui) Q V. Since a compact topological space is normal, there exists a partition of the identity corresponding to the covering {Ui }, therefore there exist continuous functions f l . . . . . fn in K ( V ) such that 3~ ~> 0, Y~in_=l j~ ~ 1, f/(x) - - 0 if x q~ Ui. Consider a map 4~" K ( V ) --+ E defined by q~(x) -- y~in 1 fi(x)Aix. The right hand side of this equality is a convex combination of elements from the convex set V, therefore 4~ (x) 6 V if x E K (V). The composition K 4~ of the maps K and 4~ is a continuous map of the convex compact K (V) into itself. By the Schauder-Tikhonov fixed point theorem the map K4~ has a fixed point, therefore there exists an element xo ~ K ( V ) such that K ~ ( x o ) = xo and Y~in=l K 3 ~ ( x o ) A i x o - xo. The operator K1 -- ~Qinl ~ ( x o ) K A i is compact, is contained in R and has the fixed vector x0. Recall that x0 7~ 0 because K (V) does not contain the zero vector. [3 C O R O L L A R Y 1.4. Let A be a linear continuous operator on a complex locally convex space E which is different from a scalar operator and commutes with a nonzero compact operator in E. Then the operator A has a nontrivial closed hyper-invariant subspace, that is a subspace invariant for all operators commuting with A.


537

This seems to be the first invariant subspace result for general locally convex spaces. PROOF. Let R be the set of all operators from L ( E ) commuting with A. Assume that R is transitive. Then according to Lemma 1.3 there exists a compact operator K E R and an element x0 6 E such that K xo = xo ~: O. Let E1 be the linear subspace of E consisting of all fixed vectors of the operator K. It is clear that E1 is a finite dimensional subspace invariant for the operator A. Therefore A has an eigenvector in E with an eigenvalue )~. Let E2 be the subspace consisting of all eigenvectors of the operator A corresponding to the eigenvalue )~. This subspace is not empty and does not coincide with E because the operator A is not scalar. It is clear that this subspace is invariant for R, so R is not transitive, ff] COROLLARY 1.5 (Schur's lemma). If a transitive algebra of linear continuous operators in a complex locally convex space contains a nonzero compact operator, then the commurant of this algebra consists of scalar operators. LEMMA 1.6. If a transitive algebra R C L ( E ) contains a nonzero finite dimensional operator K, then this algebra contains a nonzero one dimensional operator F(e ~, e) such that F(e ~, e)x -- (e',x)e, where e ~ ~ E*, e, x E E. PROOF. Put E1 -- K (E). From Lemma 1.1 it follows that vectors of form K A x , where A 6 R, generate a dense set in E1 so that operators of form KA restricted to E1 generate a transitive algebra of operators in El. According to the classical Burnside theorem this algebra coincides with L (El). Therefore there exists an operator A 6 R such that K Ax = (e', x)e for any x c El, where e 6 E l , e' 6 E*. [] LEMMA 1.7. If a transitive algebra R C L ( E ) contains a nonzero finite dimensional operator, then the closure of the algebra R in the weak operator topology coincides with L(E). PROOF. Let F(et, e) be an operator which exists by Lemma 1.6. Since AF(et, e ) = F(e', Ae) and F(e ~, e)A = F(A*e ~, e), where A* is an operator dual to A, from Lemma 1.2 it follows that the closure of the algebra in the weak operator topology contains any one dimensional operator and therefore any finite dimensional operator. Finite dimensional operators are weakly dense in L ( E ) , so the lemma is proved. [-q LEMMA 1.8. Let K be a compact operator in E, p ( K ) its spectral radius. If p ( K ) < 1, then the sequence {K n } of powers of the operator K converges to zero in the strong operator topology. PROOF. Let U be an absolutely convex neighborhood of zero in E which the operator K maps into a relatively compact set and let p be the Minkowsky functional corresponding to this neighborhood. Let E u be the quotient space E~ ker(p). The norm in E u is generated by the functional p. Let the Banach space E u be the completion of the space E u . The canonical map 1 7 : E --+ E u is continuous. Since the kernel of the operator K contains

538


the kernel of the map H in the space E u , the operator K 1 is well defined by the formula Kl x = H K H - 1x. The operator K1 is extended continuously to an operator K2 acting in the space Eu. It is easy to check that the operators K1 and K2 are compact and K, KI and K2 have the same nonzero eigenvalues. Therefore the spectral radii of these operators coincide. From Gelfand's spectral radius formula it follows that A

lim p ( K ~ / T x ) - -

n - + oc~

lim p ( K ~ H x ) =

n-+oo

lim

n - + cx3

p(Knx)=O

(1)

for each element x E E. For this reason Knx ~ U, Kn+lx ~ K ( U ) for sufficiently large n and the set {Knx} is relatively compact. Let S be the set of the condensation points for the sequence {Knx}. It is clear that K ( S ) -- S and that the set S is contained in the kernel of /7 since l i m n ~ p ( K n x ) = 0. This implies that S E ker(K) and consequently S = {0}. 7q LEMMA 1.9. If a transitive algebra R contains a nonzero compact operator, then it contains a sequence which converges in the strong operator topology to a nonzero finite dimensional operator PROOF. By Lemma 1.3 the algebra R contains a compact operator K with an eigenvalue

)~l -- 1. Let )~l . . . . . )~n be all eigenvalues of K which are no less then 1 by modulus, let/Zl . . . . . /Zn be the maximal dimensions of Jordan cells in the corresponding spectral invariant subspaces and let M be the linear span of those subspaces. By the Leray-Riesz spectral decomposition Theorem [15] the space E is the direct sum of M and another subspace N, which is also invariant under K, and the spectral radius of the restriction KIN is strictly less than 1. Let Q be the projection operator onto M parallel to N. Put P()~) = (~. - 1) # 1 - 1 1 - - I i n _ 2 ( ) ~ )~i) lzi. It is clear that the operator P ( K ) has a nonzero restriction P(K)]M and P ( K ) K n x -- K n P ( K ) X = P ( K ) x for each x 6 M and for all nonnegative integers n. For each x E E put x = x l + x2 for x l 6 M, x2 E N. Then -

-

P ( K ) K n x : P ( K ) K n x l -+- P ( K ) K n x 2 = P ( K ) x l + P ( K ) K n x 2 . The second term in this formula converges to zero by Lemma 1.8. So the sequence P (K) K n converges in the strong operator topology to the finite dimensional nonzero operator P ( K ) Q . It is clear that P ( K ) K n E R, which finishes the proof. D Combining Lemmas 1.7 and 1.9 completes the proof of Theorem 1.1.

2. Generalizations of Burnside's theorem in Banach spaces In 1991 Theorem 1.1 was generalized for operator algebras in the following way [25]. In Theorem 2.1 below, let B be a Banach space, L ( B ) the bounded linear operators on B and K (B) the compact operators on B. Let IIIA III denote the essential norm of A, that is the distance from A to the space of compact operators.


539

THEOREM 2.1. Let R be a weakly closed sub-algebra of L(B), R ~ L(B). Then there exists x E B** and y ~ B*, x ~ 0 and y ~ O, such that for every A E R ](x, A'y)[ ~< IIIAIII.

(1)

For a given subset S C L ( B ) and a vector x 6 B put Sx UA6S Ax and S' = U A*. We say that S is transitive if it does not have a nontrivial invariant subspace and S is essentially transitive if the conclusion of Theorem 2.1 is false for S. We say that a subalgebra R C L (B) has the Pearcy-Salinas (PS) property if there exists net {Ac~} C R and a nonzero operator A 6 L (B) such that: =

lim(x , A ~*y ) - - ( x , A * y)

(2)

Ol

for every vector x E B** and every functional y 6 B* and lim IIIAc~ III - 0.

(3)

ot

Of course, every bounded operator is a weak limit of finite dimensional ones, so it is the assumption {A~ } C R that makes the condition nontrivial. COROLLARY 2.2. Let R be a weakly closed proper sub-algebra of L ( B ) with the (PS) property. Then the algebra R ~ is non-transitive. PROOF. Let x and y be as in Theorem 2.1. Then for every pair of operators T1, T2 in R we have

I(x,

T~A*TOy)I -

liml(x,

T2A~TOy)I 0 and

Ilrxll < or. Ilxll

sup-

~L

(7)

Assume y E o-(T) and lyl > o~. Then y E CrF(T). LEMMA 2.1 2. If R is an essentially transitive sub-algebra, then R contains an operator T with thefollowingproperties" 1 E OfF(T) and ifF E cr(T)\CrF(T) then lyl < 1/2. LEMMA 2.13. Let R be a uniformly closed essentially transitive sub-algebra of L(B). Then R contains a nonzero finite dimensional projection. PROOF. Let T be an operator as in L e m m a 2.12. Since {9/" (T - F I ) - I E R} is a component of the resolvent set of T it follows that if (T - )/I)-1 exists and [VI ~> 1/2 then ( T - y i ) - I E R. By L e m m a 2.12 there exists a circle cr C C such that 1 is the only point of or(T) inside c~ and for every point y E o~, (T - y I) -1 E R. By the Riesz theorem P - -27ri 1 for (T -

V 1) -1 dy is a non-zero finite dimensional projection and P E R ~. If P -- P~ then P1 is finite dimensional projection in R. []


542

Now to finish the proof of Theorem 2.1 we need the following well-known fact [29]. LEMMA 2.14. I f a transitive algebra R contains a non-zero finite dimensional operator, then R is weakly dense in L ( B ). Essential transitivity implies transitivity, so by combining Lemmas 2.13 and 2.14 we finish the proof of Theorem 2.1. Of course Lemma 2.13 gives the possibility of obtaining different results on density. For example one of them is THEOREM 2.15. I f H is a Hilbert space and R is a uniformly closed essentially transitive sub-algebra of L ( H ) then R contains all compact operators in L ( H ) . Now we will prove the equivalence of Theorem 2.1 and Theorem 2.4 in Hilbert space. We first prove that Theorem 2.4 implies Theorem 1 if B is Hilbert space. Assume that Theorem 2.4 holds. Let C = lirl~ Ix~ I" lY~ I. If A 6 R then 0 = lim(Ax,~, y~) -- lim(A(x,~ - x), (y~ - y)) - (Ax, y) + lim(Ax, yot) 13l

lY

+ lim(Ax~, y), (y

where lim(Ax, y~) = lim(Ax~, y) = (Ax, y). ly

o/

Thus (Ax, y) = lim(A(x - x~), (y~ - y)). If K is a compact operator, then

l(Ax,

Y)I ~< li-~ I((A- g)(x -x=),

(y~ -

Y))I + l~ml(g(x -x~),

(y~ -

Y))I

I I A - KII4C. Now let z = y / ( 4 C ) . Then I(Ax, z)] ~< i~cf IIA - gll = IIIAIII, so Theorem 2.1 follows from Theorem 2.4. To prove that Theorem 2.1 implies Theorem 2.4 we use the following result of Glimm [20]. B is assumed to be Hilbert space. LEMMA 2.16. Let qJ ~ L(B)* and q/ ~ K (B) • Then there is a pair ofbounded nets (x~) and (y~) such that


543

(1) w-limaXa = 0 , w-lima Ya = 0 . (2) For every A ~ L ( B ) , q-/(A) = lirna(Axa, Ya). Now let q-'l (A) -- ( A x , y) where x, y are as in Theorem 2.1 and A 6 R. Theorem 2.1 gives that qq can be extended to Span(R, K ( B ) ) by putting q-tl ( T ) - - 0 for T E K ( B ) . Now let q-' be a Hahn-Banach extension of the qq to all of L (B). We now use L e m m a 2.5 and we get Theorem 2.4 with xa = x 4- Xa and Ya = Y - Ya. 7] Finally we will mention that in the case of a non-reflexive Banach space Theorem 2.1 gives invariant subspace corollaries only in the dual space B*. Since the first author in 1976 [16,17] showed that there are counter examples to the invariant subspace problem in general Banach spaces, this may be a sign that the following result is true: CONJECTURE 2.17 ([25]). I f A is a bounded linear operator in a Banach space then A* has a non-trivial invariant subspace. The known counterexamples do not contradict this conjecture. By Corollary 2.2, this would be true if the following is true: If A is a bounded linear operator in a Banach space then there exists an algebra with the (PS) property which contains the operator A. S.W. Brown [11] independently obtained similar results in the case of a commutative algebra.

3. Generalizations of Burnside's theorem in Hilbert spaces Given an operator A on a Hilbert space we define Re A - - A + A2 * and I m A A-A* With 2 " the normalizing conditions Ixl = lYl = 1 inequality (1) from Section 2 has the form -

I(x, A *y)[ IIIRe

AIIIAI(f).

Then there exists a positive number ~ > O, together with a weak neighborhood W of f , such that for every h ~ W U S: AA(h) > IIIReAIIIAI(h)I +S. PROPOSITION 3.5. Let A ~ F~(TY) be a convex subset of the bounded linear operators acting on a real or complex Hilbert space H. Fix a unit vector fo E H and choose a positive number r 6 (0, 1). Suppose that for every vector g _J_fo and I[gl[ ][[Re A [[[(1 - [[g [[2).


545

Then A contains an operator Ao with an eigenvector in the set S={f6Hlllfo-fll~ I11Re AIII. PROPOSITION 3.6. Suppose H is a real or complex Hilbert space, and )~ ~ C is a point in the spectrum of the operator A ~ L ( H ) such that I ReZl > Ill Re AIII. Then the norm closure of the algebra generated by A contains a nonzero finite-rank operator Similar techniques can be used to prove PROPOSITION 3.7. Suppose H is an infinite-dimensional complex Hilbert space and let R be a convex set of commuting essentially selfadjoint operators. Then the set of vectors x with non-dense orbits Rx is dense in H. COROLLARY 3.8 ([26]). Every essentially selfadjoint operator on infinite-dimensional complex Hilbert space has a nontrivial closed real invariant subspace. From 3.7 one can derive the remarkable THEOREM 3.9. Every essentially selfadjoint operator on infinite-dimensional real Hilbert space has a nontrivial closed invariant subspace.

4. Extremal vectors Extremal vectors were introduced in Ansari and Enflo [2] and Enflo [19]. For results on special operators see also Spalsbury [38]. Extremal vectors provide a tool to find hyperinvariant subspaces for compact and normal operators on Hilbert space in a unified, more constructive way. They also provide a tool to strengthen Theorem 2.4 in some special cases. For most of the proofs we refer the reader to [2]. Since Theorem 4.8 is presented here for the first time, we give a complete proof. We feel that the extremal vector technique for obtaining invariant subspace results is still in its beginning. The origin of this is to study best approximate solutions of the equations Tny = xo for the case when T has dense range and x0 is not in the range of T. More precisely, one considers, for e > 0 and for each n, the Yn of minimal norm such that

II

- xo 11

It turns out that T n Yn in many cases converges to a vector which is hyper-noncyclic for T, that is non-cyclic for all operators commuting with T.

546


4.1. Definitions and some basic properties o f extremal vectors H will denote a Hilbert space over the field of complex numbers, R ( T ) will represent the range of T. Backward minimal vectors. Let T ' H ~ H be a bounded operator with dense range. Let x0 6 H and e > 0 with e < [Ix0ll. There is a unique vector Yn,xo such that T Y n , x o - x0l] ~< e and

]lYn,.0II- inf{llyl[-

[[Zny

-

x0]] ~ e}.

The points yen,xo are called backward minimal vectors. Forward minimal vectors. Let T" H --+ H be an injective bounded operator. Let xo 6 H and e > 0 with e < Ilxoll. There is a unique vector V.,x E o such that IIV n~, x o -- XOII ~< ~ and

I[Tn Un,xo ~ II =

inf{ I1Tn v 11" 11v - x0 l[ ~<e } 9

The points T n Un,xo e a r e called forward minimal vectors. W h e n there is no ambiguity, we will drop some of the superscripts and subscripts from Yne,xo and Vne,x0. Sometimes the vectors T n Yn and Vn will also be referred to as backward, respectively, forward minimal vectors. It is clear that [[T n Yn,xo e - xo 11= e and [IU n - - XO 11 -e. The minimality gives easily Orthogonality relations.

If rn I Yn then Tnrn I T n Yn - x o . And we get THEOREM 4.1 (Orthogonality equations). There exist 6~ < 0 such that ye = ~ T* (Tye - xo) a n d (possibly another) 6~ < 0 such that (rE - xo) -- 6~ T * T v e .

REMARK 1. From the orthogonality equations we obtain Tnyn = - S n ( I - 8 n T n T * n ) - l T n T * n x o

and

Vn : (I - 6 n T * n T n ) - l x o .

Let An - - 6 n ( I - 6 n T n T * n ) - l T n T *n and Bn - - ( I 6nT*nTn) -1. Then An and Bn are positive operators. ][An][ < 1 and [[Bn[[ = 1 for all n. Let f n ( z ) - 1-8,.~ -~,.z . Then fn (z)


547

is analytic in a neighborhood of a ( T n T *n) and a ( A n ) = f n ( a ( T n T * n ) ) . The maximum -~n'llrnll2 . It follows that point of f ( a ( T n T * n ) ) is i---~.:ii~l~

[IAnll =

- a n . IIT n I12 1 -

~n"

< 1.

IITn II2

1 Then gn is analytic in a neighborhood of a ( T n T *n) and Now let g n ( z ) 1-~n.z a ( B n ) = g n ( a ( T n T * n ) ) . Since 0 E a ( T n T *n) it can be verified that the maximum point of a (T n T *n) is 1. Hence l]Bn l] = 1. The proofs of the following results can be found in [2]. .

THEOREM 4.2. Let xo E H and xo =/: O. The functions e w+ ye and e w-~ ve are analytic on (0, Ilxoll). THEOREM 4.3. For any b o u n d e d operator T with dense range, if xo q~ R(T), then Tyen-xo

Tyen

\

IIT y~, - xoll' IIT y~, II f o r some sequence 6n ~

/ --~0

O.

4.2. Invariant subspaces and a two sequences theorem LEMMA 4.4. I f T is quasi-nilpotent then IlYnk II/llYnk+l II ~ 0 as k --+ cx~f o r some subsequence nk. This follows easily from the spectral radius formula. In the following theorem, let K be a quasi-nilpotent, compact operator without eigenvector and let (nk) be as in L e m m a 4.4. We have THEOREM 4.5. For every weak accumulation point z o f K nk Ynk, K z is hyper-noncyclic f o r K and { T z [ K T = K T } is a hyper-invariant subspace f o r K. PROOF. By passing to a subsequence, we can assume K nk+l Ynk + 1 -'-+ YO weakly. We have T K n k + 1Ynk --+ T K z

in norm.

Put Tynk -- OtkYnk+l + rk rk 3_ Ynk+l. Then, obviously, Olk --+ 0 as k --+ cx~. We get T K nk + 1Ynk --- OtkK nk + 1Ynk + l -+- Knk + l rk 9

548


Scalar multiplying both sides by K nk + 1Ynk+ 1

--

XO

gives

(TKnk+lyn k, Knk+lyn~+l -- xo) = Otk(Knk+lynk+l, Knk+l ynk+l -- XO) + (Knk+l , Ynk+l Knk+lynk+l -- XO).

The first term on the fight-hand side --+ 0 since otk --+ 0. The second term is 0 because of the orthogonality relations 9 The left-hand side --+ ( T K z , Yo - xo) as k --+ cx~. Since the yf angle between Knyn and x o - Knyn is/> ~, Ilyo][2 ~< liminflIKnynll 2 ~ I[xoll2 - e 2. Thus Yo - xo # 0 and the theorem is proved. O We also have THEOREM 4.6. Assume that T is normal with dense range R ( T ) and that T has a cyclic vector. Assume xo ~ R ( T ) 9 Then f o r each e > 0 Tny rel , X 0 converges in norm to a hypernoncyclic vector f o r T. To prove the following two sequences theorem we use the following THEOREM 4.7. Let xo ~ H, xo q~ R ( T ) . I f f o r every 61 > 0 and all 62 with 0 < 62 < 61, there exists r > 0 with (T n Yne, Tn Yne -- xo) ~ 1 and all e, $2 ~ 6 ~ SI, then xo is hyper-noncyclic f o r T.

From this we derive the following THEOREM 4.8. Let R be a commutative algebra of operators on H which contains a O9 O) quasi-nilpotent operator. Then there exist two sequences Xn --+ x # 0 and Yn ~ Y # 0 such that f o r every bounded sequence An of operators in R, (Anxn, Yn) --+ O. PROOF. Let T E R be quasi-nilpotent. Assume xo ~ R ( T ) . Then by Theorem 4.7 either x0 is hyper-noncyclic or, for some e > 0 and some subsequence (n~) of the integers

(1)

(T n~ Yn~ , Tn~ Yn~ - xo) --+ O,

where (Yn~) are the backward minimal vectors with respect to x0 and e. We write Tn~yn~ = ( 1 - e2)x0 + y E V / 1 - e2s0 + V / 1 - y 2 E v / 1 - e2sv,

where Ilsoll = I l s ~ l l - 1 and sv -~ O. (1) then gives ((1 - e2)xo + yew/1 - e2so + V/1 - y2eV/1 - e2s~, e x o - yv/1 - e2xo

-,/1-

o.

549

Some aspects of the invariant subspace problem Now, let Av be a bounded sequence of operators in R, let

xv = ( 1 - e2)xo + y e ~ / 1 - e2so + V / 1 - v2ev/1 - e2sv and let

yv -- exo - Y v / 1 (2)

Then xv --+ (1

-

g2)x0

-+-

e2so - ~ / 1 - y 2 v / 1 -

ge~/1

e2sv. O)

-

g2s0

--

X and yv --+ exo - y~/1

-

62s0

=

y. Moreover

I(A~x~, yv)l -- [(A~T~Y.~, T"~Y.~ xo)l- [(T"~A~y.~, T"~Y.~ -

-

I(z~(~y~

-

xo)l

+ rn~), T n ~ y n ~ - xo)]

l+ [(r" v rn~ ,

Tnv

Yn~ - xo)] .

(2)

Since Oln~ is bounded, by (1) the first term on the right-hand side of (2) --+ 0 as v ~ oc. The second term on the right-hand side of (2) is 0, because of the orthogonality relations. This proves the theorem. []

5. Operators without invariant subspaces 5.0. Introduction and history In this chapter we will present examples of operators on Banach spaces with only trivial closed invariant subspaces. We will refer to them as transitive operators. In this section we will give a short history. The first construction of a transitive operator was given by the first author, who completed and circulated in preprint form a first manuscript in 1975. An outline of the construction was presented in Seminaire Maurey-Schwarz in the Spring of 1976 [16]. (See, also the survey article from 1982 by Radjavi and Rosenthal [30].) An improved manuscript of the same construction appeared in the Institute MittagLeffler report series in 1980 and was submitted to Acta Math. in 1981. This version with some improvements in the notation and presentation- appeared in Acta Math. in 1987. In 1984, B. Beauzamy [7] and C. Read [31] presented constructions of transitive operators on some Banach spaces using a similar approach and basic considerations but with the technical aspects worked out in different ways - a discussion and comparison will be given below. Beauzamy's construction gave the first example of an operator T on a Banach space B where all non-zero vectors are super-cyclic: for every y 6 B, y 5/=0,

{t~TKy l t~ 6 R, K 6 N } -- B. A slight modification of Read's first construction gave a transitive operator on s

550


Simplifications of the construction of a transitive operator on s were given by A.M. Davie (see Beauzamy [8]) and by C. Read [33], whose example we will present below. This example has recently been used by V. Troitsky to test several conjectures on invariant subspaces. Later, C. Read [34-36] constructed stronger examples as well as transitive operators on new spaces and we present some of his results below. But, as we have seen above, there is no example of a transitive operator which is the adjoint of another operator, a n d - in p a r t i c u l a r - no example of a transitive operator on a reflexive space. In Section 5.1, we outline the construction in [16-18]; in Section 5.2, we compare it to the constructions in [7] and [31]; in Section 5.3, we give the stronger examples of Read, and in Section 5.4, we discuss the recent work of Troitsky.

5.1. Outline of the original considerations and construction In this section we will outline the original considerations and construction of an operator with only trivial invariant subspaces on a Banach space. The Banach space will be constructed at the same time as the operator and will be non-reflexive. There are very serious difficulties in carrying out a similar construction in a reflexive Banach space. So we feel that the construction gives some support to the conjecture that every operator on a Hilbert space has a non-trivial invariant subspace. We now turn to the basic considerations behind this approach. It is clear that every operator with a cyclic vector on a Banach space can be represented as multiplication by x on the set of polynomials under some norm. So what we will do is to construct a norm on the space of polynomials and prove that multiplication by x under this norm has only trivial invariant subspaces. Our next basic consideration comes from the fact that one can have an operator with a dense set of cyclic vectors without having all vectors cyclic. In order to be able to make some limit procedure work, we will construct the operator so that it has the following uniformity property: Let 1 be a cyclic vector of norm 1 in B. Let (p j) be a sequence which is dense on the unit sphere of B. For every m there is a positive number Cj,m such that for every Pn with [[Pj -- Phil < 1/2 m+4 there is a polynomial s in T with [[s en3 . . . . . Also we assume deg qn 2. el . . . . . ~k and C1 . . . . . Ck will determine a number ak+l inductively as explained below and we define a sequence of norms as in the following definitions. DEFINITION. For any polynomial p, consider all representations p -- ~ a i , ~ x i g.~l I . . . g,~nn and put [Plopn -- inf Z where cients.

lai,3l 2i (Clel l1 )~' 9. . (Cnl~ n l1 )~n

I1~ denotes the usual el norm equal to the sum of the absolute values of the coeffi-

REMARK. In the final norm the operator x will have norm ~< 2, and multiplication by s norm ~< Ck I~11. DEFINITION. For any p, consider all representations

p -- r + ~

Sk (g~kakqk -- 1). 1

Put Ipl ~ = inflr]l -+- Z~lSklopnek. Put Ipl ~ = IPll, and let a~ be determined inductively by the condition la~qk I~- 1 = 1. REMARK. [g~kakqk -- 1[ n < ek and clearly the operator norm of multiplication by g is [g[opn. We see that [[n is the maximal norm satisfying the following four properties:


552

(A1) II n

~lll, (A2) le~a~q~ - 11n ~ Ek, k = 1,2 . . . . . n, (A3) [gklop 4zr,

(8)

f o r 0 ~< ~ ~< 27r.

(9t

1

(~92(~) § (~92(~ -- 27r) = ~-y

Let q/0 (t) be the function whose Fourier transform equals 69 (~). First let us observe that for a function f 6 L2(R) the following conditions are equivalent: (a) the system { f (x - n)}ncZ is orthonormal, (b) ~ n ~ Z I f ( ~ + 2krr)l 2 : 1/2zr a.e. This observation follows from the following calculation which uses Plancherel's Theorem:

Z n

an f (x

-

=

n)

2 --

f(~)

d~

~[f(~

§

an ein~ ~

n

E a n e in~ n

d~.

keZ

Since (5)-(9) immediately give ~ 169(~ + 2kzr)l 2 - 1/2zr we see that {q/~ - n)}nzZ is orthonormal in L2(•). Let V0 denote s p a n { q / ~ n)}nzZ. If f 6 V0, then f = ~ n a n ~ O ( 9- n), so f ( ~ ) = (y~nzzanein~)o(~). Thus V0 can be characterized as the space of all f 6 L2(R) such that f ( ~ ) - m ( ~ ) O ( ~ )

for some 2re-periodic, locally L2

Special bases in function spaces

567

function m(~) on R. From this and (7) and (8) we infer that the space V1 := { f E L2(R): f ( 2 . ) E V0} contains the space V0. Let m(~) be a 2Jr-periodic function on • which on [-Jr, 7r] equals x/T~69(2~) and let Z,(~) = ei~/Zm(~/2 + 7r)O(~/2). Clearly, supp,~ C [ - 8zr, - 2 7 r ] tO [~Jr, 3s--Tr].Let (P 1(t) be the function whose Fourier transform equals ,~ (~). Looking at the Fourier transform one can check that the system {q/1 (x - k)}kcz is an orthonormal basis in the space V1 O V0. Note that for any f 6 V1 O V0 with [If ][2 - - 1 the system {2J/Zf(2Jx)}j~z is orthonormal. From this it easily follows that t/-/1 is a wavelet on R. Elementary properties of the Fourier transform yield that q/1 is a real-valued C ~ function satisfying !P 1( _ 89_ x) - ~ 1( _ 89+ x). The above construction leaves a lot of possibilities to choose the function 69. Observe that if 69 is infinitely differentiable (and one can easily construct it in such a way) then also ,~, is C ~ . Since it is also compactly supported we infer that in this case both q/0 1 and q/1 are from the Schwartz class. Another extremum is to put 69(~) -- ~-l[-~r,~r](~) which gives ,~(~) -- ~ 1

ei~/2[l[_2rr _:r] + l[:r,2jr]]. This gives the Shannon wavelet

S ( x ) - (rc(x + 1 ) ) - I [sin27r (x + l ) _ sinn'(x + 1)].

(10)

The way to construct a corresponding wavelet set is analogous to (4). We put q/a(tl . . . . . td) -- q/a, (tl) . . . . . q/a~ (td),

(11)

where a 6 E. This is Meyer's wavelet set on iRd. If 69 is infinitely differentiable then all ~ a ' s are from the Schwartz class. This procedure to pass from one dimensional wavelets to several dimensional systems is standard in wavelet theory (cf. [73, 5.1]). Note that it involves an extra function 4) (in our examples q/0 and 1[0,1] respectively) which is called a scaling function. Its main property is that the system (4)(x - n)),~z is an orthonormal basis in the space span{ tllj,k } j /r,l I

T. Figiel and P. Wojtaszczyk

572

This leads directly to the required estimate I ~< CXllflll, because we prove below that there is a C depending only on r/such that sup

Z

r,1 r'>r,l'

2 r f rI (2rt -- 1)rI (2r't -- 1') dt ~ C. d

Using the substitution 2rt - 1 = u, we can deduce the latter estimate from the following lemma. Fq 4. There is C < ~ depending only on rl such that, if mutually disjoint dyadic intervals with IIj,kl = 2 - j ~ 1, then LEMMA

Z (j,k)~S

F

(Ijk)(j,k)~S

is any family of

r l ( u ) r l ( 2 J u - k ) d u ~ C.

PROOF OF THE LEMMA. Clearly, it will suffice to produce two integrable functions on R, say 4~0, 4)1 such that, for any dyadic interval Ijk with j ~> 0, if Jjk -- [k2-J - 1, k 2 - J + 1], then

lo - fjj ~ n(u)n(2au - k)du ~

.

(13)

Since If(x)l ~< )~ for x 6 F we get fF If(x)l 2 dx ~< )~llflll, so from Fact 1 we infer that Ilgl12 ~< C~/)~llfll~. Using this we get

{

x Is

/

g(x)l>5

"J

x Is g(x)l

4

4

C

~< ~_~[[U,g[[2_ ~_~11gl[2 ~< ~-[[fl[1.

(14)

Letus denote A -- ~ r l 10./rZ. Then IAI ~< ~ Ilflll. On the other hand,

f, \A

rl

\A

IU~Q~(fr,>l 0 such that for n big enough we have degfn ~> ( 89+ e)n cf [61 ] and also [69]. If the system (fn)n~=O is an unconditional basis in Lp (~) for some p, 1 89 cf [70]. REMARK. In [70] an unconditional basis in Lp(T), 1 ~ (1 + e)n for n big enough. Conversely, given an e > O, we can find a basis (fn)n~176 0 in C [ - 1, 1] (and another basis in L 1 [ - 1, 1]) consisting of algebraic polynomials and such that deg fn ~< (1 + e)n.

Special bases in function spaces

579

In the case of L p norm the above substitution introduces a weight which creates a problem in the translation of the results. Nevertheless it is known that for each p there exists a system of Jacobi polynomials which is a basis in L p[ 1, 1]. Let (fn)n=0 c~ oc be the system of algebraic polynomials with deg fn~ = n for n = 0, 1, 2 , . . . that is orthonormal on [ - 1 , 1] with respect to the measure (1 - x2) ~ dx. This definition requires oe > - 1 . A special case of a theorem of Muckenhoupt [52] can be formulated as follows: -

THEOREM 14. Let - 1 < ol < oo and let 1 < p < oo. The system (fn~)~_o is a basis in L p [ - 1 , 1] if and only if 1

l+~ < p 4

when - l < c~

1

(28)

2' 1

2"

(29)

For unconditional bases the results are weaker and less explicit. The best published result seems to be the following theorem of Canturija [ 18]. T H E O R E M 15. For any p, 1 0 there exists an orthogonal system o f algebraic polynomials (fn)n~=o which is an unconditional basis in L p [ - 1, 1] and satisfies

deg fn < n j+e f o r n big enough.

2.7. Rational bases The question of existence of polynomial bases considered in previous section is a particular case of a general (open ended) question of existence of bases whose elements belong to some analytically defined family. A natural question of this type which has been around for some time is: does there exist a basis in C[0, 1] consisting of rational functions of uniformly bounded degree. A very general approach to this and other similar questions has been proposed recently by E Petrushev [58]. It is based on the following perturbation result: Let us take a Meyer's wavelet ~P E L2(]1{) O ,5' and take a function r such that [tP(J)(t) - q:,(J)(t) [ 0 are properly chosen. Then the system cI)j,k(x) := 2J/ZcI)(2Jx - k) is an unconditional basis in L2(N) with biorthogonal functionals having very good decay. This result leads to the fact that the system {(I)j,k}j,kcZ is a basis (unconditional basis) in many function spaces, just like wavelets. An analogous construction can be performed on the unit interval. It is possible to find a rational function q~(x) satisfying (30) and since translations and dilations do not change the degree of rational function we obtain THEOREM 16 ([58]). There exists a system o f rational functions o f uniformly bounded degree which is a basis in C[0, 1] and an unconditional basis in Lp[O, 1], 1 < p < oc. The above ideas can be also applied for functions on IRn (see [39]).

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T. Figiel and P Wojtaszczyk

3. Negative results There are natural restrictions on properties of bases a given space may have. We will list several results in this direction. The following is true for classical spaces. THEOREM 17 ([35]). The space Lp[0, 1]for 1 < p < oo, p ~ 2, does not have a subsymmetric basis because every normalized unconditional basis contains a subsequence equivalent to the unit vector basis in g.p. THEOREM 18. I f the space Ll (lz) has a normalized unconditional basis (en)n~=l, then this basis is equivalent to the unit vector basis in ~ 5. In particular lZ has to be a purely atomic measure, so LI[0, 1] does not have an unconditional basis. I f the space C ( K ) has a normalized unconditional basis (en)n~__l, then this basis is equivalent to the unit vector basis in co. In particular C[0, 1] does not have unconditional basis (see [42] or [72]). REMARK. Actually spaces L 110, 1] and C[0, 1] do not embed into spaces with unconditional bases. As we know neither the Walsh system nor the trigonometric system is a basis in L 1 [0, 1]. This is connected with the fact that those systems are uniformly bounded. Note that the unit ball from L ~ is a weakly compact subset in L m[0, 1], and thus does not contain a subsequence equivalent in L 110, 1] to the unit vector basis from el. This shows that the following theorem implies in particular that there is no uniformly bounded, normalized basis in L 1[0, 1]. THEOREM 19 ([66]). Every normalized basis in the space L1 [0, 1] contains a subsequence equivalent to the unit vector basis in g.1. For every normalized basis (fn)n~=l in C[0, 1] there exists a sequence of integers nk such that the map (y~n~=l an fn) ~ (ank) is onto co. This in particular implies that there is neither p-besselian basis in C[0, 1] nor a philbertian basis in L m[0, 1]. Let us recall that a normalized basis (Xn)n~=l in a Banach space X is oo lan ip ) 1/p , - p-besselian (for given p E [1, cx~) ) if II Y~n~=l anXnll ~ C ( Y~n=l - p-hilbertian (for given p E (1, ~ ) ) i f II ~ n ~ l anxnll 0. We shall describe a construction of U ( M ) which is good in the case of Sobolev spaces, i.e., . T - Wpk, and s , where k ~> 0, s > 0 and 1 0, 1 ~ p q ~< ec satisfies for each integer m > s The Besov s p a c e B p,q the relation s Bp,q (M) -- (W 0 (M), % m (M))s/m, q . o

The related space

s (M) is defined as the interpolation space Bp,q

o

o

o

s Bp,q(M)-(W 0 (M), Wpm(M)) s/m,q' if 0 < s < m, 1 ~< p, q ~< c~ (choosing another m leads to the same space up to equivalence o

but its range is not closed in of norms). There is a natural map B sp,q (M) ---> Bp,q(M), s Bp,qS(M) for some values of s The result for Besov spaces makes use of the sequence space bPp,q which is defined for any real p and 1 ~ p, q ~< co. A sequence a - (an)n~=l is in bPp,q provided that

IlallbPp,q
~ 0, and Ix(K) = 1. Furthermore, for any continuous linear functional f on X, we have f x f dIx - Y~Ixi f (xi) = f (Y~Ixixi) = f (x). The equality f ( x ) -- f x f dIx (for all f 6 X*) is what we mean when we say that Ix represents x. Since the continuous linear functionals f separate points of X, this is equivalent to saying that x = ~ #ixi. DEFINITION 2.1. Suppose that K is a nonempty compact subset of a locally convex space X, and that IX 6 P ( K ) , the set of all regular Borel probability measures on K. A point x in X is said to be represented by IX if f (x) = f/( f dIx for every continuous linear functional f on X. (We will sometimes write Ix ( f ) for f/( f dix, when no confusion can result.) (Other terminology: "x is the barycenter of IX", "x is the resultant of IX".) The restriction that X be locally convex is simply to insure the existence of sufficiently many functionals in X* to separate points; this guarantees that there is at most one point represented by IX. Note that any point x in K is trivially represented by ex; the interesting (and important) fact brought out by Minkowski's theorem above is that in the special case of a compact convex subset K of a finite dimensional space, each x in K may be represented by a probability measure which is supported by the extreme points of K. To obtain such a representation in infinite dimensional spaces, a necessary first step is to show (below) that every probability measure on K has a resultant. PROPOSITION 2.2. Suppose that A is a compact subset of a locally convex space X, and that the closed convex hull K of A is compact. I f # is a probability measure on A, then there exists a unique point x in K which is represented by Ix, and the function Ix --+ (resultant of ix) is an affine weak* continuous mapfrom P ( A ) into K. PROOF. We want to show that the compact convex set f ( x ) = fA f d # for each f in X*. For each f , let H i closed hyperplanes, and we want to show that (-]{Hi: compact, it suffices to show that for any finite set f/, . . . , is nonempty. To this end, define

T" K ~ R n

K contains a point x such that = {y: f ( y ) -- Ix(f)}; these are f ~ X*} A K # 0. Since K is fn in X*, the set (']in_l Hfi A K

by Ty = ( f l ( y ) , f2(y) . . . . . fn(Y));

then T is linear and continuous, so that TK is compact and convex. It suffices to show that p E TK, where p = ( # ( f l ) , # ( f 2 ) . . . . . # ( f n ) ) . If p q~ TK, by the separation theorem there exists a linear functional on R n which strictly separates p and TK; representing the functional by a = (al, a2 . . . . . an), this means that (a, p) > sup{(a, Ty): y ~ K}. If we define g in X* by g = ~ ai fi, then the last assertion becomes fa g dix > sup g ( K ) . Since A C K and # ( A ) = 1, this is impossible, and the first part of the proof is complete. It is straightforward to verify that the resultant map is affine, and an argument using the compactness of K shows that it is continuous. U]

Infinite dimensional convexity

605

A simple, but useful, characterization of the closed convex hull of a compact set can be given in terms of measures and their barycenters. PROPOSITION 2.3. Suppose that A is a compact subset of a locally convex space X. A point x in X is in the closed convex hull K of A if and only if there exists a probability measure It on A which represents x. PROOF. If # is a probability measure on A which represents x, then for each f in X*, f ( x ) = # ( f ) ~< sup f ( A ) n-1 }. It is easily checked that each Fn is closed, and that a point x of K is not extreme if and only if it is in some Fn. Thus, the complement of the extreme points is an F,,. D THEOREM 2.10 ([27,29]). Suppose that K is a metrizable compact convex subset of a locally convex space X, and that xo is an element of K. Then there is a probability measure # on K which represents xo and is supported by the extreme points of K (i.e., #(ext K) -- 1). The proof (below) is due to Herv6 [86] and Bonsall [ 14]. It uses the concept of the upper envelope f of a real-valued function f on K. DEFINITION 2.11. If f is a real-valued function on K and x 9 K, let f ( x ) = inf{h(x)" h 9 A ( K ) and h ~> f}; this is called the upper envelope of f . The function f has the following useful properties, whose proofs are elementary: (a) f is concave, bounded whenever f is bounded, and upper semicontinuous (hence Borel measurable). (b) f ~< f and if f is concave and upper semicontinuous, then f = f . (c) If f, g are bounded, then f + g ~< f + ~ and I f - g[ ~< [If - g[], while f + g -f + g if g 9 A ( K ) . If r > 0, then r f = r f . PROOF OF THEOREM 2.10. Since K is metrizable, C ( K ) (and hence its subspace A ( K ) ) is separable. Thus, we can choose a sequence of functions {hn} in A ( K ) such that Ilhnll~ - 1 and the set {hn}neC__lis dense in the unit sphere of A(K); in particular, it separates points of K. Let f0 -- ~ 2-nh2; the series converges uniformly, hence is in C ( K ) and it is a strictly convex function in C(K). (Indeed, if x 7~ y, then hn(x) r hn(y) for some n, so hn is nonconstant and affine on the segment [x, y]. It follows that h 2 is strictly convex on [x, y] and therefore fo is strictly convex on [x, y].) Let B denote the subspace A ( K ) + R f o of C ( K ) generated by A ( K ) and f0. From property (c) above, it follows that the functional p defined on C ( K ) by p(g) = g,(xo) (g E C(K)) is subadditive and positivehomogeneous. Define a linear functional on B by h + rfo --+ h(xo) + r fo(xo) (h in A ( K ) , r real). We will show that this functional is dominated on B by the functional p, i.e., that


609

h(xo) + r fo(xo) ~ 0, then h + r fo -h + rfo, by (c), while if r < 0, then h + rfo is concave, and hence by (b) h + rfo = h + rfo ~ h + r fo. By the H a h n - B a n a c h theorem there exists a linear functional m on C ( K ) such that m(g) ~ g'(x0) ) m(g), i.e., m is nonpositive on nonpositive functions and hence is continuous. By the Riesz representation theorem, there exists a nonnegative regular Borel measure # on K such that m(g) -- lz(g) for g in C ( K ) . Since 1 6 A ( K ) , we see that 1 -- m(1) - #(1), so # is a probability measure. Furthermore, # ( f o ) -- m ( f o ) -- fo(xo). Now, fo ~< fo, so # ( f o ) ~< # ( f o ) . On the other hand, if h ~ A ( K ) and h ) fo, then h >~ fo, and consequently h(xo) = m(h) -- # ( h ) ~> # ( j ~ ) . It follows from the definition of fo that fo(xo) >1 # ( f o ) , and therefore # ( f o ) -- # ( f o ) . This last fact implies that # vanishes on the complement of X - {x" f o ( x ) - fo(x)}. We complete the proof by showing that A" is contained in the set of extreme points of K. Indeed, if x - 89 y + 89 z, where y and z are distinct points of K, then the strict convexity of fo implies that f o ( x ) < 89 fo(Y)+

89 fo(z) 0 and let {fi} be a norm dense subset of ext K. For each i, let Bi denote the intersection with K of the closed ball of radius e/3 centered at j~. Thus, each Bi is weak* compact and convex and U Bi D ext K. Let f be a point of K and let/z be a maximal probability measure on K with resultant r ( # ) = f . Since U Bi is a weak* F~ set, we have # ( U Bi) -- 1. Let n be a positive integer such that, if D = /7 U i = I Bi, then # ( D ) > 1 - e / 3 M . T h e n / z can be decomposed as # = )~#l + (1 - )~)#2, where )~ = # ( D ) and # l , #2 are probability measures on K defined by ~#l = #ID

and

(1 - )~)#2 = # I ( K \ D ) -

(If)~ = 1, let # 2 be an arbitrary probability measure on K.) Then f = r ( # ) = )~r(/zl) + (1 - ~,)r(/z2). Since r ( # 2 ) E K we have oe

II f - )~r (/zl

- 0 and every finite set F C G there is a unit vector x e H so that lirr(g)x - xll ~< e for every g e F. The group G is said to have property T of Kazhdan if every representation Jr of G which has almost invariant vectors actually has a nonzero vector which is invariant with respect to Jr (g) for all g e G. Commutative groups and more generally amenable groups fail to have property T. On the other hand, SL(n, Z) with n ~> 3 has property T. We can now state the following dichotomy result. THEOREM 3.1 1 ([76]). Let G be a countable discrete group. (i) If G has property T, then for any action of G on a compact metric space I2 the set

of G-invariant measures on 1-2 (if nonempty) is a Bauer simplex. (ii) If G does not have property T, then for the natural action of G on the compact metric space S-2- {0, 1}G the set of G-invariant measures on 12 is the Poulsen

simplex.

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By an action of G on a space s we mean a homomorphism from G into the group of homeomorphisms of S-2. In the paper [76] an analogous result is also proved for actions of topological (nondiscrete) groups. The Poulsen simplex arises also as the set of invariant states for many examples of quantum spin systems. Mathematically, this involves the study of G-invariant states of a C*-algebra. For a presentation of these results we refer to [153,23,161] and [162]. An interesting and useful property of compact simplices is exhibited in the following separation result. PROPOSITION 3.12 ([51]). Let K be a compact simplex and let f and - g be convex upper semicontinuous real valued functions on K with f (k) 0, n = 1,2 . . . . , )--~n~_-I•n ~< 1} is a Bauer simplex whose extreme points are {Xn}n~=l and 0. By using a category argument (on {0, 1}s~ = the set of all subsequences of the integers) Christensen shows that any linearly independent sequence {Xn}n~=l tending to 0 can be split into two subsequences {Xn,}n,~=l and {Xn,,}n,~,_l both of which are strongly independent. This concludes the proof. The following structure theorem for general compact metric simplices is proved by a more sophisticated application of the selection Theorem 3.18. THEOREM 3.22 ([113]). For every compactmetric simplex K there is a sequence {Sn }n~__l of n-dimensional simplices and surjective affine maps qgn "Sn+l --+ Sn so that K is the inverse limit of the system

S1 ~ 82 ~ $3 0 },

where q~c~,k is an affine function of the form ~ , k ( x ) -- x*,k + c~,k with

IIx~*,~II - 1 and is

chosen so that {x E F~,k: q~,k(x) > 0} is non-empty and of diameter smaller than 2 -k. Such a choice is possible as long as F~,k r 0, since every closed convex subset of Bx is dentable. For every V a limit ordinal we put Fy,k -- ~ 0} (in other words x 9 Fc~,k if and only if ot ~< otk(x)). For every positive integer j let Rj be a 2 - J - n e t in [ - 1 , 1]. We are next going to find a sequence {en }n~__l of unit vectors in X and finite subsets An C X* so that the following hold: (i) SUPx,ea n x*(x) >/[Ixll(1 - 2 -n) for every x 9 span{ei}in=l . (ii) x*(en+l) = 0 for x* 9 An. n ( n (iii) Otk(Y~i=l ziei -+-Zen+l) ~ 0tk ~i=1 ziei) for every 1 ~< k ~< n, every choice of zi 9 Ri and every z with Izl ~< 1. We choose as el any vector of norm 1. We then choose A1 so that (i) holds for n = 1. In general, having chosen {ei}in=l we choose An so that (i) holds for n. We show how to choose en+l once An is defined.

V.P Fonf et al.

636

We choose en+l of norm 1 so that (ii) holds and so that for each of the finite number of functionals X*

n

0~k (~--~i = 1 ziei),k'

l (1). Suppose to the contrary that X D I1. We shall construct an equivalent norm on X such that (4) does not hold even for B = w * - c l e x t B x , . We consider just the particular case X = ll. (For the general case X D ll see [78].) We construct the desired norm on the complex space ll and then, by a standard procedure (see the proof of Theorem 5.10), we deduce the result for the real space 11. Let E be a Sidon set which is not an interpolation set and let Y = C E ( T ) . (In particular, x --+ } is an isomorphism of Y onto the complex space l l (E).) We will show that the norm on l l which comes from Y via this isomorphism works. By a result of Kahane [97], the set E is an interpolation set iff each bounded function u(n) E lot(E) may be extended from E to Z as an almost periodic function on Z. Let ~ :AP(Z) -+ loc = C (I3 Z) be the natural embedding. With the help of the above mentioned result of Kahane and the TietzeUrysohn lemma it is not difficult to see that E is an interpolation set iff the restriction gz*l~E is a homeomorphism of fiE onto bE. (The set fiE is just the w*-closure of E in C*(fiZ) and b E is the w*-closure of E in AP*(Z).) Clearly the restriction ~P*I~E is a homeomorphism iff for each A C E there is a function p of the form (5.3) such that inf{[p(n) - p(k) 1: n ~ A, k ~ E \ A} > 0. As is well known, for each g E ext By, there exist a t ~ T and a complex number a, la] = 1 such that g(x) = ax(t) for each x ~ Y. Therefore each g E span{w*-cl ext By, } has a representation g(x) -- ~-,k=l m akx(tk), where tk ~ T. Put g , ( n ) - - ff-~km=l a k e intk . Clearly s is of the form (5.3). Since E is not an interpolation set, there is an A C E such that inf{lp(n) - p(k)]: n ~ A, k ~ E \ A} = 0 for each p of the form (5.3). In particular, the characteristic function of A does not belong to [w*-cl ext B x , ] (we identify here the space C*E(T ) with lee(E)), i.e. the subspace [w*-clextBy,] is a proper subspace of Y*. (1) => (2). Suppose to the contrary that K is not w*-compact. Then there is an f e w*-cl K \ K. Let u c Bx** be such that

u ( f ) > oe > sup{u(g): g e K}

(5.4)

for some ol. Since X 75 11, by the Odell-Rosenthal theorem [134] there exists a sequence {Xn}neC__l C BX such that u -- w*-limxn. Put C -- {x ~ Bx" f (x) ~ ol}, B -- K and apply Proposition 5.13. (We may clearly assume that Xn 6 C for all n.) We have sup{u(g)" g 9 B} >~ inf{sup{g(x)" g e B}" x ~ C} >~ inf{f(x)" x ~ C} >~ ce, which contradicts (5.4). The other implications are trivial. Next we consider the question how properties of a Banach space may influence properties of the boundaries for this space. THEOREM 5.15 ([121]). Let X be an infinite-dimensional reflexive Banach space. Let B C Sx be a boundary f o r X*. Then B is uncountable. In particular ext B x is uncountable.

V.P. Fonf et al.

646

PROOF. Assume to the contrary that B = {Xi}~=1. Put Fi = { f 6 BX*" f ( x i ) - - I l f l l } , oo

i = 1, 2 . . . . . By the reflexivity of X and by the definition of boundary, Bx, -- U i = l Fi. By the Baire category theorem, there is a j such that the set Fj contains a w*-relatively open set in Bx,. Thus, there are a functional f ~ Fj and a finite set {Yk}~_-i C X such that

g ~ Bx*" max

Ig{Y~)- f(yk)l

< 1} c Fj.

Without loss of generality we may assume that Ilfll - 1 - e, ~ > 0. Put L = ( [ y k ] L 1 ) • n [xj] • It is clear that ( f + L) n Bx, C Fj. Since d i m L > 0 there is an h 6 ( f + L) n Sx,. Therefore 1 = h(xj) -- f ( x j ) = IIfll -- 1 - e < 1, a contradiction. O Actually the same proof is valid for any CCB (not necessary symmetric) body in a reflexive Banach space X. Recall that the set of extreme points of a metric convex compact in a topological vector space is always a Ga set, and hence homeomorphic to a Polish space (see Section 2). Therefore, by Theorem 5.13, if X is an infinite-dimensional separable reflexive Banach space, then for each CCB body V C X, the set ext V (in the weak topology) is homeomorphic to an uncountable Polish space. The converse also holds. PROPOSITION 5.16 ([112]). Each uncountable Polish space is homeomorphic to the set of extreme points (in the weak topology) of some CCB body in 12. This proposition complements Haydon's theorem (Theorem 3.6) and its proof uses this result. We mention that in [57] a stronger version of Theorem 5.15 is proved which actually gives a characterization of reflexivity. In [121] an equivalent norm on Hilbert space was constructed for which the set strexp Bx is countable. The intermediate set exp Bx is the subject of Theorem 5.19 and Corollary 5.23 below. For our next result we need a new notion. Recall that a subset C C X* of a dual space is called r-norming (0 < r 0. A subspace E C X* is called norming if its unit ball BE is a norming subset. Each norming subspace is total (i.e., f (x) = 0 for all f 6 E implies x = 0) but the converse is not true. Indeed, the following proposition holds. PROPOSITION 5.17 ([38]). A Banach space X is quasireflexive (i.e., dim X** / X < c~) iff

every total subspace of X* is norming. DEFINITION 5.18 ([62]). A subset C C X* is called thin if it can be covered by an increasing union of non-norming sets, i.e. oo

ccUci, i--1

where inf{sup{[f(x) l" f E Ci }" x E Sx } -- O, i = 1, 2 . . . . .

647

Infinite dimensional convexity It is clear that if X is infinite-dimensional, then each countable set in X* is thin.

THEOREM 5.19 ([66]). Let X be a separable Banach space. The following assertions are equivalent: (1) The space X contains an isomorphic copy o f co. (2) There exists an equivalent norm on X such that the set w*-exp B x , is thin. To prove Theorem 5.19 we need some auxiliary results. We start by a useful characterization of thin sets. LEMMA 5.20 ([68]). Let X be a Banach space and C be a bounded subset o f X*. The following assertions are equivalent: (1) The set C is thin. (2) There exists a linear bounded one-to-one operator T : X --+ E into some Banach space E such that the inverse operator T -1 is u n b o u n d e d a n d T * ( E * ) D C. Let T : X --+ E be a linear bounded one-to-one operator from a Banach space X into a Banach space E such that the inverse operator T -1 is unbounded. We call the topology on X generated by the sets of the form T - I ( G ) , where G ranges over the open subsets of E, the E-topology of X. LEMMA 5.21 ([66]). Let T : X --+ E be a linear bounded one-to-one operator from a separable Banach space X into a Banach space E and let C C Sx, M T* E*. Set

Z - - {x E Sx" 3 f E C f ( x ) - -

l}.

Then Z can be represented as a countable union o f subsets o f Sx which are E-closed in B x . PROOF. Without loss of generality we can assume that T has a dense range and therefore E is separable (as well as X) and T* is a one-to-one operator. Write

Vn--{xESx"

3f ESx, MT*(nBE,)with f(x)--l},

n--1,2 .....

It is clear that Z C U Vn. We shall show that each Vn is E-closed in Bx. Let E- l i m x j = x, xj E Vn, x ~ B x . By the definition of Vn there exists a sequence of functionals { f j } C Sx, M T * ( n B E , ) such that f j ( x j ) = 1, for all j. Let gj = T * - l ( f j ) , j = 1,2 . . . . . Since the sequence {gj} C nBE, is bounded and E is separable, we can assume that there exists g E n BE,

such that w*- lim gj

=

g.

(5.5)

Since l i m T x j = T x it follows from (5.5) that g ( T x ) = l i m g j ( T x j ) = lim f j ( x j ) and therefore g ( T x ) = 1. It is clear that f -- w * - l i m T * gj = T* g E B x , M T * ( n B E , ) . Since f ( x ) = 1 and x E B x we deduce that I l x l l - I l f l l - - 1 and thus x E Vn. D The following proposition is the main step in the proof of Theorem 5.19.

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PROPOSITION 5.22 ([66]). Let X be a separable Banach space and r be a Hausdorff topology on X that is consistent with the linear structure o f X and is strictly weaker than the norm topology. I f sm Sx C ~_J~ Vn, where each Vn is a r-closed in B x subset o f S x , then X contains an isomorphic copy o f co. PROOF. Without loss of generality we can assume that the sequence {Vn }n~__lis increasing. Let T:12 --+ X be an arbitrary compact linear operator with a dense range such that the image of the unit ball K1 = TBI2 is contained in int B x . Since K1 N V1 = 0 and K1 is compact, there exists a r-closed balanced r-neighborhood of zero G1 such that (K1 + G1) N V1 = 0. Since the set G1 is unbounded, there exists an element Xl e r a n t G1 such that, for K2 ---]-Xl § K1, (1) K2 C intBx, (2) K2 C K1 § r a n t G1, (3) dll.ll(K2, S x ) < 2 -2. By K2 A V2 = 0 (see (1)), compactness of K2 and (2), there exists a r-closed balanced rneighborhood of zero, G2, such that (K2 + G 2 ) C (K1 + r a n t G1) and (K2 + G2) f'l V2 = 0. Since the set G2 is unbounded there exists an element x2 e r a n t G2 such that for K3 = -f-x2 § K2, K3 C int B x , K3 C K2 + r a n t G2, dll.ii (K3, S x ) < 2 -3. In this way we construct a sequence of elements {Xn }, a sequence of compact sets Kn y.~,~-I +xi + K1, n -- 1, 2 . . . . . and a sequence of r-closed sets {Gn} such that (Kl + G1) (K2 § G2) D (K3 § G3) D ..- and (Km § Gm) f-) Vm ----13 for every integer m. Thus, (a) (y~7 +n -']-Xi § gl) C (y~n~ -+-Xi + g l § Gm+l), m , n - 1, 2 . . . . . (b) (y~n -+-xi § KI + Gm+l) f') Vm+l -- 0, m = 1, 2 . . . . . (c) dll.ii( y ~ n -+-xi + K1, S x ) < 2 -m, m = 1, 2 . . . . . Let us assume that the Banach space X does not contain co. Then, by the BessagaPetczyfiski theorem [9], the series Y~xi converges unconditionally and the set F {~_,~ aixi" ai -- -4-1} is compact. By the construction, F + Kl C B x . From (a) and (b) (sending n to infinity) we get (F + K1) A Vm+l -- 0 for all m. Therefore (F + K1) A sm Sx - 0. From (c), using the compactness of the set F + K l, we deduce that (F + K 1 ) N S x ~ 0. Letx e F, y e K1 be such that (x + y ) e Sx. S i n c e ( F + K 1 ) C B x we have y e OaK1 (OAK1 is the algebraic boundary of the set K1). Since x + y is not a smooth point of Sx there exist two distinct functionals f, g e S x , such that 1 - f (x + y) - g(x + y). It is clear that the functionals f and g are supporting functionals for the set x + K1 (recall that (F + K1) C B x ) . But every support point of the set x + Kl is a smooth point of Sx, since K1 is an image of the unit ball of Hilbert space and linK1 is dense in the space E. This contradiction completes the proof. [2 THE PROOF OF THEOREM 5.19. (2) => (1). The proof of this implication is just a direct combination of Lemmas 5.20 and 5.21, equality (5.1) and Proposition 5.22. The inverse


649

implication (1) =~ (2) is a simple consequence of Sobczyk's theorem: co is complemented in each separable Banach space which contains it (for details see [66]). D REMARK. The proof of Theorem 5.19 works just as well if we replace in (2) the unit ball of an equivalent norm by a general CCB body (i.e., not necessary symmetric). By applying Theorem 5.19 to reflexive spaces we get, in particular COROLLARY 5.23 ([66]). Let W C X be a CCB body in an infinite-dimensional reflexive Banach space. Then the set exp W is uncountable. It is perhaps worthwhile to state the following reformulation of Theorem 5.19. THEOREM 5.24. Let X be a separable Banach space which does not contain an isomorphic copy of co and let the set w*-exp B x , be represented as a union of an increasing sequence of sets: oo

w*-exp B x , -- U Di. i=1

Then there exists an integer j such that the

set Dj

is norming.

Theorem 5.24 looks like the Baire Category theorem. This similarity is strengthened if we recall that a set D is norming iff the set w*-cl co{+D} contains some ball. The following consequence of Theorem 5.24 may be considered as a more precise version of the Banach-Steinhaus theorem, for separable Banach spaces that do not contain co. COROLLARY 5.25 ([66]). Let X be a separable Banach space which does not contain an isomorphic copy of co and let A be an unbounded subset of X. Then there exists an f E w*-exp B x , such that the set f ( A ) is unbounded. PROOF. Suppose to the contrary, that for each f E w*-exp B x , , the set f ( A ) is bounded. By writing Dn = {f E w*-expBx,: sup If(A)l < n}, we have w*-exp B x , = U Dn, where {Dn } is an increasing sequence of sets. By Theorem 5.24, there exists an integer m such that the set Dm is norming, i.e.,

3 c >OVx E x ,

Ilxll

C up{lf(x>l f EDm}.

Thus sup{ Ilx II: x E A} ~< Cm, a contradiction. We conclude this section with an application to classical analysis. The following result may be considered as a generalization of a well-known theorem of Zygmund on lacunary trigonometric series. We call a sequence {Xn(t)}n~__l of functions in C[0, 1], a generalized Sidon system if it is equivalent to the natural basis of l l.

V.P Fonf et al.

650

COROLLARY 5.26 ([62,66]). Let {Xn}n~__l C C[0, 1] be a generalized Sidon system and X = [Xn]n~=l. Assume that X separates the points of [0, 1]. Let B C [0, 1] be the set of all peakpoints for the subspace X. If the partial sums of the series ~-,n=l anXn(t) are bounded oo at each point t E B, then the series converges absolutely, i.e., Y~n=l [anl < oo. PROOF. As usual, we identify the interval [0, 1] with a subset of C*[0, 1] and moreover, with a subset of X*. By definition, w*-exp Bx, = +B. Put k

Dn -- { s E B" supk ~ a i x i ( s )

n=l,

....

i=1

It is clear that the sets Dn form an increasing sequence of sets and that w*-exp Bx, = Un%l i D n . Since X, being isomorphic to ll, does not contain co, it follows from Theorem 5.24 that there exists an index m such that the set Dm is norming (say r-norming). In particular, for each n

rll

aixi

i--1

{ I k aixi (S) " s E D m }

~< sup s~p Z

~m.

i=1

Since {Xn } is equivalent to the natural basis of 11, we deduce that ~n~176lan I < cx).

[-1

6. Convex polytopes In this section we study the class of infinite-dimensional polytopes. We shall consider only closed convex bounded (CCB) sets in a Banach space. DEFINITION 6.1. A closed convex bounded set P in a Banach space X is called a polytope if every finite-dimensional section of it is a (usual finite-dimensional) polytope. The Banach space X is called polyhedral if its unit ball Bx is a polytope. This definition was introduced by Klee [105] (who actually considered only unit balls of Banach spaces). There are in the literature other definitions of polytopes. Some of them are isomorphically equivalent to Definition 6.1 (see [45] and [71]), while the others are completely different, see, e.g., Definition 6.27. It seems however, that only the notion we have just defined has led to an interesting theory. In [103] Klee proved that a finite-dimensional CCB set is a polytope iff every twodimensional section of it is a polygon. Thus in Definition 6.1 it is enough to consider only two-dimensional sections of P (and if P is symmetric, just two-dimensional central sections). This observation will simplify several proofs below. It is reasonable to expect that a discussion of polytopes will start with some examples of such sets in "nice" Banach spaces (for example Hilbert space). It will soon become evident that this is impossible; there are no infinite-dimensional polytopes in reflexive Banach spaces. On the other hand, the unit ball of co is easily seen to be a polytope. In some sense

651


it is a typical example of an infinite-dimensional polytope. All this will be clarified by the results below. We start with a result which shows, in particular, that sections cannot be replaced in Definiton 6.1 by continuous images. THEOREM 6.2 ([117]). Every infinite-dimensional Banach space X has a two-dimensional quotient space whose unit ball is not a polygon. The idea of the proof is simple. We start with any two-dimensional quotient which is a polygon and show that we can perturb it slightly to get a two-dimensional quotient with more vertices. In this way we get a sequence of two-dimensional quotient spaces whose unit balls have a monotonely increasing number of vertices which converge to a limiting two-dimensional quotient space whose unit ball is not a polygon. This approximation argument has to be done somewhat carefully since the number of vertices of a polygon is only a lower semi-continuous function (in the Hausdorff metric). In order to present the details of the argument we need three simple lemmas. The first two are simple facts about finite-dimensional polytopes. We shall use below the following notation. For a set C we denote its cardinality by ICI. For a set A in a Banach space and e > 0 we denote by A~ the set {x: d ( x , A) ~< e}. The first lemma is completely elementary. LEMMA 6.3. Let P be a symmetric polygon in the plane (taken, f o r instance, with the usual Euclidean norm). Then there is an e = ~(P) > 0 so that f o r any convex body C in theplane with OC C (OP)~ we have lextC] ~> ]extP]. LEMMA 6.4. Let X be a finite-dimensional polyhedral Banach space. Let T be a linear map from X t o I[~ 2 o f rank 2. Assume that there is a hyperplane Y C X so that T B x = T By. Then f o r every e > 0 there is a linear W : X - - + ~2 SO that IITIy - WlYll ~ s,

OWBx C (OTBx)e,

lext W B x l > l e x t T B x l .

171 PROOF. Let {Vj }j__l be the set ext T B x . If, for some j , T -1Vj 0 B x is not a singleton let ~o be a functional on X which takes on this set both positive and negative values and let u be a vector in the direction of a line in the plane which touches T Bx just at vj. The operator W x = T x + O~p(x)u will have all the desired properties for small enough positive 0. We may thus assume that for every j there is a unique xj ~ ext B x such that T x j = vj. Clearly Y D {Xj}jm=l 9Let T1 be an operator with liT1 - TII ~< e / 2 so that T1 is one-to-one on ext B x . If lext T1Bxl > lext T Bxl we can take W = T1. If this is not the case (and e is small enough) it follows that ext T1Bx - - {Tlxj}jm=l 9Let u be a vector in the plane so that no vector of the form {/'1 (p - q): p, q 6 extBx} is in the direction u, let q9 be a non-zero functional which vanishes on Y and consider T2,z(x) = Tlx + )@(x)u. Clearly for all )~, T2,~. is one-to-one on ext B x . Let )~0 = max{)~ ~> 0: T2,~B x C T1Bx }. For sufficiently small positive 0 the operator W -- T2,)~0+0 will have the desired properties. D

LEMMA 6.5. Let X be an infinite dimensional Banach space and let T be a quotient map from X onto a two-dimensional polyhedral space L. Let 2n = [ext BL] and let m ~ n. Then

652

V.P. Fonf et al.

there is an (m + 1)-dimensionalquotient space Z o f X and quotientmaps Tt: X --+ Z and V" Z --+ L so that T = V T I.

PROOF. Let ext BE = { - q - u i } ~ . Put cr -- {i" T - l u i 0 S X - - ~ } . If cr -- 0 let Y' be any subspace of Ker T of codimension m + 1 in X and put Z -- X~ Y'. If o- 7~ 0, we can choose for every i 6 cra sequence {xJ}~_l- and vectors xi so that T(xi + x j ) -- Txi -- ui for all j and limj ]]xi + x j 11- 1. Since the sequence {xJ}j_l has no w-cluster points we can assume (by passing to a subsequence if needed), that {x j }~c_1 is a basic sequence. By passing to further subsequences we may assume that Y1 -- [x/]iEcr, j--1 .... is of infinite co-dimension. Take as Y' any subspace with Y1 C Y' C ker T so that the co-dimension of Y' in X is (m + 1) and put Z - X~ Y'. [3 PROOF OF THEOREM 6.2. Assume to the contrary that all two-dimensional quotient

spaces of X are polyhedral. By [105] all finite-dimensional quotient spaces of X are polyhedral too. We start with an arbitrary two-dimensional quotient space L 1 of X with the quotient map 7'1 : X --+ L1. Put 2nl = ]ext BLI] and apply L e m m a 6.5 for L -- L1, T = 7'1, n = n,, m = n l + 1. By the choice of m there is a hyperplane (say Y1) in Z1 such that V1Br~ contains ext BL~ and thus V1Br~ = BE1. Put 2el = e(BCl) (the number given by L e m m a 6.3). Next, choose points {xl}l ' in Sx and 61 > 0 such that T(xli 6 Y, for e v e r y / and such that, whenever Ilui - Tlx 1 II < 61 for every i, we have 0 co{u]}l 1 c (Bcl)~I. By L e m m a 6.4 there exists an operator W1 : Z1 --+ L1 such that 2 n 2 - - l e x t W l B g l l > 2n,, and

[IWIT;x: - T, xlll < 6 , / 2

Vi

OWIBz1 C (SL1)ej.

Let T2 - - W 1 T( and L2 be the two-dimensional Banach space with the unit ball BL2 = W1BzI. Apply L e m m a 6.5 again for T = T2, L -- L z, n -- n z, m -- n l + n2 + 1. Let Y2 be a hyperplane in Z2 with T2x ] E Y2 for every i -- 1 . . . . . n l and VzBr2 = B L 2 . Take e2 so small that (BLz)E2 C (Bcl)~I and e2 < e(BL2)/2. Choose 32 > 0 and {X2}l 2 C Sx as in the first step and then by L e m m a 6.4, find an operator W z ' Z 2 --+ L2 such that IIWzTJxki -- Tzx/kl[ < & for k = 1,2 a n d / = 1 . . . . . nk, OWzT~Bx C (Sc2)~2 and such that 2n3 - l e x t W2T~Bxl > 2n2. Continuing in this manner we get sequences {nk}, {6~}, {ek}, {Tk}, {Lk} and {x/k} such that (1) nl < n 2 < . . . . (2) ]ext B/~I - 2n~, k - 1, 2 . . . . . (3) (Sck)ek C (SLk-1)~k-1, e~ < e~-l, k = 2, 3 . . . . . (4) For every symmetric convex body C in the plane such that OC C (Sck)zEk we have lextCI ~> 2nk. (5) {x/~}Tk 1 C Sx and whenever I l u i - T~x~ill < 6k for i - - 1 . . . . . n~, we have O(co{+ui 1i=1) ~k C (OTkBx)~k.

/

(6) IIT~x - T~-lX J II < a j / 2 ~-~ i = 1

nj j -- 1

k-

1


653

By (3), IITkll ~ (1 + el)liT111 for every k and hence, by w* compactness of the unit ball B x , , the sequence {Tk} has a limit point T in the strong operator topology. By using (3) again we get that OTBx C (1 + ek)OTkBx for each k. Also, by (6), IlZxki -- Tkxkill < 6k for every i and k and hence by (5),

T B x ~ co{+Txki }i%, ~ ( 1 - - e k ) B L , . We have thus that O T B x C (SLk)2ek for every k and therefore, by (1) and (4), the set of extreme points of cl T Bx is not finite, i.e., this set is not a polygon. This contradiction completes the proof. D

No infinite-dimensional dual (and in particular reflexive) Ba-

COROLLARY 6.6 ([117]).

nach space is polyhedral. The next proposition is of a general nature but we shall see that it is very useful in the study of polytopes. PROPOSITION 6.7 ([69]). Let V be a CCB symmetric subset of an infinite-dimensional Banach space X such that int V = 0. Then there exists a subspace E C X such that the section V f-) E is infinite-dimensional and compact. PROOF. Without loss of generality we may assume that X is separable. Therefore there exists a countable norming subset {gk} C X* for X. Let Y be the Banach space span V with the norm generated by V as the unit ball and let T : Y --+ X be the natural embedding. Thus the image T By = V is closed and by the H a h n - B a n a c h Theorem T ' X * is a 1-norming linear manifold. Also, it is clear that the restriction of T to any finite codimensional subspace of Y is not an isomorphism. By using these two facts we construct a sequence {Yk} C Sy as follows. Fix a sequence {Ej }oc j=0 of positive numbers tending fast to zero with e0 = 1/2. Take Yl E Sy with []Tyll] < 2 -1 and let hl E ST*X* be such that h l ( y l ) > 1 - E l . Choose y2 E Sy f-) [hi, T*gl] • with ]]Tyzl] < 2 -2 and put E2 -- [Yl, Y2]. Find {h2 . . . . . hm2} C BT*X* such that {hzlE2 . . . . . hmz[E2} is an ez-net for BE~. Take Y3 E Sy A [T*gl, T*gz, hl . . . . . hm2] • with I[Ty3[[ < 2 -3, and continue in an obvious way. A standard calculation shows that {yk } is a basic sequence and that for the partial sum operators we have [ISn I[ l

-

Ix;(ry)l-

Ilry

llly;(y)l

y,

(3.2)

where # is a probability measure, M is a subspace of L ~ ( # ) , and Mp is a subspace of L p (/z) containing M, or alternatively Mp is the closure of M in L p (/z). For p ~> 1, T is said to be strictly p-integral if it factors through the entire inclusion L ~ ( # ) r LP(lz), that is, we can take M -- L ~ ( / z ) and Mp = LP(I z) in (3.2). This notion differs slightly from that of a p-integral operator, as defined in Basic Concepts. However, the two notions coincide if, say, Y is reflexive. As explained in Basic Concepts, for p ~> 1 every p-summing operator on C(K) is strictly p-integral, so that the p-summing and the strictly p-integral operators on C (K) coincide.

678

T. W. Gamelin and S. V. Kislyakov

We recall that any absolutely summing operator is weakly compact. For 1-integral operators, we can use the Dunford-Pettis property of L 1(#) to say more. LEMMA 3.1. A (strictly) 1-integral operator from X to a reflexive Banach space is compact. In particular, an absolutely summing operator from C (K) to a reflexive Banach space is compact. To see this, consider the factorization (3.2) above, with p - 1, M -- L ~ ( # ) , Mp = L p (#). Since Y is reflexive, V is weakly compact. By the Dunford-Pettis theorem, V maps weakly compact subsets of L 1(#) to norm-compact subsets of Y. Since bounded subsets of L ~ (#) are weakly precompact in L 1 ( # ) , the composed operator T maps bounded subsets of X into norm-compact subsets of Y, and T is compact. The prototype for an absolutely summing operator that is not 1-integral is the Paley operator P on the disk algebra CA. The Paley operator assigns to a function f on the unit circle F the sequence of 2kth Fourier coefficients {f(2k)}~_l . Paley's inequality is

cxz

)1/2 CEil fill,

k=l

fell

l(m),

for some constant ce > 0. In other words, the restriction of the Paley operator P to H 1(m) is a bounded operator from H 1(m) to g2. For the proof, see [24,53]. Let M be the closed linear span in L2(m) of the exponential functions exp(i2k0), k ~> 1. It is a classical fact (see [53]) that the LP-norms on M are equivalent, for 0 < p < oo. Further, for 1 M

> g2,

and P operating on C ( F ) is p-integral for 1 < p < oo. On the other hand, P is not compact, so P is not absolutely summing on C (F). The story is different if we restrict P to CA. Paley's inequality yields the factorization P ),~A ~ - +n"tl ' m "

v

2

where V is the Paley operator on Hi(m). Thus P is absolutely summing o n CA. On the other hand, P maps the exponential functions exp(i2k0), k ~> 1, to the standard basis vectors of g2, so that P is not compact, and P is not 1-integral. This shows incidentally that CA is not complemented in C ( F ) , nor even isomorphic to any quotient space of a C(K)-space, or else the composition of the projection and P would produce an absolutely summing operator on C (K) that is not compact. Our aim is to transfer the Paley operator and this final observation to an arbitrary uniform algebra. Paley's inequality transfers directly, as follows.

Uniform algebras as Banach spaces

679

LEMMA 3.2. Let A be a uniform algebra on K. Let f E A satisfy Ilfll ~ 1, and let r be any measure on K orthogonal to A. Then the sequence

P(r)-{f

f2k dr ]k~__l

belongs to ~2, and IIP(v)II2 ~ cP Ilvll, where cp is the best constant for Paley's inequality. The estimate persists for f E H ~ ( I r l ) , the weak-star closure of A in L ~ ( I r l ) .

PROOF. Define U ' C ( F ) --+ C ( K ) by (Ug)(s) - ~ ( f ( s ) ) for g E C ( / ' ) and s E K, where is the Poisson integral of g. Then IIUII ~< 1, U(z n) -- f n , and U(~ n) = f n . Now U * ' M ( K ) --+ M ( F ) sends A • to C ) , which by the E and M. Riesz theorem is identified with H~(m). Thus U * ( r ) - h d m for some h E H i ( m ) , and

J K j g2k dr -- f r ~2kh(•) dm,

k~>l.

Paley's inequality for h yields IIP(v)II2 ~ cpIIhlll = cpIIU*(v)ll ~ cpIIvll. This proves the first statement of Lemma 3.2, and the second is obtained by applying the first to the uniform algebra H ~ ( I r l ) and noting that r generates a functional on L ~ ( I r l ) orthogonal to H ~ ( I r l ) . D THEOREM 3.3 (Kislyakov [29]). If A is a proper uniform subalgebra of C(K), then there is an absolutely summing operator from A to ~2 that is not compact, hence not 1-integral. The proof depends on the Paley operator associated with an extremal function F for a certain dual extremal problem. Since A is proper, there is a measure/z on K such that # 2- A but the complex conjugate /2 o f / z is not orthogonal to A. We assume that the functional f ~ f f d/2 on A has unit norm. By the Hahn-Banach and Riesz representation theorems, there is a measure )~ on K such that IlZll = 1 and )~ - / 2 2_ A. Let {fn} be a sequence of functions in A such that IIf~ II ~< 1 and f fn d/2 --+ 1, and let F E H ~ (llzl + I~1) be a weak-star limit point of the sequence {fn}. Then IFI ~< 1, and f F d)~ = 1, from which it follows that IFI-- 1 a.e. d)~. Now f()~ - / 2 ) 2_ A for all f E A, hence for all f E H ~ ( I # I + I~1). In particular, F2()~ - / 2 ) 2_ A, and r = / z + F2()~ - / 2 ) 2_ A. We define

gdr

/

,

gEA.

(3.3)

k=l

By Lemma 3.2, applied to F E H ~ (I/~1 + I~1) and the orthogonal measure gr, the sequence T(g) is square summable and IIT(g)ll2 ~< cllgrll -- c f Igl dlrl. Thus T can be factored through the closure H j (Irl) of A in L1 (Irl), T'A ~

Hl(Irl)

> g2,


680

and T is absolutely summing. To see that T is not compact, we compute the kth component of T ( F 2k-1) (to be rigorous, rather we must consider l i m n ~ T(f2k-1))"

T(F2~-~)~= f F2~-~-ff2~dr--f F2k+~-ff2kdX + f F2~-~-ff2*d# _ f F2k+~-ff2kdfz.

(3.4)

Let E be the set on which ]F] -- 1. Since )~ is carried by E, the first integral on the right is f F d)~ - 1. Since ]F]n --+ 0 off E as n --+ cx~, (3.4) tends to

l + f E f f d # - f E F d [ z - - l + 2 i I m ( f E f f d# ),

(3.5)

which is not zero. Since the kth components of the vectors T (Fn), n ~ 1, do not tend to zero uniformly in n, the vectors T (F n) do not lie in a compact subset of ~2, and then neither does the image of the unit ball of A under T. Thus T is not compact, and by Lemma 3.2, T is not 1-integral on A. If we analyze the proof of Theorem 3.4, we find that it extends to any closed subspace B of a uniform algebra A providing there is a function f E B such that f A _ B while f ~ A. Indeed, let I be the set of all f satisfying f A ___B. This is a closed ideal in A. We choose # _1_A such that/2 generates a norm-one functional on I, and we proceed as before. For some time it was an open problem, known as the Glicksberg problem, as to whether a proper uniform algebra on a compact space K can be complemented in C (K). Theorem 3.4 settles the Glicksberg problem, and it does even more. THEOREM 3.4. If A is a proper uniform subalgebra of C(K), then A is not isomorphic to a quotient ofa C(J) space. In particular, A is not complemented in C(K). Indeed, suppose A is the quotient of C (J). If we compose the operator T from Theorem 3.3 with the quotient map, we obtain an operator

C(J)

>C ( j ) / Z ~ A

T> ~2

that is absolutely summing, hence compact, by Lemma 3.1. Since the projection is an open mapping, the operator T must be compact, and this contradicts Theorem 3.3. We mention some further conclusions that can be drawn from this circle of ideas. Recall (see Basic Concepts) that a Banach space X has Gordon-Lewis local unconditional structure (GL 1.u.st.) if, roughly speaking, its finite dimensional subspaces are well embeddable in spaces with unconditional basis. This occurs if and only if X** is a complemented subspace of a Banach lattice. Thus C (K) has GL 1.u.st., as do all LP-spaces, 1 ~ s

However, in this way we only obtain an operator quite similar to P (but not P itself). We redefine F to be a weak-star limit point of the sequence {fn } in A**, and we use the realization of A** as a weak-star closed subspace of C** described in the next section. The operator T** may still be defined by (3.3), where now g belongs to A**, and the functions being integrated in (3.3) are the projections of the functions in C** into L ~ (It 1). The correspondence p ( Z ) ~ p ( F ) mapping a polynomial in the coordinate function Z to p ( F ) is of norm at most 1, hence extends to a bounded operator U from CA into A**. Now the hypothesis of GL 1.u.st. implies by the Gordon-Lewis theorem (see [ 15, Theorem 17.7]) that the absolutely summing operator T** factors through an L 1-space, and we obtain

T**U "CA u > A** v > L 1(v) w>

s

.

By Bourgain's extension of the Grothendieck theorem (Theorem 6.5), the composition V U mapping CA into an L 1-space is 2-summing, hence weakly compact. By the DunfordPettis theorem, the weakly compact operator W maps weakly compact subsets of L l(v) to norm-compact subsets of s Thus the composition WVU is compact, contradicting the noncompactness of T**U (see (3.4)-(3.5)). Along similar lines, it can also be proved that if a proper uniform algebra A is a quotient of a Banach space X having GL 1.u.st., then X contains a complemented copy of 11. The crucial observation here is that if X fails to have a complemented copy of 11, then every operator from X to L 1 (13) is weakly compact. To see this, combine the Petczyfiski property of L ~ (v) with [33, Proposition 2.e.8]. In another direction, Garling [21 ] showed that the dual A* of a proper uniform algebra is not a subspace of the dual of a C*-algebra. The proof is modeled on an earlier argument for CA and uses the basic objects ()~, #, and F) appearing in the proof of Theorem 3.4.

4. Tight subspaces and subalgebras of C (K) The classical Hankel operator corresponding to a function g on the unit circle operates from HZ(d0) to HZ(d0) • sending f to g f - P ( g f ) , where P is the orthogonal projection from LZ(d0) onto H2(d0). The Hankel operator is equivalent to the operator f --> g f + H 2 from H 2 to the quotient space L 2 / H 2. The analogue of these operators, acting on subspaces of C (K), has proved a key to understanding uniform algebras. Let A be a closed subspace of C(K). To each g ~ C ( K ) we associate a generalized Hankel operator Sg from A to the quotient Banach space C ( K ) / A by

S g f - - g f -+-A,

f eA.

682

T.W. Gamelin and S. V. Kislyakov

We say that A is a tight subspace of C(K) if the operators Sg are weakly compact for all g ~ C(K). We say that A is a compactly tight subspace of C(K) if Sg is compact for all g E C(K). Tightness was introduced in [12]. Our discussion is based on that paper, and on [46,47]. We will use the representation of the bidual C** of C = C(K) as a uniform algebra. This representation is realized as follows. The dual space of C is the space M(K) of finite (regular Borel) measures on K, with the total variation norm, and this can be regarded as the direct limit of the spaces L 1 ( # ) , # E M(K). The bidual C** is then represented as the inverse limit of their dual spaces L ~ ( # ) , # E M(K). A "simple-minded" way to express this is to say that each element F 6 C** determines for each # 6 M(K) a function Fu E L~ and these satisfy the compatibility condition that Fv = Fu almost everywhere with respect to v whenever v 0 is a weight, f w dO = 1. We arrive at the following analogs of (6.1) and (6.2):

T" H2(wdO) 7r2(T)> y,

(6.3)

T.H~(wdO)

(6.4)

IITII>y.

The question now is whether we can interpolate between (6.3) and (6.4) as we did between (6.1) and (6.2). The answer is that we can replace w by a weight v ~> w, f v ~< C, such that for this new weight the above interpolation is possible. This will follow from results in the next section. Indeed, since L 1(d0) is BMO-regular (see Proposition 7.4), there is a majorant v for w such that log v belongs to BMO, and on account of Theorem 7.7 the desired interpolation holds for this majorant v. This proves Bourgain's theorem. In a standard way, Bourgain's theorem implies that every operator from C~ (or from L 1/ H~) to 12 is absolutely summing. Then from the relations CA ~ (CA ~ CA ~ ' " ")co,

L'/H

IL'/H 9 L /H0' 9 ) , ,

(see [51, III.E. w 12]), it is also standard to conclude that L 1 / H I and C~ are of cotype 2. See [51, III.I, w 14] for more details. We mention another approach to the Grothendieck theorem, due to Maurey. This method gives some information about operators T : C ( K ) ---> Y, where Y is a space of arbitrary finite cotype. In particular, it applies to Y -- LP for any 1 ~< p < c~. For the proof of the following theorem, see [15, Chapter 10] or [30]. THEOREM 6.7 (Maurey). For 1 Y is (q, p)summing if and only if T factors through the inclusion C ( K ) ~ Lq,1 (#) (the Lorentz space) for some probability measure # on K. If Y is of cotype q, the identity operator of Y is (q, 1)-summing. Thus Maurey's theorem shows that every operator from C ( K ) to Y is (q + e)-summing. We recover the Grothendieck theorem by setting q = 2 and applying Lemma 6.3. The following theorem allows us to transport these results to the disk algebra CA.


691

THEOREM 6.8 (Kislyakov). For an arbitrary Banach space Y and q > p ~ 1, every

(q, p)-summing operator T : CA ---> Y extends to a (q, p)-summing operator from C(F) toY. According to the Mityagin-Petczyfiski theorem, the theorem remains true if q = p > 1. The remainder of this section is devoted to an outline of a proof of Theorem 6.8, with some simplifications compared to the exposition in [30]. LEMMA 6.9. Under the conditions of Theorem 6.8, the operator T is lOq-summing. We break the proof of L e m m a 6.9 into four steps. First note that the family of (q, p)summing operators grows as p decreases, so we may assume that p = 1. We assume also that 7rq,1 (T) = 1. When convenient, we use (., -) to denote the pairing between vectors and functionals. Step 1. There is a probability measure )~ on F such that

IlZxll q ~ < q [ ( 1 - ~ , ~ > l

forallx, rpECA satisfying I x l + l ~ l ~ 1.

(6.5)

To see this, we use a trick invented by Pisier to prove his characterization of (q, p)summing operators on C(K) mentioned above. Let

9 X l .....

XnE CA, Z I xj(t) l ~ 1}.

Clearly C n / z zrq,1 (T) -- 1. We choose 6~ "x 1 and for every n find xl n) . . . . . x(~n) E CA .,. (n) . (n) such that ~ j Illxj IIq > 1 and y~'j I~j (t)l ~< 6,/Cn. Then we c h o o s e / ~: (j ' ) E Y* such that ~ j

I1~: ~.!n)) _ 1 , and define a functional )~n on CA by the ~ j !') IIq' ~< 1 and ~ j (TxJ n) ~ ~j

formula )~n ( ~ ) -- ~j(T(Trx~. ")), "J~:!'))" Then )~, (1) -- 1 and IIZ, II ~< ~,. We consider a weak-star limit point of the sequence {)~}. This is a functional on CA. We extend it to C(F) with preservation of norm, obtaining a measure )~ on F . Since )~(1) = 1 and I1~11 ~< lim 6n -- 1, )~ is a probability measure 9 Now let x, 99 E CA satisfy Ixl + I~ol ~< 1. We define Yl . . . . . Yn+I 1 ~< j ~< n and Y,+I = x . Then

( Z l l T y j l l q ) 1/q ~< Cn+, s u p ~ t

E CA

by yj

--

qgx)n) for

ly/(t)] 1 - q l ( 1 -qg,)~)l, from which (6.5) follows. Step 2. We apply the absolute continuity principle. It can be assumed that )~ = v dO, f v dO -- 1, and that (6.5) is valid for x, rp E H a (dO). (Again, we can convolve with Fej6r kernels to ensure this.)


692

Step 3. There is a ~> v,

fa

~< C, such that

1-1/8q 1/8q IITxll ~ 0, and also ot ~< Cllxl since Ixl ~< 1/2. We define successively -

~p --

c~4b -+- i7-/( ~

)

,

4~ -- ~p

1/4

,

q) -- e x p ( - A 4 5 ) ,

b + iT-/(b) where the constant A > 0 will be chosen momentarily. Since 7t is the quotient of functions with values in the right half-plane, it omits the negative axis, and we choose the branch of 45 whose argument ranges between-+-zr/4. Then Req~ ~> 1~1/~/2 and l 1 - qgl ~< C21r Now 17~1 >/or4/( 1 -+- Co), so R e r ~> C3ot. We set A = 1/C3, and then I~01= e x p ( - A R e q ~ ) ~< e x p ( - A C 3 o t ) = 1 - Ixl. Thus we may apply (6.5) with this q9 and )~ = v dO:

IIrxllq f II- t dO

f I*ladO C4 I*tadO

Now, 1~lSa : 1~12b 2 ~ (ot4b) 2 + 7"-/(ot4b)2. Using the L2-estimate for 7-/, we obtain

f

IqblSad0 ~
0 and x E X0 + X1 by

K ( x , t ; Xo, X1) : inf{llxollo + tllxllll" xo-+-xl - - x , xo e Xo, x l

EX1}.

For 0 < 0 < 1 and 0 < q ~< cx~, we define the interpolation space (X0, X1)O,q t o consist of x 6 X0 + X1 such that t o K(x, t; Xo, X1) belongs to L q (dt/t), and we define the norm


695

of x in (X0, X1)O,q to be the norm of t o K ( x , t; X0, X1) in L q (dt/t). Actually the specific expressions for the K-functional and the norm will not play a role for us. Let Y0 C X0 and Y1 C X1 be closed subspaces. We say that the couple (Y0, Y1) is Kclosed in (Xo, X 1) if there is C > 0 such that any decomposition y = x0 + x 1 of an element y E Y0 + Y1 with xi E Xi can be modified to a decomposition y = Y0 + Yl with Yi E Yi and [[Yi 11/ ~ C]]xi 11i, i = 0, 1. In this case we have

(Yo, Y1)o,q = (Yo -+- Y1) A (Xo, Xl)o,q, with equivalence of norms, and the interpolation properties of the couple (X0, X1) and its subcouple (Y0, Y1) are identical. Our basic problem can be formulated as follows. PROBLEM. When is the couple

(XA, YA) K-closed in (X, Y)?

We shall see that this happens fairly often. We begin with a useful duality result. Assume that X0 and X1 are Banach spaces and that X0 A X1 is dense in both X0 and X1. Then, in a natural way, the spaces X 0 and X 1 are included in (X0 A X1)* and, consequently, form a compatible couple. If Yi C Xi (as above), we denote by Yi• the annihilator of Yi in X*, that is, the set of L 6 Xi* such that

L = O o n Yi. LEMMA 7.1. The couple (Yo, Y1) is K-closed in (Xo, X1) if and only if the couple

(Y~-, Y(-) is K-closed in (X~, X~). The proof is left to the reader (see [31,38]). If X is a Banach lattice of measurable functions on ( F • 12, m • #), it often happens that under the duality (f, g) = f f f g dm d#, X* is also a lattice of measurable functions on the same measure space. We will assume that this is the case, and further that both X and X* satisfy the conditions (i)-(iii) above. Then one easily sees that, as in the classical case of the HP-spaces on the circle, we have X ] -- Z(X*)A, where Z is the coordinate function on F . Thus L e m m a 7.1 relates interpolation properties of the couples (XA, YA) and ((X*)A, (Y*)A), X and Y being two lattices as above. The class of BMO functions will play an important role in what follows. Recall that a function f on F is in BMO if f = u + 7-/v, where u, v 6 L ~ . As usual, we disregard the constant functions and define IIf IlBgO to be the infimum of Ilu II + Ilvll over all such representations, where II 9II is the norm in L ~ modulo the constants. LEMMA 7.2. Let w > 0 be a measurable function on F. Then log w ~ BMO if and only if there exist constants C > 1, 0 0 such that w~ C 0 such that any uniform algebra B that is (1 + e) isomorphic to A is actually isometrically isomorphic to A. Thus the algebras C ( K ) are stable, while the annulus algebras are not. Rochberg [43] proved in 1972 that the disk algebra CA is stable, and he went on to study the perturbations of the algebras A ( K ) for K a finitely connected subset of the complex plane with smooth boundary, or more generally for K a finite bordered Riemann surface. The flavor of his work is given by the following result. THEOREM (Rochberg). Let K be a finite bordered Riemann surface. Then for e > 0 small, any uniform algebra B that is (1 + e)-isomorphic to A (K) has the form A (J) for a compact bordered Riemann surface J that is a deformation of K by a quasi-conformal homeomorphism with dilatation tending to 0 with e. For expository accounts of these results, see [44,45]. More recently, Jarosz [26] was able to prove that the nonseparable algebra H ~ ( A ) is also stable. Meanwhile there is currently no known compact set K in the complex plane with nonempty interior such that CA (K) can be shown to be linearly nonisomorphic to the disk algebra. Conformal mapping theory and the relation CA ~ (CA ~3 CA @ " ")co proved by Wojtaszczyk (see [51, III.E, w12]) suggest that a compact set K for which CA (K) (or P ( K ) , or R ( K ) ) is proper but is not isomorphic to the disk algebra should not be too simple (if it exists).

10. The dimension conjecture It is natural to ask how linear topological properties of an algebra of analytic functions of several complex variables reflect the geometry of the underlying domain. The oldest problem along these lines is to determine what effect the number of variables (dimension) has. DIMENSION CONJECTURE. If G1 and G2 are bounded domains in C n and C m respectively, with n ~: m, then the spaces A(G1) and A(G2) are not linearly homeomorphic, nor are H ~ ( G 1 ) and H ~ ( G 2 ) . We discuss briefly some results related to the dimension conjecture. The main references are [11] and [39, w 11]. It is most natural to examine the dimension conjecture first for the polydisks A n and the balls Bn. In [11 ] it was proved that A (A n) is not linearly homeomorphic to A (Am)

704


if m r n. The invariant distinguishing the spaces (in fact, their duals) is the behavior of certain vector-valued multiindexed martingales on the measure space F ~ • ... x F ~ with natural filtration; the martingales in question must have some additional complex analytic structure. The method in [ 11 ] yields the following theorem. THEOREM 1 0.1. Let U1 . . . . . Un, V1 . . . . . Vm be strictly pseudoconvex domains with C2-smooth boundary (the dimension may vary f r o m one domain to another). I f m > n, then A ( V l x . . . x Vm)* does not e m b e d in A ( U ! x . . . x Un)* as a closed subspace. Theorem 10.1 includes the previously known result that the spaces A ( B m ) and A ( A n) are not linearly isomorphic for m, n ~> 2. It had been shown that A (An) * does not embed in a direct sum of an L 1-space and a separable space (see [39, w 11]), whereas A ( B m ) * is such a direct sum by Theorem 4.3. Very little is known beyond Theorem 10.1. Currently it is not even known whether the ball algebras A (Bm) are mutually nonisomorphic for m / > 2. They are all distinct from A ( B 1 ) -- A ( A ) = CA. The latter space is a subspace of C ( F ) with separable annihilator, whereas A ( B m ) for m ~> 2 does not embed in C ( K ) as a subspace with separable annihilator (see [39, w 11 ]). The series { H ~ ( A n ) } seems to be quite similar to {A(An)}; however, in general the method of [11] is not applicable to H ~ ( A n ) . It is known only that H ~ ( A ) differs from H ~ ( A n) for n ~> 2. Again, see [11,39] for proofs. For comparison, we describe the situation concerning the Hardy spaces H 1( F n) and H I ( O B n ) . The spaces of the first series are not isomorphic to one another (see [6,5]), whereas those of the second series are all isomorphic (see [52]). More recently, it was shown that for any strictly pseudoconvex domain with smooth boundary the corresponding space H 1 is isomorphic to the classical H 1 in the disk (see [2]). Finally, note that in contrast to the current state of affairs in one complex variable, it is possible to find in several complex variables many examples of nonisomorphic spaces A (G) where the underlying domains G have the same dimension. For instance, from Theorem 10.1 it follows that A(A 4) is not linearly isomorphic to A ( B 2 x B2).

References [1] H. Alexander and J. Wermer, Several Complex Variables and Banach Algebras, 3rd edn, Springer-Verlag (1998). [2] H. Arai, Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains, Tohoku Math. J. 46 (4) (1994), 469-498. [3] R.M. Aron, B.J. Cole and T.W. Gamelin, Weak-star continuous analytic functions, Canad. J. Math. 47 (1995), 673-683. [4] J. Bergh and J. Lrfstrrm, Interpolation Spaces, Springer-Verlag(1976). [5] J. Bourgain, The nonisomorphism of H 1-spaces in one and several variables, J. Funct. Anal. 46 (1) (1982), 45-57. [6] J. Bourgain, The nonisomorphism of H 1-spaces in a different number of variables, Bull. Soc. Math. Belg., Ser. B 35 (2) (1983), 127-136. [7] J. Bourgain, On weak completeness of the dual spaces of analytic and smooth functions, Bull. Soc. Math. Belg., Ser. B 35 (1) (1983), 111-118.


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[8] J. Bourgain, New Banach space properties of the disc algebra and H ~ , Acta Math. 152 (1984), 1-48; 387-391. [9] J. Bourgain, Bilinearforms on H ~ and bounded bianalyticfunctions, Trans. Amer. Math. Soc. 286 (1984), 313-337. [10] J. Bourgain, The Dunford-Pettis property for the ball-algebras, the polydisc-algebras and the Sobolev spaces, Studia Math. 77 (1984), 245-253. [11] J. Bourgain, The dimension conjecture for polydisk algebras, Israel J. Math. 48 (4) (1984), 289-304. [12] B.J. Cole and T.W. Gamelin, Tight uniform algebras and algebras of analytic functions, J. Funct. Anal. 46 (1982), 158-220. [13] G. David, Op~rateurs integraux singuliers sur certaines courbes du plan complexe, Ann. Sci. l~cole Norm. Sup. 17 (1) (1984), 157-189. [ 14] A.M. Davie, Bounded approximation of analytic functions, Proc. Amer. Math. Soc. 32 (1972), 128-133. [ 15] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge University Press (1995). [16] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag (1999). [ 17] EL. Duren, Theory of H P Spaces, Academic Press (1970). [ 18] T.W. Gamelin, Uniform algebras on plane sets, Approximation Theory, G.G. Lorentz, ed., Academic Press (1973), 101-149. [19] T.W. Gamelin, Uniform Algebras, 2nd edn, Chelsea (1984). [20] T.W. Gamelin, Analytic functions on Banach spaces, Complex Function Theory, Gauthier and Sabidussi, eds, Kluwer Academic (1994), 187-233. [21] D.J.H. Garling, On the dual of a proper uniform algebra, Bull. London Math. Soc. 21 (1989), 279-284. [22] J.B. Garnett, Bounded Analytic Functions, Academic Press (1981). [23] J. Guti6rrez, J. Jaramillo and J.L. Llavona, Polynomials and geometry of Banach spaces, Extracta Math. 10 (2) (1995), 227-228. [24] K. Hoffman, Banach Spaces of Analytic Functions, Prentice-Hall (1962). [25] K. Jarosz, Perturbations of Banach Algebras, Lecture Notes in Math. 1120, Springer-Verlag (1985). [26] K. Jarosz, H ~ ( D ) is stable, J. London Math. Soc. 37 (1988), 490-498. [27] N.J. Kalton, Complex interpolation ofHardy-type subspaces, Math. Nachr. 171 (1995), 227-228. [28] S.V. Kislyakov, Bourgain's analytic projection revisited, Proc. Amer. Math. Soc. 126 (11) (1988), 33073314. [29] S.V. Kislyakov, Proper uniform algebras are uncomplemented, Soviet Math. Dokl. 40 (3) (1990), 584-586. [30] S.V. Kislyakov, Absolutely summing operators on the disc algebra, St. Petersburg Math. J. 3 (4) (1992), 705-774. [31] S.V. Kislyakov, Interpolation of HP-spaces: some recent developments, Function Spaces, Interpolation Spaces, and Related Topics, Israel Math. Conf. Proceedings 13, Amer. Math. Soc., Providence, RI (1999). [32] S.V. Kislyakov and Q. Xu, Interpolation of weighted and vector-valued Hardy spaces, Trans. Amer. Math. Soc. 343 (3) (1994), 1-34. [33] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Ergebn. Math. Grenzgeb. 92, Springer-Verlag (1977). [34] H. Milne, Banach space properties of uniform algebras, Bull. London Math. Soc. 4 (1972), 323-326. [35] J. Mujica, Complex Analysis in Banach Spaces, North-Holland, Amsterdam (1986). [36] M. Nagasawa, Isomorphisms between commutative Banach algebras with application to rings of analytic functions, Kodai Math. Sem. Rep. 11 (1959), 182-188. [37] D. Oberlin, A Rudin-Carleson theorem for uniformly convergent Fourier series, Michigan Math. J. 27 (1980), 309-313. [38] G. Pisier, Interpolation between Hardy spaces and noncommutative generalizations. I, Pacific J. Math. 155 (1992), 341-368. [39] A. Peiczyfiski, Banach Spaces of Analytic Functions and Absolutely Summing Operators, Conf. Board of Math. Sciences, Vol. 30, Amer. Math. Soc. (1977). [40] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag (1992). [41 ] IT Privalov, Boundary Properties of Analytic Functions, GITTL, Moscow, Leningrad 1950 (Russian). [42] R.M. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, SpringerVerlag (1986). [43] R. Rochberg, The disk algebra is rigid, Proc. London Math. Soc. 39 (1979), 119-129.

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[44] R. Rochberg, Deformation theory for uniform algebras: an introduction, Proceedings of the Conference on Banach Algebras and Several Complex Variables (New Haven, Conn., 1983), Contemp. Math. 32, Amer. Math. Soc. (1984), 209-216. [45] R. Rochberg, Deformation theory for algebras of analytic functions, Deformation Theory of Algebras and Structures and Applications (I1Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 247, Kluwer (1988), 501-535. [461 S.E Saccone, The Petczyhski property for tight subspaces, J. Funct. Anal. 148 (1) (1997), 86-116. [47] S.F. Saccone, Function theory in spaces of uniformly convergent Fourier series, to appear. [48] T.P. Srinivasan and J.-K. Wang, Weak* Dirichlet algebras, Function Algebras, E Birtel, ed., Scott-Foresman (1966), 216-249. [49] E.L. Stout, The Theory of Uniform Algebras, Bogdon and Quigley (1971). [50] J. Wermer, Function rings and Riemann surfaces, Ann. Math. 67 (1958), 45-71. [51] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge University Press (1990). [52] T. Wolniewicz, On isomorphisms between Hardy spaces on complex balls, Ark. Math. 27 (1) (1989), 155168. [53] A. Zygmund, Trigonometric Series, Vol. I, II, Cambridge University Press (1959).

CHAPTER

17

Euclidean Structure in Finite Dimensional Normed Spaces

Apostolos A. Giannopoulos Department of Mathematics, University of Crete, Heraklion, Greece E-mail: apostolo @math. uch.gr

Vitali D. Milman* Department of Mathematics, Tel Aviv University, Tel Aviv, Israel E-mail: vitali@math, tau.ac, il

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Classical inequalities and isotropic positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Classical inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Extremal problems and isotropic positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Background from classical convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Steiner's formula and Urysohn's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Geometric inequalities of "hyperbolic" type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Continuous valuations on compact convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Dvoretzky's theorem and concentration of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Concentration of measure on the sphere and a proof of Dvoretzky's theorem . . . . . . . . . . . . . 4.3. Probabilistic and global form of Dvoretzky's theorem . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Applications of the concentration of measure on the sphere . . . . . . . . . . . . . . . . . . . . . . 4.5. The concentration phenomenon: L6vy families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Dvoretzky's theorem and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Isomorphic versions of Dvoretzky's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The low M*-estimate and the quotient of subspace theorem . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The low M*-estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The g-position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

709 710 710 711 718 726 726 729 732 735 735 737 740 743 744 746 748 749 749 751

*The authors acknowledge the hospitality of the Erwin SchrSdinger International Institute for Mathematical Physics in Vienna, where this work has been completed. The second named author was supported in part by the Israel Science Foundation founded by the Academy of Sciences and Humanities. H A N D B O O K OF THE GEOMETRY OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 707

708

A.A. Giannopoulos and V.D. Milman

5.3. The quotient of subspace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Variants and applications of the low M*-estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Isomorphic symmetrization and applications to classical convexity . . . . . . . . . . . . . . . . . . . . . 6.1. Estimates on covering numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Isomorphic symmetrization and applications to classical convexity . . . . . . . . . . . . . . . . . . 7. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. The hyperplane conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Geometry of the B a n a c h - M a z u r compactum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Symmetrization and approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Quasi-convex bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Type and cotype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Nonlinear type theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

752 754 756 756 759 763 763 765 768 769 769 771 772

Euclidean structure in finite dimensional normed spaces

709

1. Introduction In this article we discuss results which stand between geometry, convex geometry, and functional analysis. We consider the family of n-dimensional normed spaces and study the asymptotic behavior of their parameters as the dimension n grows to infinity. Analogously, we study asymptotic phenomena for convex bodies in high dimensional spaces. This theory grew out of functional analysis. In fact, it may be viewed as the most recent one among many examples of directions in mathematics which were born inside this field during the twentieth century. Functional analysis was developed during the period between the World Wars by the Polish school of mathematics, an outstanding school with broad interests and connections. The influence of the ideas of functional analysis on mathematical physics, on differential equations, but also on classical analysis, was enormous. The great achievements and successful applications to other fields led to the creation of new directions (among them, algebraic analysis, noncommutative geometry and the modem theory of partial differential equations) which in a short time became autonomous and independent fields of mathematics. Thus, in the last decades of the twentieth century, geometric functional analysis and even more narrowly the study of the geometry of Banach spaces became the main line of research in what remained as "proper" functional analysis. The two central themes of this theory were infinite dimensional convex bodies and the linear structure of infinite dimensional normed spaces. Several questions in the direction of a structure theory for Banach spaces were asked and stayed open for many years. Some of them can be found in Banach's book. Their common feature was a search for simple building blocks inside an arbitrary Banach space. For example: does every Banach space contain an infinite unconditional basic sequence? Is every Banach space decomposable as a topological sum of two infinite dimensional subspaces? Is it true that every Banach space is isomorphic to its closed hyperplanes? Does every Banach space contain a subspace isomorphic to some gp or co ? This last question was answered in the negative by Tsirelson (1974) who gave an example of a reflexive space not containing any gp. Before Tsirelson's example, it had been realized by the second named author that the notion of the spectrum of a uniformly continuous function on the unit sphere of a normed space was related to this question and that the problem of distortion was a central geometric question for approaching the linear structure of the space. Although Tsirelson's example was a major breakthrough and introduced a completely new construction of norm, the search for simple linear structure continued to be the aim of most of the efforts in the geometry of Banach spaces. We now know that infinite dimensional Banach spaces have much more complicated structure than what was assumed (or hoped). All the questions above were answered in the negative in the middle of the 90s, starting with the works of Gowers and Maurey, Gowers, Odell and Schlumprecht. Actually, the line of thought related to Tsirelson's example and the concepts of spectrum and distortion were the most crucial for the recent developments. The systematic quantitative study of n-dimensional spaces with n tending to infinity started in the 60s, as an alternative approach to several unsolved problems of geometric functional analysis. This study led to a new and deep theory with many surprising consequences in both analysis and geometry. When viewed as part of functional analy-

710


sis, this theory is often called local theory (or asymptotic theory of finite dimensional normed spaces). However, it adopted a significant part of classical convexity theory and used many of its methods and techniques. Classical geometric inequalities such as the Brunn-Minkowski inequality, isoperimetric inequalities and many others were extensively used and established themselves as important technical tools in the development of local theory. Conversely, the study of geometric problems from a functional analysis point of view enriched classical convexity with a new approach and a variety of problems: The "isometric" problems which were typical in convex geometry were replaced by "isomorphic" ones, with the emphasis on the role of the dimension. This change led to a new intuition and revealed new concepts, the concentration phenomenon being one of them, with many applications in convexity and discrete mathematics. This natural melting of the two theories should perhaps correctly be called asymptotic (or convex) geometric analysis. This paper presents only some aspects of this asymptotic theory. We leave aside typecotype theory and other connections with probability theory, factorization results, covering and entropy (besides a few results we are going to use), connections with infinite dimension theory, random normed spaces, and so on. Other articles in this collection will cover these topics and complement these omissions. On the other hand, we feel it is necessary to give some background on convex geometry: This is done in Sections 2 and 3. The theory as we build it below "rotates" around different Euclidean structures associated with a given norm or convex body. This is in fact a reflection of different traces of hidden symmetries every high dimensional body possesses. To recover these symmetries is one of the goals of the theory. A new point which appears in this article is that all these Euclidean structures that are in use in local theory have precise geometric descriptions in terms of classical convexity theory: they may be viewed as "isotropic" ones. Traditional local theory concentrates its attention on the study of the structure of the subspaces and quotient spaces of the original space (the "local structure" of the space). The connection with classical convexity goes through the translation of these results to a "global" language, that is, to equivalent statements pertaining to the structure of the whole body or space. Such a comparison of "local" and "global" results is very useful, opens a new dimension in our study and will lead our presentation throughout the paper. We refer the reader to the books of Schneider [177] and of Burago and Zalgaller [43] for the classical convexity theory. Books mainly devoted to the local theory are the ones by: Milman and Schechtman [150], Pisier [164], Tomczak-Jaegermann [195].

2. Classical inequalities and isotropic positions 2.1. Notation 2.1.1. We study finite-dimensional real normed spaces X = (•n, I1" II). The unit ball K x of such a space is an origin-symmetric convex body in R n which we agree to call a body. There is a one to one correspondence between norms and bodies in Rn: If K is a body, then Ilxll = min{)~ > 0: x 9 ;~K} is a norm defining a space XK with K as its unit ball. In this way bodies arise naturally in functional analysis and will be our main object of study.


711

If K and T are bodies in ~n we can define a multiplicative distance d(K, T) by

d(K, T) -- inf{ab: a, b > O, K c bT, T cC_.aK}. The natural distance between the n-dimensional spaces XK and XT is the Banach-Mazur distance. Since we want to identify isometric spaces, we allow a linear transformation and set

d(XK, Xv) - inf{d(X, uT)" u E GLn }. In other words, d ( X x , XT) is the smallest positive number d for which we can find u ~ GLn such that K c_ uT c dK. We clearly have d(XK, XT) ~> 1 with equality if and only if XK and XT are isometric. Note the multiplicative triangle inequality d(X, Z) 1 X I K I 1/n -t- (1 - ~,)ITI ~/n.

(2)

Then, the arithmetic-geometric means inequality gives a dimension free version: I~.K -t-(1 - ~ . ) T I > IKIXITI ~-~.

(3)

There are several proofs of the Brunn-Minkowski inequality, all of them related to important ideas. We shall sketch only two lines of proof. The first (historically as well) proof is based on the Brunn concavity principle: Let K be a convex body in ~n and F be a k-dimensional subspace. Then, the function f " F • --+ R defined by f (x) -- [K A (F + x)[ 1/~ is concave on its support.

The proof is by symmetrization. Recall that the Steiner symmetrization of K in the direction of 0 E S n-1 is the convex body So (K) consisting of all points of the form x + )v0, where x is in the projection Po ( K ) of K onto 0 • and [)v[ ~< 89• length(x + 1R0) N K. Steiner symmetrization preserves convexity: in fact, this is the Brunn concavity principle for k = 1. The proof is elementary and essentially two dimensional. A well known fact which goes back to Steiner and Schwarz but was later rigorously proved in [45] (see [43]) is that for every convex body K one can find a sequence of successive Steiner symmetrizations in directions 0 E F so that the resulting convex body K has the following property: K A ( F + x) is a ball with radius r ( x ) , and [K A (F + x)l -- [K A (F + x)l for every x ~ F • Convexity of K implies that r is concave on its support, and this shows that f is also concave. 71 The Brunn concavity principle implies the Brunn-Minkowski inequality. If K, T are convex bodies in IRn, we define K1 = K • {0}, T1 -- T • {1 } in IRn+l and consider their convex hull L. If L(t) -- {x ~ IRn" (x, t) ~ L}, t ~ IR, we easily check that L(0) -- K, L(1) = T, and L(1/2) = K+T Then, the Brunn concavity principle for F -- ]1~n shows that 2 " K+T 2

1/n

1

1

> -~IKI ~/~ + -~ITI~/n

[3

(4)

A second proof of the Brunn-Minkowski inequality may be given via the KnOthe map: Assume that K and T are open convex bodies. Then, there exists a one to one and onto map cp:K --+ T with the following properties: (i) q~ is triangular: the ith coordinate function of 4~ depends only on Xl . . . . . xi. That is, ~ ( x l . . . . . x~) - ( ~ l ( x l ) , ~ : ( x l , x2) . . . . . ~ ( x l . . . . . x~)).

(5)


713

0q~i (ii) The partial derivatives 7Z/are nonnegative on K, and the determinant of the Jacobian of 4) is constant. More precisely, for every x E K [-I Od/)i (det J~)(x) -- i=1

~-X/(x) -

ITI

(6)

Igl

The map 05 is called the Kn6the map from K onto T. Its existence was established in [103] (see also [ 150, Appendix I]). Observe that each choice of coordinate system in ~n produces a different Kn6the map from K onto T. It is clear that (I + 4~)(K) _ K + T, therefore we can estimate IK + T] using the arithmetic-geometric means inequality as follows:

IK+TI

~>f(/

+4~)(K)

-

dX--fKldetJ~+r + Oxi/dx>

(1

+ detJ2/n) n dx

i=1

=

1+

]K]

K J/n

= (Igll/n + Irll/n) n. This proves the Brunn-Minkowski inequality.

(7) D

Alternatively, instead of the Kn6the map one may use the Brenier map O:K -+ T, where K and T are open convex bodies. This is also a one to one, onto and "ratio of volumes" preserving map (i.e., its Jacobian has constant determinant), with the following property: There is a convex function f c C2(K) defined on K such that gr = V f . A remarkable property of the Brenier map is that it is uniquely determined. Existence and uniqueness of the Brenier map were proved in [42] (see also [ 126] for a different proof and important generalizations). It is clear that the Jacobian Jo = Hess f is a symmetric positive definite matrix. Again we have (I + 7r)(K) ___K + T, hence

IK-+-TI >1~ldetJ/+g~(x)ldx- s det(I + H e s s f ) d x /7

(8)

=s t=l

where/~i (X) are the nonnegative eigenvalues of Hessf. Moreover, by the ratio of volumes preserving property of gr, we have Ui%l )~i(x) --]TI/IK] for every x 6 K. Therefore, the


714

arithmetic-geometric means inequality gives

IK + TI >1 fK

( In ] 1+

)U(X)

1/n)ndx =

(IKI 1/n + IZl~/n) n

Fq

(9)

i=1

This proof has the advantage of providing a description for the equality cases: either K or T must be a point, or K must be homothetical to T. Let us describe here the generalization of Brenier's work due to McCann: Let #, v be probability measures on ~n such that # is absolutely continuous with respect to Lebesgue measure. Then, there exists a convex function f such that V f:It~ n --+ R n is defined # almost everywhere, and v(A) = # ( ( V f ) - l (A))) for every Borel subset A of IRn ( V f pushes forward/z to v). If both/z, v are absolutely continuous with respect to Lebesgue measure, then the Brenier map V f has an inverse (V f ) - 1 which is defined v-almost everywhere and is also a Brenier map, pushing forward v to #. A regularity result of Caffarelli [44] (see [5]) states that if T is a convex bounded open set, f is a probability density on IRn, and g is a probability density on T such that (i) f is locally bounded and bounded away from zero on compact sets, and (ii) there exist Cl, C2 > 0 such that Cl ~ g(y) ~ C2 for every y e T, then, the Brenier map V f : (IRn, f dx) --+ (IRn , g dx) is continuous and belongs locally to the H61der class C c~ for some ot > 0. The following recent result [5] makes use of all this information: FACT 1. Let K1 and K2 be open convex bounded subsets of 1Rn with IKll = IK2I = 1.

There exists a Cl-diffeomorphism r : K1 --+ K2 which is volume preserving and satisfies K1 +)vK2 = {x + X ~ ( x ) : x E K1},

& > 0.

(10)

PROOF. Let p be any smooth strictly positive density on ~n. Consider the Brenier maps

~i = V fi " (]t~n, p d x ) --->.(Ki, dx),

i = 1, 2.

(11)

Caffarelli's result shows that they are C 1-smooth. We now apply the following theorem of Gromov [82] (for a proof, see also [5]): FACT 2. (i) Let f : ]~n __~ ]I~ be a C2-smooth convex function with strictly positive Hessian. Then, the image of the gradient map Im V f is an open convex set. (ii) If f l, f2 are two such functions, then I m ( V f l + V f2) = I m ( V f l ) + Im(Vf2). It follows that, for every )~ > 0, K1 + )vK2 -- { V f l (x) + ~Vf2(x)" x E I[~n }.

(12)

Then, one can check that the map r = ~P2 o (l~rl) - 1 :K1 -+ K2 satisfies all the conditions of Fact 1.


715

The existence of a volume preserving l/r" K1 --+ K2 such that (I + ~ ) (K1) = K1 -~- K2 covers a "weak point" of the Kn6the map and should have important applications to convex geometry. We discuss some of them in Section 3.2. (b) Consequences o f the Brunn-Minkowski inequality ( b l ) The isoperimetric inequality f o r convex bodies. body K is defined by

O(K)-

lim

[K + e D n ] -

s--+O+

IK]

e

The surface area O(K) of a convex

.

(13)

It is a well-known fact that among all convex bodies of a given volume the ball has minimal

surface area. This is an immediate consequence of the Brunn-Minkowski inequality: If K is a convex body in R" with ]K] -- ]r Dn ], then for every e > 0 (14)

IK + eDnl '/n >1 IKI '/n + s l O , I~/" = (r + s)ID~I ~/~. It follows that the surface area O(K) of K satisfies 0 ( K ) --

lim s--+O+

[K +

eDn[-

[K[ /> lira

e

s-+O+

(r + e) n - r n

ID, I

= nlDnll/nlK[(n-l)/n

(15)

with equality if K -- r Dn. The question of uniqueness in the equality case is more delicate. (b2) The spherical isoperimetric inequality. Consider the unit sphere S n-1 with the geodesic distance p and the rotationally invariant probability measure or. For every Borel subset A of S " - 1 and for every s > 0, we define the s-extension of A" Ae -- {x ~ S n-l" p(x, A) O. This means that if A _c S " - I and (r(A) = cr(B(x0, r)) for some xo E S n-1 and r > 0, then a ( A s ) >~ cr(B(xo, r + e))

(17)

for every s > 0. Since the a - m e a s u r e of a cap is easily computable, one can give a lower bound for the measure of the s-extension of any subset of the sphere. We are mainly interested in the case or(A) = 1/2, and a straightforward computation (see [61]) shows the following:


716 THEOREM

2.2.1. If A is a Borel subset of S n+l and or(A) - 1/2, then (18)

cr(Ae) >~ 1 - x / ~ e x p ( - e 2 n / 2 ) f o r every e > O.

(The constant ~/-~/8 may be replaced by a sequence of constants an tending to 1/2 as n -----~ o ~ . )

The spherical isoperimetric inequality can be proved by spherical symmetrization techniques (see [ 176] or [61 ]). However, it was recently observed [ 11 ] that one can give a very simple proof of an estimate analogous to (18) using the Brunn-Minkowski inequality. The key point is the following lemma: LEMMA. Consider the probability measure lz(A) -- IAI/IDnl on the Euclidean unit ball Dn. If A, B are subsets of Dn with # ( A ) >~ or, # ( B ) ~ el, and if p ( A , B) -- inf{la - bl" a A, b E B} = p > O, then

oe t})dt

(24)

and estimates # K ( { x E K: If(x)[ ~> t}) for large values of t using Borell's lemma with say A -- {x ~ Rn: If(x)[ ~< 3[[fllq}. The dependence of Cp on p is linear as p --+ ec. This is equivalent to the fact that the L 0~ (K) norm of f

IIf lIL e', (K) = i n f , ~, > 0: IX[

e x p ( l f ( x ) [ / X ) ~< 2

/

(25)

718


is equivalent to II f Ill. The question to determine the cases where c(p) ~_ ~ as p --+ ec in (23) is very important for the theory. This is certainly true for some bodies (e.g., the cube), but the example of the cross-polytope shows that it is not always so. Inverse H61der inequalities of this type are very similar in nature to the classical Khinchine inequality (and are sometimes called Khinchine type inequalities). In fact, the argument described above may be used to give proofs of the Kahane-Khinchine inequality (see [150, Appendix III]). Khinchine type inequalities do not hold only for linear functions. For example, Bourgain [28] has shown that if f :K --+ R is a polynomial of degree m, then Ilfllp ~< c(p, m)llfll2

(26)

for every p > 2, where c(p, m) depends only on p and the degree m of f (for this purpose, the Brunn-Minkowski inequality was not enough, and a suitable direct use of the Kn6the map was necessary). It was also recently proved [108] that (23) holds true for any norm f on R n. Finally the interval of values of p and q in (23) can be extended to ( - 1, +cx~) (see [ 146] for linear functions, [88] for norms).

2.3. Extremal problems and isotropic positions In the study of finite dimensional normed spaces one often faces the problem of choosing a suitable Euclidean structure related to the question in hand. In geometric language, we are given the body K in R n and want to find a specific Euclidean norm in ]~n which is naturally connected with our question about K. An equivalent (and sometimes more convenient) approach is the following: we fix the Euclidean structure in R n, and given K we ask for a suitable "position" u K of K, where u is a linear isomorphism of R n . That is, instead of keeping the body fixed and choosing the "right ellipsoid" we fix the Euclidean norm and choose the "right position" of the body. Most of the times the starting point is a question of the following type: we are given a functional f on convex bodies and a convex body K and we ask for the maximum or minimum of f (u K) over all volume preserving transformations u. We shall describe some very important positions of K which solve such extremal problems. What is interesting is that there is a simple variational method which leads to a description of the solution, and that in most cases the resulting position of K is isotropic. Moreover, isotropic conditions are closely related to the Brascamp-Lieb inequality [41] and its reverse [20], a fact that was discovered and used by K. Ball in the case of John's representation of the identity. For more information on this very important connection, see the article [ 18] in this collection.

(a) John's position. A classical result of E John [94] states that d(X, g~)

(3)

n

for some contact point x of K and D~. Let e > 0 be small enough. From (1) we have tr S III + ~SII ~ [ d e f t / + sS)] l / n = 1 -Jr-~-- - } - 0 @ 2 ) .

(4)

n

Let xe E S n-1 be such that IIx~ -+- eax~ll -- III -+- call. Since D~ ___ K, we have IIx~ll ~ 1. Then, the triangle inequality for II- II shows that

IlSx~ II/>

tr S n

+ O(e).

(5)

We can find x E S n-1 and a sequence em ---> 0 such that xe m --> x. By (5) we obviously have IISx II/> ~.t~s Also, Ilxll - lim IIx~m + emSx~m II - IIIII - 1. This proves (3). Now, let T E L ( R n, R n) and write S - I + e T , e > 0. We can find xe such that IIx~ II Ixe I - 1 and

IIx~ + e Txe II/>

tr(I + eT) n

tr T -- 1 + e ~ . n

(6)


720

Since IIx~ + eTx~ll -- 1 + e(Vllx, II, Txe) -+- O(62), we obtain (Vllx~ II, Txe) ) ~ + O(e). Choosing again em -+ 0 such that xe m --->x 9 S n- 1, we readily see that x is a contact point of K and Dn, and trT ~ ~ .

(7)

But, V Ilx II is the point on the boundary of K ~ at which the outer unit normal is parallel to x (see [177, pp. 44]). Since x is a contact point of K and Dn, we must have V llx II - x. This proves the theorem. D As a consequence of T h e o r e m 2.3.1 we get John's upper bound for d (X, s THEOREM 2.3.2. Let X be an n-dimensional normed space. Then, d(X, s

~ ~/-n.

PROOF. By the definition of the B a n a c h - M a z u r distance we may clearly assume that the unit ball K of X satisfies the assumptions of T h e o r e m 2.3.1. In particular, ]ix ]l ~< Ix] for every x 9 R n . Let x 9 ]~n and consider the map T y = (y, x ) x . T h e o r e m 2.3.1 gives us a contact point z of K and Dn such that

(z, Tz) ~>

tr T

=

H

Ix 12

.

(8)

/7

On the other hand, (z, Tz) = {z,x) 2 ~ Ilzll 2, IIx II2 = Ilx II2 ,

(9)

since one can check that Ilzll, -- 1. Therefore, Ilxll ~ Ixl ~ ~/-~llx II. This shows that Dn K C_ ~/~Dn. D REMARK. The estimate given by John's theorem is sharp. If that d (X, s - v/ft.

X =

s or s

one can check

T h e o r e m 2.3.1 gives very precise information on the distribution of contact points of K and Dn on the sphere S n-1 , which can be put in a quantitative form: THEOREM 2.3.3 (Dvoretzky-Rogers lemma [54]). Let Dn be the maximal volume ellipsoid o f K. Then, there exist pairwise orthogonal vectors Yl . . . . . Yn in IRn such that

n-i

+ 1)1/2 /,/

]]Jill ~ l y i l - 1,

i -- 1 . . . . . n.

(10)


721

PROOF. We define the yi's inductively. The first vector Yl can be any contact point of K and Dn. A s s u m e that yl . . . . . yi-1 have been defined. Let Fi --span{yl . . . . . Yi-| }. Then, t r ( P F l ) -- n -- i + 1 and using T h e o r e m 2.3.1 we can find a contact point xi for which

n-i+1 ~ - - . n

[PFixXi[ 2 -- (Xi, PF.•

(11)

We set yi - PF~Xi/IPF~Xil. Then,

<xi, yi}

1 - - lyil ~ Ily/ll-

Ily/ll 9 Ix/ll,/> (n--i+l) 1/2

= [PFi•

>/

9

n

D

(12)

Finally, a separation argument and T h e o r e m 2.3.1 give us John's representation of the identity: THEOREM 2.3.4. Let Dn be the maximal volume ellipsoid of K. There exist contactpoints X l . . . . . Xm of K and Dn, and positive real numbers )~1. . . . . )~m such that m

I -- Z )~iXi | Xi. i=1 PROOF. Consider the convex hull C of all operators x @ x, where x is a contact point of K and Dn. We need to prove that I / n E C. If this is not the case, there exists T 6 L (R n , R/7) such that

(13)

(T, I / n ) > (x @ x, T)

for every contact point x. But, (T, 1/n) -- tr__TTand (x | x T) -- (x Tx) This would contradict T h e o r e m 2.3.1. D DEFINITION. A Borel m e a s u r e / z on S n-1 is called isotropic if

is

(X, 0} 2 d/z(x) -- # ( s n - 1 ) n-I

for every 0

n

6 S n-1 .

John's representation of the identity implies that m

Z

i=1

)~i (Xi, 0} 2 -- 1

(14)

722


for every 0

6 S n-1 .

This means that if we consider the measure v on S n-j which gives

m a s s )~i at the point xi, i = 1 . . . . . m, then v is isotropic. In this sense, John's position is

an isotropic position. One can actually prove that the existence of an isotropic measure supported by the contact points of K and Dn characterizes John's position in the following sense (see [ 16,68]):

"Assume that Dn is contained in the body K. Then, Dn is the maximal volume ellipsoid of K if and only if there exists an isotropic measure v supported by the contact points of K and Dn." NOTE. The argument given for the proof of Theorem 2.3.1 can be applied in a more general context: If K and L are (not necessarily symmetric) convex bodies in R n, we say that L is of maximal volume in K if L c_ K and, for every w E ~n and T E SLn, the affine image w + T (L) of L is not contained in the interior of K. Then, one has a description of this maximal volume position, which generalizes John's representation of the identity" THEOREM 2.3.5. Let L be of maximal volume in K. For every z E int(L), we can find contactpoints Vl . . . . . Vm of K - z and L - z, contactpoints ul . . . . . Um of (K - z) ~ and (L - z) ~ andpositive reals )~l . . . . . )~m, such that ~-~ ~.juj - - 0 , (uj, vj) -- 1, and m

I -- Z

~,jUj (~ Vj.

j=l

This was observed by Milman in the symmetric case with z - 0 (see [195, Theorem 14.5]). For the extension to the nonsymmetric case see [70], where it is also shown that under mild conditions on K and L there exists an optimal choice of the "center" z so that, setting z = 0, we simultaneously have y~ ) ~ j U j - - ~ ~ , j l ) j - - 0 in the statement above.

(b) Isotropic position - Hyperplane conjecture. A notion coming from classical mechanics is that of the Binet ellipsoid of a body K (actually, of any compact set with positive Lebesgue measure). The norm of this ellipsoid E B ( K ) is defined by IIx II2Es~K~ = Igl 1 f g I<x, y)l 2 dy.

(15)

The Legendre ellipsoid EL (K) of K is defined by

fE

L(K)

(X, y)2 dy -- fK (x, y)2 dy

(16)

for every x 6 R n, and satisfies (see [147])

Es(K)

-

-

(n + 2)I/2IEL(K)1-1 (EL(K)) ~

(17)


723

That is, EL (K) has the same moments of inertia as K with respect to the axes. A body K is said to be in isotropic position if [KI = 1 and its Legendre ellipsoid EL (K) (equivalently, its Binet ellipsoid E s ( K ) ) is homothetical to Dn. This means that there exists a constant L x such that

fg(y,

O)2dy--L2

(18)

for every 0 6 S n- 1 (K has the same moment of inertia in every direction 0). It is not hard to see that every body K has a position u K which is isotropic. Moreover, this position is uniquely determined up to an orthogonal transformation. Therefore, LK is an affine invariant which is called the isotropic constant of K. An alternative way to see this isotropic position in the spirit of our present discussion is to consider the following minimization problem: Let K be a body in R n. Minimize f , i( Ix 12dx over all volume preserving transformations u. Then, we have the following theorem [147]: THEOREM 2.3.6. Let K be a body in R n with IKI = 1. The identity map minimizes fuK Ix 12dx over all volume preserving transformations u if and only if K is isotropic.

Moreover, this isotropic position is unique up to orthogonal transformations. PROOF. We shall use the same variational argument as for John's position. Let T L(R n, R n) and e > 0 be small enough. Then, u = (I + eT)/[det(I + eT)] 1In is volume preserving, and since f , x Ix I2 dx ~> fK Ix I2 dx we get

lx + ETxl 2 dx >~ [det(I + sT)] 2In fK Ixl2 dx.

(19)

But, Ix + eTxl 2 - Ixl 2 + 2e(x, Tx) + 0(82) and [det(l + sT)] 2/n -- 1 + 2e t_~_ _+_0(82). Therefore, (19) implies

fK (x, Tx) dx >~ trT fK Ix 12dx.

(20)

n

By symmetry we see that

fK (x, Tx) dx -- trT fK Ix 12dx

(21)

n

for every T E L (R n , R n). This is equivalent to

fK

(X, O)2 d x

1 n

fK IxlZdx'

0 E S n-1

(22)


724

Conversely, if K is isotropic and if T is any volume preserving transformation, then

fTK lXl2 dx -- fK lTXl2 dX -- fK(X' T*Tx) dx ---- tr(T*T) fK lXl2 dx >~fK lXl2

(23)

which shows that K solves our minimization problem. We can have equality in (23) if and only if T ~ O (n). D It is easily proved that L/~ ~> Lon ~> c > 0 for every body K in ~;~n, where c > 0 is an absolute constant. An important open question having its origin in [26] is the following: PROBLEM. Does there exist an absolute constant C > 0 such that LK ~< C for every body K? A simple argument based on John's theorem shows that L/~ ~< c~/-n for every body K. Uniform boundedness of L/~ is known for some classes of bodies: unit balls of spaces with a 1-unconditional basis, zonoids and their polars, etc. For partial answers to the question, see [ 13,47,48,98,99,105,147]. The best known general upper estimate is due to Bourgain [28]: L x /0 (see [43,177]). That is, ItlKl + ' "

+ tmKml -

Zin~m V(Kil . . . . .

Kin)til ""tin,

1~ 0, can be expanded as a polynomial in t:

]g-~-tOn[--~(~)wi(g) ti,

(1)

i=0

where Wi (K) = V (K; n - i, D,; i) is the i th Quermassintegral of K. It is easy to see that the surface area of K is given by

O(K) = n W I ( K ) .

(2)

Kubota's integral formula

ID,I

Wi (K) -- ID~-i I,-i

fG

IP~KI,-i dvn,n-i(~) ....

(3)

i

applied for i = n - 1 shows that IDnl

Wn-l(K) = ~w(K). 2

(4)

3.1.2. The Alexandrov-Fenchel inequalities constitute a far reaching generalization of the Brunn-Minkowski inequality and its consequences: If K, L, K3 . . . . . Kn ~ 1Cn, then

V ( K , L, K3 . . . . . Kn) 2 >~ V ( K , K, K3 . . . . . K n ) V ( L , L, K3 . . . . . Kn).

(5)

The proof is due to Alexandrov [6,7] (Fenchel sketched an alternative proof, see [58]). From (5) one can recover the Brunn-Minkowski inequality as well as the following generalization for the quermassintegrals:

W i ( K -+- L) l/(n-i) > / W i ( K ) 1~(n-i) -k- Wi(L) 1~(n-i),

i --0 ..... n-

1

(6)

for any pair of convex bodies in R n . If we take L = t Dn, t > 0, then Steiner's formula and the Brunn-Minkowski inequality give

~(~) wi(g) t i =

i=0

IDnl

IOnl

\ \ [--~[

----~(~)( Igli--0]---~nl)(n-i)/n t i

+ t (7)

A.A. Giannopoulosand V.D.Milman

728

for every t > 0. Since the first and the last term are equal on both sides of this inequality, we must have

W( ~n (- K ~ ) /~n(lIDKnI l)

(8)

which is the isoperimetric inequality for convex bodies, and

w(K)__2Wn-I(K)>~2(IKI)I/n IDnl

-~

Urysohn's inequality. Both inequalities

which is

,

(9)

inequalities are special cases of the set of Alexandrov

(Wi(K)) 1~(n-i) (Wj(K)) 1~(n-j) [Dn[ 3.1.3.

>/

[Dn[

, n > i > j >~O.

(10)

Let K be a body in R n . We define

M*(K)

= f s ,,-, I l x l l , o ( d x ) - -w(K) -~.

(11)

The Blaschke-Santal6 inequality asserts that the volume product [KIIK~ is maximized over all symmetric convex bodies in R n exactly when K is an ellipsoid:

IKIIK~ ~< IDnl 2.

(12)

A proof of this fact via Steiner symmetrization was given in [ 12] (see also [ 130,131 ] where the nonsymmetric case is treated). H61der's inequality and polar integration show that

1

M*(K) ~ F I [ d e t A ] | / n

i=1 Also, the following concavity principle (reverse triangle inequality) is true: The function [detA] l/n is concave in the positive cone of Sn. This is in fact easy to demonstrate directly. We want to show that, if A|, A2 are positive then [det(A| + A2)] 1/n >/[detA|] |/n + [detA2] 1/n.

(11)

We may bring two positive matrices to diagonal form without changing their determinants. Then, we should show that for )~i, #i > 0,

(n

)in (H)ljn (H)

1 7 ()~i + #i ) i=1

|/n

~

)~i i=1

_qt_

#i

(12)

i=1

which is a consequence of the arithmetic-geometric means inequality. 3.2.3. We now return to convex sets. The results of Sections 3.2.1 and 3.2.2 have their analogues in this setting, but the parallel results for mixed volumes are much more difficult and look unrelated. Even the fact that the volume of tl K1 + . . . § tm Km is a homogeneous polynomial in ti ~ 0 is a nontrivial statement, while the parallel result for determinants follows by definition. To see the connection between the two theories we follow [5]. Consider n fixed convex open bounded bodies Ki with normalized volume IKi] = 1. As in Section 2.2(a), consider the Brenier maps

~i "( I~n, Yn) ~ gi,

(13)

where Yn is the standard Gauss|an probability density on R n. We have ~ i -- V j~, where fi are convex functions on IRn. By Caffarelli's regularity result, all the 7ti's are smooth


732

maps. Then, Fact 2 from 2.2(a) shows that the image of (R n, Yn) by ~ ti ~i is the interior of y~ ti Ki. Since each Oi is a measure preserving map, we have d e t ( 02fi ) OxkOxl

(x) -- y~(x),

i = 1 . . . . . n.

(14)

It follows that

• i=1

ti Ki

-- f R n d e t ( ~ t i ( i=l

--

t i , " " tin

02fi ) ) dx OxkOxl D

-~xk O~ . . . . .

02fin(X)) -O-~k-O~

dx .

(15)

il ..... in--1

In particular, we recover Minkowski's theorem on polynomiality of ] y~ ti Ki 1, and see the connection between the mixed discriminants D(Hessfil . . . . . Hess fin) and the mixed volumes

V (Kil . . . . . Kin ) -- f~n D (Hess j~, (x) . . . . . Hess fin (x)) dx.

(16)

The Alexandrov-Fenchel inequalities do not follow from the corresponding mixed discriminant inequalities, but the deep connection between the two theories is obvious. Also, some particular cases are indeed simple consequences. For example, in [5] it is proved (as a consequence of (16)) that

V(K1 . . . . . Kn) ~ ~-I [Kill/n

(17)

i=1

3.3. Continuous valuations on compact convex sets

(a) Polynomial valuations. We denote by K~n the set of all nonempty compact convex subsets of ~n and write L for a finite dimensional vector space over IR or C. A function go : K2n --+ L is called a valuation, if go(K 1 U K2) -+- go(K1 A K2) = go(K1) + go(K2) whenever K1, K2 E K~n are such that K1 U K2 E ~ n . We shall consider only continuous valuations: valuations which are continuous with respect to the Hausdorff metric. The notion of valuation may be viewed as a generalization of the notion of measure defined only on the class of compact convex sets. Mixed volumes provide a first important example of valuations. A valuation go: K~n ~ L is called polynomial of degree at most I if go(K + x) is a polynomial in x of degree at most I for every K E K~n. The following theorem of Khovanskii and Pukhlikov [ 102] generalizes Minkowski's theorem on mixed volumes (see also [ 127,2]):


733

THEOREM 3.3.1. Let qg" ]~n ---> L be a continuous valuation, which is polynomial of degree atmost I. Then, if K1 . . . . . Km ~ 1Cn, cp(tlK1 4 - ' " 4- tmKm) is apolynomial in tj >/0 of degree at most n 4- I. Let K -- (K1 . . . . . Ks) be an s-tuple of compact convex sets in ~n, and F" ]1~n ~ C be a continuous function. Alesker studied the Minkowski operator M E. which maps F to M~. F'R~_ --+ C with

(M~F)(/Z1 . . . . . Xs)

--

f~_~4.<s,~.iKi

F(x)dx.

Let ,A(C n) be the Frechet space of entire functions of n variables and C r (R n) be the Frechet space of r-times differentiable functions on R n, with the topology of uniform convergence on compact sets. The following facts are established in [1 ]: (i) If F 6 .A(Cn), then M y F has a unique extension to an entire function on C S, and the operator ME, ",A(C n) --> A ( C '~) is continuous. It follows that if F is a polynomial of degree d then M~. F is a polynomial of degree at most d 4- n. (ii) If F 6 C r (Rn), then M~. F 6 C~ (R~_), and M E. is a continuous operator. Moreover, continuity of the map K w-> M E. with respect to the Hausdorff metric is established.

(b) Translation invariant valuations. A valuation of degree 0 is simply translation invariant. If ~o(uK) = ~p(K) for every K 6/Cn and every u E SO(n), we say that ~0 is SO(n)invariant. Hadwiger [89] characterized the translation and SO(n) invariant valuations as follows (see also [ 101 ] for a simpler proof): THEOREM 3.3.2. A valuation ~p is translation and SO(n)-invariant if and only if there

exist constants ci, i -- 0 . . . . . n, such that

~p(K) -- ~ ci Wi (K) i=0

(1)

for every K c 1Cn. After Hadwiger's classical result, two natural questions arise: to characterize translation invariant valuations without any assumption on rotations, and to characterize O(n) or SO(n) invariant valuations without any assumption on translations. Both questions are of obvious interest in translative integral geometry and in the asymptotic theory of finite dimensional normed spaces respectively (consider, for example, the valuation cp(K) -- fK Ixl2 dx which was discussed in 2.3(b)). It is a conjecture of McMullen [128] that every continuous translation invariant valuation can be approximated (in a certain sense) by linear combinations of mixed volumes. This is known to be true in dimension n ~< 3. In [127,128] it is proved that every transn lation invariant valuation ~o can be uniquely expressed as a sum ~0 - ~-~i=0 ~0i, where ~i


734

are translation invariant continuous valuations satisfying (,/9i (t K) = t i ~o(K) (homogeneous of degree i). Moreover, in the case L = R, homogeneous valuations ~oi as above can be described in some cases: ~00 is always a constant, ~On is always a multiple of volume, ~0n-1 is always of the form

q)n-1 (K) = fs.-' f (u) dSn-1 (K, u),

(2)

where f : S n-1 --+ R is a continuous function (which can be chosen to be orthogonal to every linear functional, and then it is uniquely determined). Under the additional assumption that ~o is simple (~0(K) = 0 if dim K < n), a recent theorem of Schneider [ 178] completely describes ~0: THEOREM 3.3.3. Every simple, continuous translation invariant valuation q) : ]~n ~ I[{

has the form q)(K) -- clKI § Ln-1 f (u) dSn-1 (K, u),

(3)

where f : S n- 1 ~ • is a continuous odd function. REMARK. McMullen's conjecture was recently proved by Alesker [3] in dimension n = 4.

Added in proofs.

Even more recently, Alesker [4] gave a description of translation invariant valuations on convex sets, which in particular confirms McMullen's conjecture in all dimensions.

(c) Rotation invariant valuations. Alesker [2] has recently obtained a characterization of O (n) (respectively SO(n)) invariant continuous valuations. The first main point is that every such valuation can be approximated uniformly on the compact subsets of K~n by continuous polynomial O(n) (or SO(n)) invariant valuations. Then, one can describe polynomial rotation invariant valuations in a concrete way. To this end, let us introduce some specific examples of such valuations. We write v for the (n - 1)-dimensional surface measure on K and n(x) for the outer normal at bd(K) (this is uniquely determined v-almost everywhere). If p, q are nonnegative integers, we consider a valuation ~p,q:]~n --~ ]I~ with

~p,q(K) -- fb

(x,n(x))Plxl 2q dr(x).

(4)

d(K)

All ~p,q are continuous, polynomial of degree at most p + 2q § n, and O(n)-invariant. Theorem 3.3.1 shows that, for every K 9 1On, ~p,q(K § eDn) is a polynomial in e ~> 0, therefore it can be written in the form

~p,q(K§

p+2q+n Z ~r(i)p,q(K)ei" i=0

(5)


735

~lr(i) are continuous, polynomial and O(n)-invariant. These particular valuations sufAll wp,q fice for a description of all rotation invariant polynomial valuations [2]:

THEOREM 3.3.4. I f n >~ 3, then every SO(n)-invariant continuous polynomial valuation (i) ~o" lCn --+ R is a linear combination o f the wp,q. Since ,tr (i) are O(n)-invariant, Theorem 3.3.4 describes O(n)-invariant valuations as wp,q well. The case n = 2 is also completely described in [2] (and the same statements hold true if • is replaced by C).

4. Dvoretzky's theorem and concentration of measure

4.1. Introduction A version of the Dvoretzky-Rogers lemma [54] asserts that for every body K whose maximal volume ellipsoid is Dn, there exist k ~_ x/-ff and a k-dimensional subspace Ek of ]1~n such that Dk ___K N Ek ___2 Qk, where Dk denotes the Euclidean ball in Ek and Qk the unit cube in Ek (for an appropriately chosen coordinate system). Inspired by this, Grothendieck asked whether Qk can be replaced by Dk in this statement. He did not specify what the dependence of k on n might be, asking just that k should increase to infinity with n. A short time after, Dvoretzky [52,53] proved Grothendieck's conjecture: THEOREM 4.1.1. Let e > 0 and k be a positive integer. There exists N = N ( k , e) with the following property: Whenever X is a normed space o f dimension n ~ N we can find a k-dimensional subspace Ek o f X with d(Ek, s ~ L } ) ~ > ~

and

o-({x" f ( x ) ~ L } ) ~ > ~ .

1

A few observations arise directly from this statement: Assume that x ~ S n- 1 has maximal norm IIx II = b. Consider the one-dimensional subspace E1 spanned by x. We have b = M(E1) c > 0 for every norm. This is of course not enough for a proof of Dvoretzky's theorem. On the other hand, recall that M >~ 1/a. By Theorem 4.2.1, every X has a subspace of dimension k >~ ce2n/(ab) 2 on which I1" II is (1 + e)-equivalent to the Euclidean norm. Since we can choose a linear transformation of K x so that ab >,( ( c e 2 ) / l o g ( 1 / e ) ) n ( L f / b ) 2 such that

~Lflxl l+s

~ Ilxll ~ (1 + e ) L f l x l

f o r every x ~ Ek. The proof of Theorem 4.2.1 is now complete. We just have to observe that if f (x) =

Ilxll on S n-l, then L f ~ M. By Markov's inequality, cr(x" f ( x ) >>,2M) ~< 1/2 and this shows that L f ~ c M as well, where c > 0 is an absolute constant [150]. It follows that we can have almost spherical sections of dimension k ~> ( ( c e 2 ) / l o g ( 1 / e ) ) n ( M / b ) 2 in Theorem 4.2.1. In order to remove the logarithmic in e term, one needs to put additional effort (see [75,174]). From Theorem 4.2.1 we may deduce Dvoretzky's theorem (Theorem 4.1.2): For every n-dimensional space X and any e 6 (0, 1) there exists a subspace Ek of X with dim Ek = k ~ ce 2 logn, such that d(Ek, g~w c log n / n . is a consequence of the Dvoretzky-Rogers lemma: There exists an orthonormal yl . . . . . Yn in ]Rn with Ilyi II >1 ((n - i + 1)/n)l/2. In particular, Ilyi II ~> 1/2, i - 1 . . . . . From the equivalence of M1 and M2 we see that

• gi (co)yi

M ~ ~

This basis n/4.

n/4

~ gi (o))yi dw

dco~>

i=1

~

~ Ixl

i=1

i>,- -c~

c" Clog n ,

fo i ~

(10)

where we have used the fact (see, e.g., [109, pp. 79]) that if gl . . . . . gm are independent standard Gaussian random variables on s then max/ 1 - ~ . n+k

(1)


741

In other words, k ( X ) is the largest possible dimension k ~< n for which the majority of kdimensional subspaces of X are 4-Euclidean. Note that the presence of M in the definition corresponds to the right normalization, since the average of M ( E k ) o v e r Gn,k is equal to M for all 1 ~< k ~< n. Theorem 4.2.1 implies that k ( X ) >~ c n ( M / b ) 2. What is surprisingly simple is the observation [152] that an inverse inequality holds true. The estimate in Theorem 4.2.1 is sharp in full generality: THEOREM 4.3.1. k ( X ) ~ 4 n ( M / b ) 2. PROOF. Fix orthogonal subspaces E 1, . . ., E t of dimension k ( X ) such that ]Kn - ~-~I= 1 E i (there is no big loss in assuming that k ( X ) divides n). By the definition of k ( X ) , most orthogonal images of each E i are 4-Euclidean, so we can find u 9 O (n) such that 1

- M l x l ~ Ilxll ~ 2MIxl, 2

x 9 uE i

(2)

for every i -- 1 . . . . . t. Every x e IKn can be written in the form x - Y~.I=1Xi' where Xi 9 u E i . Since the xi's are orthogonal, we get t

Ilxll ~

2M

~ Ixil ~ 2M~/71xl.

(3)

i--1

This means that b ~< 2Mx/t, and since t - n / k ( X ) we see that k ( X ) X][ 0 a(X,;e) ~ 0


745

as n --+ cx~. There are many examples of L6vy families which have been discovered and used for local theory purposes. In most cases, new and very interesting techniques were invented in order to estimate the concentration function c~(X; e). We list some of them (and refer the reader to [175] in this volume for more information; see also [83,84] for a development in a different direction): (1) The family of the orthogonal groups (SO(n), Pn, lZn) equipped with the HilbertSchmidt metric and the Haar probability measure is a L6vy family with constants C1 = ~ / ~ / 8 and c2 = 1/8. (2) The family X~ -- I--[ 9 9m" S ~ with the natural Riemannian metric and the product probability measure is a L6vy family with constants cl = ~/-~-/8 and c2 = 1/2. (3) All homogeneous spaces of SO(n) inherit the property of forming L6vy families. In particular, any family of Stiefel manifolds W~,k,, or any family of Grassmann manifolds G~,k, is a L6vy family with the same constants as in (1). (All these examples of normal L6vy families come from [85].) (4) The space F n = { - 1 , 1}~ with the normalized Hamming distance d(r/, r/') = #{i ~< n: r/i ~ rll }/n and the normalized counting measure is a L6vy family with constants cl - 1/2 and c2 = 2. This follows from an isoperimetric inequality of Harper [90], and was first put in such form and used in [9]. (5) The group/7~ of permutations of {1 . . . . . n } with the normalized Hamming distance d(o-, r) = #{i ~< n: or(i) -~ r(i)}/n and the normalized counting measure satisfies oe(/7~; e) ~< 2exp(-e2n/64). This was proved by Maurey [122] with a martingale method, which was further developed in [172]. We shall give two more examples of situations where L6vy families are used. In particular, we shall complete the proof of the global form of Dvoretzky's theorem using the concentration phenomenon for products of spheres.

(a) A topological application. Let 1 ~< k ~< n and Vk = {(~,x): ~ E Gn,k, x E S(~)} be the canonical sphere bundle over Gn,k. Assume that f :S n-l --+ R is a Lipschitz function with the following property: For every ~ E Gn,k we can find x E S(~) such that f (x) = O. One can easily check that Vk is a homogeneous space of SO(n) whose concentration function satisfies oe(Vk; ~) ~< x / - ~ e x p ( - - e 2 n / 8 ) . A standard argument shows that given 6 > 0, if k ~cn, and this proves the following fact [61]: THEOREM 4.6.4. Let P be a symmetric polytope in ]1~n. Then, log f (P) log v (P) ~ cn.

4.7. Isomorphic versions of Dvoretzky's theorem 4.7.1. Bounded volume ratio. quantity

v r ( K ) = inf,

-~

Let K be a body in R n . The volume ratio of K is the

. EcK

,

where the inf is over all ellipsoids contained in K. It is easily checked that v r ( K ) is an affine invariant. We shall show that if a body K has small volume ratio, then the space X/( has subspaces F of dimension proportional to n which are "well-isomorphic" to ~dimF. THEOREM 4.7.5. Let K be a body in 1Rn with v r ( K ) = A. Then, for every k C log n, every n-dimensional normed space X contains a k-dimensional subspace F for which

d(F, s

0 is an absolute constant.

Theorem 5.1.1 was originally proved in [133] and a second proof using the isoperimetric inequality on S n-1 was given in [134], where (1) was shown to hold with f()~) ) c(1 - )~) for some absolute constant c > 0 (and with an estimate f(~.) >~ 1 - )~ + o()~) as ~. --+ 0+). This was later improved to f()~) ~> c~/1 - )~ in [156] (see also [140] for a different proof with this best possible ~/1 - ) ~ dependence). Finally, it was proved in [76] that one can have 1

Geometrically speaking, Theorem 5.1.1 says that for a random ~.n-dimensional section of K x we have M ~

Kx n E c ~Dn - f ()~)

N E,

(3)

that is, the diameter of a random section of a body of dimension proportional to n is controlled by the mean width M* of the body (a random section does not feel the diameter a of K x but the radius M* which is roughly the level r at which half of the supporting hyperplanes of r Dn cut the body K x ) . The dual formulation of the theorem has an interesting geometric interpretation. A random )~n-dimensional projection of K x contains a ball of radius of the order of 1/M. More precisely, for a random E E Gn,xn we have P E ( K x ) D_

f()~) Dn N E. M

(4)

We shall present the proof from [ 134] which gives linear dependence in ~. and is based on the isoperimetric inequality for s n - l : PROOF OF THE LOW M*-ESTIMATE. Consider the set A -- {y E S n-l" obviously have a ( A ) ~> 1/2.

Ilyll, ~ 2M*}. We

Euclidean structure in finite dimensional n o r m e d spaces

751

CLAIM. For every ~. E (0, 1) there exists a s u b s p a c e E o f d i m e n s i o n k = [kn] such that (5)

E N S n - 1 c_ A ( ~ _ 5 ) , where 6 ~ c(1 - )~).

PROOF OF THE CLAIM. W e have o'(Arr/4) ~ 1 - - c ~ ' f f f o / 4 s i n n - Z t dt, and double integration through G n , k s h o w s that a random E E G n , k satisfies

~rk(nrr/4 n E ) ~ 1 - c x / n

f

Tr/4

sin n-2 t dt.

J0

(6)

On the other hand, for every x E S n - 1 n E we have

f0

sin k-2 t dt.

(7)

This means that if sin k-2 t dt ~

sin n-2 t dt,

(8)

Jr _ a) ~ 0 , and hence x E AT_~ then Arc~4 n B ( x , -~ ~ . Analyzing the sufficient condition (8) we see that we can choose 6 ~> c(1 - Z). D We complete the proof of T h e o r e m 5.1.1 as follows: Let x E S n-1 N E. There exists y E A such that sin6 ~< I(x, y)l ~< Ilyll,llxll ~ 2M*llxll,

(9)

2 ~> Ct (1 - )~), the theorem follows. and since sin 6 >~ if6

5.2. The g.-position

Let X be an n-dimensional normed space. Figiel and Tomczak-Jaegermann [60] defined the t~-norm of T 6 L (g~, X) by

e ( T ) -- ~

n-I

IITyll 2 cr (dy)

.

(1)

Alternatively, if {e j} is any orthonormal basis in ]~n, and if gl . . . . . gn are independent standard Gaussian random variables on some probability space $2, we have

g~(T)--

(11 II) E

giT(ei)

i=1

where E denotes expectation.

,

(2)


752

Let now RadnX be the subspace of L2(I-2, X) consisting of functions of the form

Y~4n_=lgi (og)Xi where xi 9 X (the notation here is perhaps not canonical, but convenient). The natural projection from L2(s

X) onto RadnX is defined by

(3)

Radn f = ~ ( fs2 gi f ) g i" i=1

We write IIRadn IIx for the norm of this projection as an operator in L2(S2, X). The dual norm e* is defined on L(X, ~ ) by g*(S) - sup{tr(ST)" T e L(e'~, X), e(T) an,

d ( E / F , e~) an and an ellipsoid s such that

C C PG(K M E) C c(1

-

ot) -1

Ilog(1 - oe)lC.

(2)

The proof of the theorem is based on the low M*-estimate and an iteration procedure which makes essential use of the e-position. PROOF. We may assume that Kx is in g-position: then, by Theorem 5.2.1 we have M ( X ) M * ( X ) )vn, and a subspace F of E*, d i m F -- k ~> ,ken, such that d ( F , t~k) ~< c(1 - X)-l log[d(X, e~) + 1]. The proof of this fact is a double application of the low M*-estimate. By Theorem 5.1.1, a random )vn-dimensional subspace E of X satisfies Clx/1 --,k

M*(X)

Ixl ~ Ilxll ~ blxl,

x E E.

(3)

Moreover, since (3) holds for a random E E Gn,zn, we may also assume that M ( E ) ~.2n and c3~/1--~.

M(X)

Ixl ~
2r, where r is the solution of the equation

M * ( K n r D n ) = g(X)r.

(4)


755

This double sided estimate provided by (1) and (4) may be viewed as an (incomplete) asymptotic formula for the diameter of random proportional sections of K, which is of interest from the computational geometry point of view since the function r --+ M* (K A r Dn) is easily computable. 4. The diameter of proportional dimensional sections of K is connected with the following global parameter of K: For every integer t >~ 2 we define r t ( K ) to be the smallest r > 0 for which there exist rotations ul . . . . . ut such that u l ( K ) A . . . A u t ( K ) ___r D n . If R t ( K ) is the smallest R > 0 for which most of the [ n / t ] - d i m e n s i o n a l sections of K satisfy diam(K A F) ~< 2R, then it is proved in [142] that rzt (K) ~< x/tRt (K). The fact that a reverse comparison of these two parameters is possible was established in [66]: There exists an absolute constant C > 1 such that Rc,(K)

(5)

~ Ctrt(K)

for every t ~> 2. 5. Fix an orthonormal basis {el . . . . . en } of R n . Then, for every non empty ~r ___{ 1 . . . . . n } we define the coordinate subspace It~~ - span{e j" j e ~r }. We are often interested in analogues of the low M*-estimate with the additional restriction that the subspace E should be a coordinate subspace of a given proportional dimension (see [63] for applications to Dvoretzky-Rogers factorization questions). Such estimates are sometimes possible [64]: If K is an ellipsoid in R n, then for every )~ e (0, 1) we can find cr ___ {1 . . . . . n } of cardinality ]or I ~> (1 - )~)n such that [)~/log(1/)O] 1/2 PR~, ( K ) D_

MK

(6)

D n AIR '~ .

Analogues of this hold true if the volume ratio of K or the cotype-2 constant of X/< is small. Finally, let us mention that Bourgain's solution of the A (p) problem [27] (see also [ 189] and [29]) is closely related to the following "coordinate" result: Let (4~i)i ~ 0 for every i ~< n, then for every p > 2 most of the subsets cr ___{ 1 . . . . . n} of cardinality [n 2/p] satisfy 1/2

(7)

for every choice of reals (ti)iecr. We refer the reader to the article [97] in this collection for the results of Bourgain and Tzafriri on restricted invertibility, which are closely related to the above.


756

6. Isomorphic symmetrization and applications to classical convexity 6.1. Estimates on covering numbers Let K1 and K2 be convex bodies in •n. The covering number N ( K 1 , K2) of K1 by K2 is the least positive integer N for which there exist Xl . . . . . xu E ]1~n s u c h that N

(1)

K1 C_ U ( x i -F K 2 ) . i=1

We shall formulate and sketch the proofs of a few important results on covering numbers which we need in the next section. The well known Sudakov inequality [179] estimates N ( K , t Dn): THEOREM 6.1.1. Let K be a body in R n. Then, N ( K , tDn) O, where c > 0 is an absolute constant. The dual Sudakov inequality, proved by Pajor and Tomczak-Jaegermann [ 155], gives an upper bound for N (Dn, t K): THEOREM 6.1.2. Let K be a symmetric convex body in •n. Then, N ( D n , t K ) ~ e x p ( c n ( M / t ) 2)

(3)

f o r every t > O, where c > 0 is an absolute constant. We shall give a simple proof of Theorem 6.1.2 which is due to Talagrand (see [ 109, pp. 82]). PROOF OF THEOREM 6.1.2. We consider the standard Gaussian probability measure Yn on R n, with density dyn - (2;r) -n/2 e x p ( - I x l 2 / 2 ) d x . A direct computation shows that Markov's inequality shows that

f Ilxll din(X) - otnM, where an/V/-n --+ 1 as n ~ ec. 1

yn(X" Ilxll ~ 2Morn) >~ -~.

(4)

Let {Xl . . . . . XN} be a subset of Dn which is maximal under the requirement that Ilxit xj II t> t, i 7~ j. Then, the sets xi + -2 K have disjoint interiors. The same holds true for the

Euclidean structure in finite dimensional normed spaces sets Yi -+-2Morn K, Yi

-

757

(4Morn/ t)xi . Therefore,

-

N

Z

Yn (Yi nt- 2Morn K)

(5)

IK A~.ID.I ~> IKI/N(K,)~ID,)

>~IKlexp(-cn/a2),

(7)

while using the dual Sudakov inequality and Lemma 3 we get

( 1 ) cn, and this suggests that by "spoiling" g~ it is possible to obtain X and Y with distance cn. The spaces which were used in [71] have as their unit ball a body ofthe form K = co{-+-ei,-+-xj: 1 0 for which there exist Z l, Z2 E ]]~n and T ~ GLn such that K - zi c_ T ( L - z2) c d ( K - zi). The question of the maximal distance between nonsymmetric bodies is open. John's theorem implies that d ( K , L) 0 in [170]. Using this fact, Rudelson showed that d ( K , L) ~ cn 4/3 log ~ n for any K, L E/Cn. See also recent work of Litvak and Tomczak-Jaegermann [116] for related estimates in the nonsymmetric case. 7. Milman and Wolfson [ 153] studied spaces X whose distance from g~ is extremal. They showed that if d ( X , ~ ) - x/n, then X has a k-dimensional subspace F with k >i c log n which is isometric to g~. The example of X -- ~ shows that this estimate is exact. An isomorphic version of this result is also possible [153]: If d ( X , ~ ) >~ otx/-fi for some c~ E (0, 1), then X has a k-dimensional subspace F (with k - - h ( n ) -+ ec as n ~ cx~) which satisfies d(F, ~ ) 0. Equivalently, if the gauge f of K satisfies (i) f ( x ) > 0 if x ~ 0, (ii) f()~x) = [,klf(x) for any x 9 R n, and (iii) f 9 C (oe), i.e., there exists c~ 9 (0, 1] such that

off(x) 1 if and only if there exists s > 0 such that T does not contain F~ 's (1 + s)-uniformly. A natural question which arises is to compare the notions of metric type and type in the case where T is a normed space. An answer to this question was given in [40], see also [163]:

THEOREM 7.6.2. Let X be a Banach space and let 1 < p < 2. (i) If X has type (respectively, metric type) p, then X has metric type (respectively, type) Pl f o r all 1 0 with the following property: every metric space T of cardinality N contains a subspace S with cardinality at least c(e) log N such that for some C ~2 with ISl - ISl we can find a bijection ~b" S --~ S with IIr -111Lip ~< 1 4- e (this means that S is (1 + e)-isomorphic to a subset of a Hilbert space). Let us finally mention an interesting connection between nonlinear problems and a more advanced form of type and cotype, the so-called Markov type and cotype which was introduced and studied by Ball [17].

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[130] M. Meyer and A. Pajor, On Santal6's inequality, Lecture Notes in Math. 1376, Springer, Berlin (1989), 261-263. [131] M. Meyer and A. Pajor, On the Blaschke-Santal6 inequality, Arch. Math. 55 (1990), 82-93. [ 132] V.D. Milman, New proof of the theorem of Dvoretzky on sections of convex bodies, Funct. Anal. Appl. 5 (1971), 28-37. [133] V.D. Milman, Geometrical inequalities and mixed volumes in the local theory of Banach spaces, Ast6risque 131 (1985), 373-400. [134] V.D. Milman, Random subspaces of proportional dimension of finite dimensional normed spaces: approach through the isoperimetric inequality, Lecture Notes in Math. 1166 (1985), 106-115. [ 135] V.D. Milman, Almost Euclidean quotient spaces of subspaces offinite dimensional normed spaces, Proc. Amer. Math. Soc. 94 (1985), 445-449. [136] V.D. Milman, Inegalit~ de Brunn-Minkowski inverse et applications ?~la th~orie locale des espaces norm~s, C. R. Acad. Sci. Paris 302 (1986), 25-28. [137] V.D. Milman, The concentration phenomenon and linear structure of finite-dimensional normed spaces, Proc. ICM, Berkeley (1986), 961-975. [138] V.D. Milman, A few observations on the connection between local theory and some other fields, Lecture Notes in Math. 1317 (1988), 283-289. [139] V.D. Milman, Isomorphic symmetrization and geometric inequalities, Lecture Notes in Math. 1317 (1988), 107-131. [140] V.D. Milman, A note on a low M*-estimate, Geometry of Banach Spaces, Proceedings of a Conference held in Strobl, Austria, 1989, EF. Muller and W. Schachermayer, eds, LMS Lecture Note Series 158, Cambridge University Press (1990), 219-229. [141] V.D. Milman, Spectrum of a position of a convex body and linear duality relations, Israel Math. Conf. Proceedings 3, Festschrift in Honor of Professor I. Piatetski-Shapiro, Weizmann Science Press of Israel (1990), 151-162. [142] V.D. Milman, Some applications of duality relations, Lecture Notes in Math. 1469 (1991), 13-40. [143] V.D. Milman, Dvoretzky's theorem- Thirty years later, Geom. Funct. Anal. 2 (1992), 455-479. [144] V.D. Milman, Isomorphic Euclidean regularization of quasi-norms in •n, C. R. Acad. Sci. Paris 321 (1995), 879-884. [145] V.D. Milman, Proportional quotients offinite dimensional normed spaces, Linear and Complex Analysis, Problem book 3, V.E Havin and N.K. Nikolski, eds, Lecture Notes in Math. 1573 (1994), 3-5. [146] V.D. Milman and A. Pajor, Cas limites dans les in~galit~s du type de Khinchine et applications g~om~triques, C. R. Acad. Sci. Paris 308 (1989), 91-96. [ 147] V.D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball ofa normed n-dimensional space, Lecture Notes in Math. 1376, Springer, Berlin (1989), 64-104. [148] V.D. Milman and A. Pajor, Entropy and asymptotic geometry ofnonsymmetric convex bodies, Adv. Math. 152 (2000), 314-335. [149] V.D. Milman and G. Pisier, Banach spaces with a weak cotype 2 property, Israel J. Math. 54 (1986), 139-158. [150] V.D. Milman and G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math. 1200, Springer, Berlin (1986). [151] V.D. Milman and G. Schechtman, An "isomorphic" version of Dvoretzky's theorem, C. R. Acad. Sci. Paris 321 (1995), 541-544. [152] V.D. Milman and G. Schechtman, Global versus Local asymptotic theories of finite-dimensional normed spaces, Duke Math. J. 90 (1997), 73-93. [ 153] V.D. Milman and H. Wolfson, Minkowski spaces with extremal distance from Euclidean spaces, Israel J. Math. 29 (1978), 113-130. [154] V.D. Milman and H. Wolfson, Topics in Finite Metric Spaces, GAFA Seminar Notes, Tel Aviv University (1984). [155] A. Pajor and N. Tomczak-Jaegermann, Remarques sur les nombres d'entropie d'un op~rateur et de son transpose, C. R. Acad. Sci. Paris 301 (1985), 743-746. [156] A. Pajor and N. Tomczak-Jaegermann, Subspaces of small codimension of finite dimensional Banach spaces, Proc. Amer. Math. Soc. 97 (1986), 637-642.

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[157] O. Palmon, The only convex body with extremal distance from the ball is the simplex, Israel J. Math. 80 (1992), 337-349. [158] G. Paouris, On the isotropic constant of nonsymmetric convex bodies, Lecture Notes in Math. 1745 (to appear). [159] C.M. Petty, Surface area of a convex body under affine transformations, Proc. Amer. Math. Soc. 12 (1961), 824-828. [ 160] G. Pisier, Sur les espaces de Banach de dimension finie a distance extremale d'un espace euclidien, S6minaire d'Analyse Fonctionelle (1978-79). [161] G. Pisier, Holomorphic semi-groups and the geometry of Banach spaces, Ann. of Math. 115 (1982), 375392. [162] G. Pisier, On the dimension of the s of Banach spaces, for 1 ~< p < 2, Trans. Amer. Math. Soc. 276 (1983), 201-211. [163] G. Pisier, Probabilistic methods in the geometry of Banach spaces, Lecture Notes in Math. 1206 (1986), 167-241. [164] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94 (1989). [ 165] S. Reisner, Zonoids with minimal volume product, Math. Z. 192 (1986), 339-346. [166] S. Rolewicz, Metric Linear Spaces, Monografie Matematyczne 56, PWN-Polish Scientific Publishers, Warsaw (1972). [167] S. Rosset, Normalized symmetric functions, Newton's inequalities, and a new set of Stringer inequalities, Amer. Math. Monthly 96 (1989), 815-819. [168] M. Rudelson, Estimates on the weak distance between finite-dimensional Banach spaces, Israel J. Math. 89 (1995), 189-204. [169] M. Rudelson, Contact points ofconvex bodies, Israel J. Math. 101 (1997), 93-124. [ 170] M. Rudelson, Distances between nonsymmetric convex bodies and the MM*-estimate, Positivity 4 (2000), 161-178. [171] J. Saint Raymond, Sur le volume des corps convexes sym~triques, Sem. d'Initiation h l'Analyse, no. 11 (1980-81). [ 172] G. Schechtman, L~vy type inequality for a class ofmetric spaces, Martingale Theory in Harmonic Analysis and Banach Spaces, Springer-Verlag, Berlin (1981), 211-215. [173] G. Schechtman, More on embedding subspaces of Lp in s Comp. Math. 61 (1987), 159-170. [174] G. Schechtman, A remark concerning the dependence on e in Dvoretzky 's theorem, Lecture Notes in Math. 1376 (1989), 274-277. [175] G. Schechtman, Concentration results and applications, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001) (to be published). [176] E. Schmidt, Die Brunn-Minkowski Ungleichung, Math. Nachr. 1 (1948), 81-157. [177] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge (1993). [178] R. Schneider, Simple valuations on convex sets, Mathematika 43 (1996), 32-39. [ 179] V.N. Sudakov, Gaussian random processes and measures of solid angles in Hilbert spaces, Soviet Math. Dokl. 12 (1971), 412-415. [ 180] A. Szankowski, On Dvoretzky's theorem on almost spherical sections of convex bodies, Israel J. Math. 17 (1974), 325-338. [181] S.J. Szarek, On Kashin's almost Euclidean orthogonal decomposition of s Bull. Acad. Polon. Sci. 26 (1978), 691-694. [ 182] S.J. Szarek, The finite dimensional basis problem, with an appendix on nets of Grassmann manifold, Acta Math. 159 (1983), 153-179. [183] S.J. Szarek, Spaces with large distance to s and random matrices, Amer. J. Math. 112 (1990), 899-942. [184] S.J. Szarek, On the geometry of the Banach-Mazur compactum, Lecture Notes in Math. 1470 (1991), 48-59. [ 185] S.J. Szarek, Computing summing norms and type constants on few vectors, Studia Math. 98 (1991), 147156. [ 186] S.J. Szarek and M. Talagrand, An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube, Lecture Notes in Math. 1376 (1989), 105-112.


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CHAPTER

18

Renormings of Banach Spaces Gilles Godefroy Equipe d'Analyse, Universite Paris VI, 4, place Jussieu, F-75252 Paris cedex 05, France E-mail: [email protected]

Contents 1. Definitions and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Separability of the dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Renormings of super-reflexive spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Smoothness of higher order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Characterizing spaces by renormings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

H A N D B O O K OF THE G E O M E T R Y OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 781

784 794 800 806 814 822 829 830


Renormings of Banach spaces

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Renorming a Banach space consists of replacing the given norm, which is usually provided by the very definition of the space, by another norm which may have better (or sometimes worse) properties of convexity or smoothness, or both. This operation is by nature geometric. We replace the original unit ball by another one, which has a different shape. In fact, geometric intuition is frequently useful and a few pictures might help in understanding certain proofs. However, the analytical point of view is of course necessary and proofs usually rely on careful computations. Computing in Banach spaces is quite difficult, since certain spaces have no basis and therefore there is no hope to use coordinates in calculations. Most results in renorming theory rely on topological assumptions. Since bases or substitutes for bases are not always available, computations have to be global, that is, to rely on coordinate-free functions such as, e.g., distances to convex sets. It turns out that such a "global calculus" suffices most of the time for constructing special norms with desirable properties. The fact that many important isomorphism classes admit natural characterizations in terms of existence of certain equivalent norms illustrates this point. Norms which enjoy good properties of convexity and/or smoothness can be computed under natural (and optimal) topological assumptions. These norms are frequently obtained by duality arguments and they are tightly connected with the linear structure. However, renormings can be used for constructing all kinds of functions on Banach spaces, by processing equivalent norms through the usual operations of analysis: linear combinations, suprema and infima, composition with real functions. Therefore they create a bridge between linear and nonlinear theory and lead in particular to a better understanding of Banach spaces considered as infinite-dimensional smooth manifolds, or as metric spaces. Finally, renorming theory is by its very definition an intermediate topic between isomorphic and isometric theory of Banach spaces. Connections work both ways. Sometimes an isomorphic assumption leads to a canonical construction of special norms, which in turn provides information on the isomorphism class under consideration. Sometimes the original norm itself, although an isometric object, informs us on the isomorphic properties of the space. Let us outline the contents of this article. Unless otherwise specified, the spaces we consider are real Banach spaces. Most of the results and proofs we display concern separable spaces. Nonseparable theory leads to delicate problems involving in particular infinite combinatorics. We refer to V. Zizler's article [ 176] "Nonseparable Banach spaces" in this Handbook for these topics. Section 1 presents the basic convexity and smoothness properties norms can have as well as the main duality results between convexity and smoothness. Some simple but important renormings of separable spaces are also shown there. In Section 2 one relates, through renormings, the existence of"nontrivial" differentiable functions on a separable space with the separability of its dual space. Variational principles are applied to subdifferentiability, when smoothness is available, while harmonic behaviour of smooth functions is displayed in nonsmooth spaces. Cantor derivations in dual spaces are an operative tool in Section 2, and in Section 3 as well, where they provide an alternative approach to uniformly convex renormings of super-reflexive spaces. This allows for dispensing with the classical martingale approach but still provides quantitative results in a canonical way. Section 4 deals with smoothness of higher order. The gist of this section is that while C 1 smoothness is available under topological assumptions, C 2 smoothness leads to much more restrictive conditions of a quantitative nature, while C ~ smoothness

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of norms forces the containment of classical sequence spaces. Section 5 displays several important isomorphism classes of separable Banach spaces which allow for simple characterizations in terms of renormings: Hilbert spaces, reflexive spaces, spaces containing I 1(N), subspaces of c0(N) and order-continuous lattices. Finally, Section 6 provides various applications of renormings: uniform approximation of continuous functions by differentiable ones, of real-valued Lipschitz functions by differences of convex functions on super-reflexive spaces, linearization of Lipschitz isomorphisms in subspaces of c0(N) and weak sequential completeness of extremely rough spaces. It should be stressed that this chapter is by no means an exhaustive work on renormings, even for separable spaces. It is rather an attempt to provide the reader with various relevant techniques and with motivations for the whole theory, in an accessible and when possible nontechnical way. Needless to say, choices had to be made and the selection of topics has been greatly influenced by the author's own taste. The sections themselves contain no references. The results are sometimes, but not always, the optimal available results. The proofs often consist of a sketch, where several computations are simply outlined. Experts might consider them as "complete" proofs, where only technical details should be added, while less experienced readers should find them helpful for understanding the proofs which are given in the referenced articles. Each section is complemented with an extensive "Notes and comments" paragraph, where references are given, related results and applications are provided, and open problems are mentioned. A bibliography section concludes the chapter. The notation and the terminology we use are classical and can be found in [130]. We should however point out the unfortunate fact that two terminologies are currently used for designating important convexity properties. We choose to use in this chapter the word "convex" instead of "rotund". So for instance, the meaning of the sentence "locally uniformly convex" is identical with the meaning of "locally uniformly rotund" in related works. Also, "smooth" and "differentiable" have the same meaning throughout this work. We refer to [ 104] in this Handbook for basic concepts and results used in this chapter.

1. Definitions and basic properties Let us recall basic definitions of special convexity properties norms can have. DEFINITION 1.1. Let X be a Banach space, and II 9 II be a norm on X. (i) The norm II 9 II is strictly convex if whenever (x, y) E X 2 are such that 2(llx II2 + IlYll2) = IIx -+- Yll2, one has x = y. (ii) The norm II 9 I1 is locally uniformly convex (1.u.c.) if whenever x E X and the sequence (Xn) C X are such that lim2(llx II2 -+-IlXn II2) --IIx + Xnll2 --0, then lim Ilx - xn II = O.


785

(iii) The norm II 9 II is uniformly convex (u.c.) if whenever (Xn) C X and (Yn) C X are bounded sequences in X such that lim2(llxn II2 -+- IlYn II2) - Ilxn -t- Yn II2 -- 0, then lim Ilxn - Yn II = 0. It is obvious from the definition that a uniformly convex norm is locally uniformly convex and that a locally uniformly convex norm is strictly convex. These notions can be alternatively defined in a n o n h o m o g e n e o u s manner: a norm is strictly convex if and only if the unit sphere Sx does not contain any nondegenerate segment. It is locally uniformly convex if the length of a segment joining any given point of Sx to another point of Sx is controlled by the norm of the midpoint. It is uniformly convex if the length of a segment joining two points of Sx is uniformly controlled by the norm of the midpoint. This last characterization naturally leads to the modulus of uniform convexity, defined for e E [0, 2] by

8x(e) -- inf{ 1 -

x+y) 2

; (x, y)

9

(Sx) 2 IIx - yll

~ e]

It is clear that II 9 II is uniformly convex if and only if 8x(e) > 0 for every e > 0. EXAMPLE 1.2. The natural norm of the Hilbert space 12 is uniformly convex, as shown by the parallelogram identity 2(llx II2 + Ilyll 2) - I I x -+- yll 2 -+-IIx - yll 2, In fact, this identity shows that

8/2 ( e ) - -

1 - (1

c2

-

-T) 1/2.

Uniformly convex spaces enjoy a simply shown but fundamental property. PROPOSITION 1.3. Any uniformly convex space X is super-reflexive. PROOF. We first show that X is reflexive. Pick t E X** with Iltll = 1. By Goldstine's theorem, the unit ball Bx of X is weak* dense in the bidual unit ball Bx**. Pick e > 0. Since II 9 II is uniformly convex, there exists 8 > 0 such that if (x, y) E (Sx) 2 are such that IIx + y II > 2(1 - 8), then IIx - y II < e. Since the bidual norm on X** is weak* lower semicontinuous, there is a convex neighbourhood V of t in (Bx**, w*) such that Ilull > 1 - 8 for all u E V. It follows that the II 9II-diameter of the nonempty set Bx M V is less than e. Since t belongs to the weak* closure of that set, we have by weak* lower semi-continuity of the norm that II 9 II - dist(t, X) ~< e. Since e is arbitrary, it follows that t E X, hence X** = X. To conclude the proof, we observe that in the above notation, the same 8 > 0 works for the same e and for every space Y which is finitely representable in X; this is indeed clear

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since the inequalities involve only 2-dimensional subspaces. It follows that any such Y is reflexive, hence X is super-reflexive. Vq It turns out that the converse to Proposition 1.3. holds true, up to a renorming (see Theorem 3.2. below). This proposition shows that separable spaces do not in general admit equivalent uniformly convex norms. However, a positive result holds true if we are ready to drop the uniformity. The proof will allow us to formulate a dual version as well. PROPOSITION 1.4. Any separable space X has an equivalent locally uniformly convex norm. Any space with separable dual has an equivalent norm whose dual norm is locally uniformly convex. PROOF. We first show that X has an equivalent strictly convex norm. Indeed, let (fn)n/>l be a sequence in Bx, which separates X; such a sequence exists since, e.g., the compact space (Bx,, w*) is metrizable and therefore separable. If N is the original norm on X, we define a new norm I . I on X by oo

Ix[2 -- N ( x ) 2 -+- Z

2-n fn (X) 2

n=l

It follows easily from the separating property of the sequence (fn) and the strict convexity of the real valued function g(x) = x 2 that [. [ is an equivalent strictly convex norm. Let n o w (Xn)n>/Obe an increasing sequence of finite dimensional subspaces of X whose union is dense in X, with X0 = {0}. Let I 9[ be a strictly convex norm on X. For any n ~> 0, we define dn on X by

dn(x) -- I 9I - dist(x, Xn), and we define II 9 II on X by the formula O43

Ilxll 2 - ~

2-ndn(x) 2

n=0

It is easily checked that 1[ . 11 is an equivalent norm on X. We pick x E X and a sequence (xk) C X such that 2(llx II2 + Ilxk II2) - l i m IIx + x~ll 2 - O.

(1)

Since the seminorms dn are convex, the corresponding expressions obtained by substituting II l[ by dn in (1) are all positive. Therefore it follows from (1) that for every n ~> 0, lim 2(dn(x) 2 -+-dn(Xk) 2) - dn(x + Xk) 2 - - 0 , k


787

and thus lim dn (xk) = dn (x)

for every n ~> 1.

k

Pick now e > 0. There is n such that dn(x) < e, hence dn(xk) < e for k large enough. Since do(xk) is bounded, the sequence (xk) is bounded. Since Xn is finite dimensional, its bounded subsets are compact and it follows that (xk) has a subsequence which is (3e)Cauchy. Since ~ was arbitrary, a diagonalization provides a Cauchy subsequence. It follows easily that the original sequence (xk) itself is convergent to a limit y. But since (1) holds true with do, we have Ix + y l 2 - 2 ( I x l 2 + lYl 2) and since the norm I 9I is strictly convex, it follows that x = y, which shows that II 9 II is locally uniformly convex. To conclude the proof, we now investigate the dual case. A first observation is that when X = Y* is a separable dual, the separating sequence (fn) can be taken within By. Hence the functionals which define I 9I are all lower semi-continuous (1.s.c.) with respect to the pointwise convergence on Y, provided that the original norm N is a dual norm, and it follows that I 9[ itself is 1.s.c. with respect to the weak* topology. Hence I 9I is dual to a norm on Y, as can be checked by the bipolar theorem. Then we observe that since I 9I is weak* 1.s.c. and the spaces Xn are finite dimensional, the seminorms dn are weak* 1.s.c. and therefore so is II 9 II. Hence the norm we construct in the dual case is a dual norm. E] REMARKS 1.5. (1) Note that this proof shows in particular that if Y is separable, there is an equivalent norm on Y whose dual norm is strictly convex. Indeed, if Y is separable and the separating sequence (fn) is contained in Y, the norm I 9I is a strictly convex dual norm. (2) Theorem 1.4 relies on the consideration of a series of convex functionals; in this case, of distances from an increasing sequence of finite dimensional subspaces. Similar series, which increase convexity, turn out to be a very efficient tool in renorming theory (see, e.g., the proofs of Theorems 3.2 and 5.4 below). They also allow for transferring locally uniformly convex norms from a space to another, which is useful in the nonseparable theory. For instance, if Y is a dual space with a dual 1.u.c. norm [I 9II and T is a weak* to weak continuous linear map from Y to X with dense range, we define a sequence of equivalent norms [ 9In on X by the formula I x l ~ - inf{ Ilx - Zyll 2 + n -1 [yl2; y 6 Y} and an equivalent norm N by oo

N(x)-Z2-nIxl2. n--1

This norm N is an equivalent locally uniformly convex norm on X. Combining such arguments with "Cantor derivation" arguments (see the proofs of Proposition 2.6 and Theo-

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rem 3.2) provides renorming theorems through a global approach, which does not refer to coordinates. Norms and convex functions on Banach spaces can enjoy two distinct fundamental properties of smoothness, which we now define. DEFINITION 1.6. A function F defined from a Banach space X into a Banach space Y is Fr6chet differentiable at x E X if there is a continuous linear map T from X into Y such that F ( x + h) = F ( x ) + T ( h ) + o(h), where limllhll__,0 IIo(h)ll/llhll - O. A norm II 9II on X is said to be GSteaux differentiable (or in short, G-smooth) at x ~ S x if there exists f E S x , such that for all y ~ S x ,

Ilxll]

lim t-1 [llx -q- t y l l t--+0

-

f(y).

(1)

If moreover the limit in (1) is uniform on y E S x , then the norm is said to be Fr6chet differentiable (or in short, F-smooth) at x. Observe that in the notation of (1) one has f (x) = 1. The linear form f is called the G~teaux or Fr6chet (in short, G- or F-) derivative of II 9 II at x. Obviously a norm is F - s m o o t h if and only if it is Fr6chet differentiable as a function from X\{0} to R. Hence, when a norm or a function is simply called differentiable, it will always be meant in the Fr6chet sense. Gfiteaux smoothness of convex functions refers to directional differentiability. By homogeneity, and since obviously no norm is smooth at 0, smoothness of norms is usually considered only on the unit sphere. We will usually say that a norm is G - s m o o t h or F - s m o o t h if it is so outside {0}. EXAMPLE 1.7. (1) The parallelogram identity and the chain rule immediately imply that the natural norm of the Hilbert space 12 is F-smooth. (2) Let l l be the space of all absolutely convergent series, equipped with its natural norm I[ 9Ill, that is, if x = (xi)i >>.1, oo

Ilxlll-~_~lxil. i=1

This norm is G-smooth at x

=

(xi) E Sll if and only if Xi ~ 0 for all i. Indeed, if Xi = 0

then clearly the norm is not smooth in the direction of ei, with e i ( j ) - 6/. On the other hand, if xi =/=0 for all i and h = (hi) E 11, we have that

l i m t -1 t--+O

[xi -t- thil - Ixi[ - sign(xi)thi i--1

Indeed, if i is such that Ithil ~ Ixi I, then

Ixi -t- thi[ -

Ixil = sign(xi)thi.

-0.

(1)

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If J denotes the set of all its such that Ithil > Ix/I, then

~l

Ixi-+-thil - Ixil -sign(xi)thil ~ 4 ~ l t h i l .

i~J

ieJ

It follows that the norm is smooth in the direction h. We now claim that the natural norm of 11 is nowhere F-smooth. Indeed, if it were F - s m o o t h at x = (xi), its F-derivative would coincide with its G-derivative, i.e., with the linear form f ( h ) - Z i ~ = l sign(xi)hi. To check that the limit in (1) is not uniform in h 9 Sll, it suffices to consider the sequence en defined above. If we substitute h = en and tn = - 2 x n in (1), the left-hand side of the equation equals -2.sign(xn), and this contradicts uniformity since lim tn = 0. On the infinite-dimensional Banach space l l, we thus have a continuous (even Lipschitz) function, namely II 9 II1, which is nowhere F-differentiable. We shall see later that the existence of such a function defined on a Banach space X is closely related to the structure of X. DEFINITION 1.8. A norm II 9 II on a Banach space X is uniformly F - s m o o t h (in short, UF-smooth) if the limit limt~0 t-l[llx + thll- Ilxll] exists uniformly on (x, h) 9 ( S x ) 2. Let us mention that there is a weaker notion of uniform smoothness, which is called uniform G~teaux-smoothness. A norm II 9 II is uniformly G~teaux-smooth (in short, UG-smooth) if for every h 9 Sx, the limit in Definition 1.8. exists uniformly in x 9 Sx. This notion is somehow trivial in separable spaces since any separable space has an equivalent uniformly G~teaux-smooth norm. The nonseparable theory, for which we refer to [176], is in this respect much more interesting. The modulus of smoothness p x ( r ) is defined for any r > 0 by the formula

px(r)

--supJ/ IIx + rYll +

2

IIx - rYll

- 1- Ilxll = Ilyll-

1}.

It is easily checked that II 9II is UF-smooth if and only if lim

r--+0

px(r) "g

= 0.

EXAMPLE 1.9. The space 12 is UF-smooth. Indeed, since ]Ix + h[[2 - ((x -k- h,x + h)) 1/2, it follows that (1 +

_ 1.

There is a natural duality between convexity and smoothness. Let us first consider the superreflexive case, where this duality can be expressed in quantitative form. The modulus of convexity of X and the modulus of smoothness of X* equipped with the dual norm are related through the following formula, which follows from Fenchel duality.

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THEOREM 1.10. One has p x , (r) = s u p { r e / 2 - 6x(e); 0 1/2 is given [81]. When such a space has the 00-1-B.A.E, its dual space has the metric approximation property. Remark 1.5 presents what is known as the transfer method, which originates in [74] and has been subsequently developed in a series of papers [83,84,57,139]. Roughly speaking, the idea consists of considering a hereditary property (P) of norms, and to check whether (P) can be "transferred" from X to Y when there is a linear operator T :Y --+ X such that T** is one-to-one. This technique is useful in the nonseparable theory, and we refer to V. Zizler's article [ 176] in this Handbook for some of its applications. Uniform Fr6chet smoothness leads to super-reflexivity (see, e.g., [12] or [90]) and is by now well-understood, but uniform G~teaux smoothness (UG) is much more delicate to handle. It is known that any space with a UG-smooth norm is a subspace of a weakly compactly generated space [60], see [ 176] for the definition of weakly compactly generated spaces. The duality formula (Theorem 1.10) which relates the moduli of convexity and smoothness is due to Lindenstrauss [128]. Several moduli, which refer to weaker "asymptotic" notions are defined in [138]. They are relevant to appropriate extensions of uniform con-


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vexity or smoothness (in a weaker, asymptotic sense) to certain nonreflexive spaces, such as Theorem 5.7 below. These notions have recently been applied to nonlinear theory [78, 107], following the pioneering work of Heinrich and Mankiewicz [96] showing that superreflexivity as well as moduli of convexity and smoothness of equivalent renormings are stable under uniform homeomorphisms. A complex analogue of uniform convexity is studied in [33], under the terminology "PL-convex". An interesting point is that many natural nonreflexive complex spaces enjoy this property, such as duals of C*-algebras, which turn out to be 2-uniformly PL-convex. This result due to Haagerup (see [33]) improves on a previous work of [168]. We refer to [148, Chapter 9] for applications of this notion to the noncommutative version of Grothendieck's theorem. Extensions of given norms from a subspace to the whole space, while keeping special properties of the given norm, is usually possible when qualitative or quantitative rotundity properties are concerned [59,167]. Smoothness properties behave differently, and in fact G-smoothness cannot be preserved even in the separable case (see [46, Section 2.8]). It is not known whether a space, such that every G-smooth norm can be extended in a G-smooth way to an equivalent norm on any separable super-space, is isomorphic to c0(N). It is shown in [142] (cf., e.g., [48]) that the Hilbert space has a larger modulus of convexity and a smaller modulus of smoothness than any normed space. The exact value of the modulus of convexity (and by duality of the modulus of smoothness) is actually an isometric characterization of the Hilbert space 12. In fact, the values of the modulus on certain sequences suffice for characterizing the Hilbert space: it is shown in [4] that if 32 denotes the modulus of convexity of 12 and if ~x(e) = ~2(e) for some e > 0 such that e/2 is not the sine of an even divisor of zr, then X is isometric to a Hilbert space. We refer to [5] for many isometric characterizations of the Hilbert space, and to [ 13] for a related very versatile "modulus for all seasons". Extremal structure of convex sets is studied for quite a long time and the literature on this topic is huge. We refer to the chapter "infinite dimensional convexity" by V. Fonf, J. Lindenstrauss and R.R. Phelps in this Handbook [69] and to [32] for integral representation theory, which refines the classical Krein-Milman theorem by representing any point in a metrizable compact convex set as a barycenter of a probability measure carried by the extreme points. Weak-star exposed points turn out to be more difficult to handle than extreme points, since even in the metrizable case their topological structure can be quite complicated, e.g., they can form a non-Borel set ([115]; see [38]). Investigating exposition and its strong version in convex sets leads to "dentability" of convex sets and to the Radon-Nikodym property of Banach spaces, for which we refer to [50,24]. The fundamental duality Lemma 1.14 is due to Smulyan [161 ]. The examples mentioned in Remark 1.17 can be found, e.g., in [175] or in [22] where "generic" constructions lead to a precise computation of the topological complexity of the set of counterexamples. Fr6chet smooth norms on spaces with a separable dual have been constructed independently in [109] and [ 116]. Note that it is shown in Section 2 below that the second part of Proposition 1.18 has a converse (Theorem 2.2). Let us quote for the record [46, Theorem 2.7.1 ] which gathers what can be done under separability assumptions: any separable Banach space has an equivalent norm which is simultaneously locally uniformly convex, uniformly convex in every direction and uniformly

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G~teaux smooth. Any Banach space with separable dual has an equivalent norm which is simultaneously locally uniformly convex, weakly uniformly convex, Fr6chet smooth, uniformly G~teaux smooth and whose dual norm is locally uniformly convex.

2. Separability of the dual space We first recall a classical notation. DEFINITION 2.1. Let X be a Banach space. A real-valued function b defined on X is called a bump function if the set supp(b) = {x 6 X; b(x) 7/= 0} is a bounded nonempty subset of X. A real-valued function is called C 1 o n X if it is differentiable (in the sense of Definition 1.6) and if its differential f ' is continuous from X to X* equipped with their norm topologies. Note that by Remark 1.15 a convex F-smooth function is C 1. We state now a fundamental characterization of spaces with separable dual. THEOREM 2.2. Let X be a separable Banach space. The following assertions are equivalent. (i) The dual X* is separable. (ii) There exists a C 1-smooth bump function on X. (iii) There exists an equivalent Frdchet differentiable norm on X. PROOF. (i) :=~ (iii): this was shown in Proposition 1.18. (iii) = , (ii): let b0 be a C ~ smooth bump function from the real line to itself such that supp(b0) C [1, 2]. If II 9II is an equivalent F-smooth norm on X, then b(x) = b0(llxll) is a C 1 smooth bump function on X. (ii) = , (i): let b be a C 1-smooth bump function. Pick any f E X*. We let 7r = b -2 - f . The function ~ takes values in R U {+oo}, it is bounded from below and lower semicontinuous. By Ekeland's variational principle, there is a slight perturbation of ~p which attains its minimum. In other words, for any e > 0, there is xe in X such that 7r (xe + h) 7z(x~)/> ellhll for all h E X. This implies that for all h 6 X and t > 0, one has t -1 [ b - 2 ( x e -+- th) - b - 2 ( x e ) ] ~ f ( h ) - ellhll. Letting t tend to 0, we obtain by the chain rule

l[-2b-3(x~)b'(x~) - f l ~ E and it follows that the linear span of the set D = {b'(x); x E X} is norm dense in X*. Since X is separable and b ' is continuous, D is separable and it follows that X* is separable as well. This concludes the proof of Theorem 2.2. U]

795


A first consequence of this result is a subdifferentiability property of lower semicontinuous functions on spaces with separable dual. We say that a function g is subdifferentiable at x E X if there exists f E X* such that when lim IIh II -- 0, one has

g(x + h) >~g(x) + f (h) + o(llhll). COROLLARY 2.3. Let X be a Banach space such that X* is separable. Every lower semicontinuous function defined on X is subdifferentiable at every point of a dense subset of X. PROOF. Let g : X --+ R be a 1.s.c. function, and U be a n o n e m p t y open subset of X. Since g is 1.s.c., there is a n o n e m p t y open ball B -- B(x0; 5) in U and an a E R such that g (x) > a for all x E B. Let ot = inf{g(x); Ilx - x011 = 5}. T h e o r e m 2.2 shows that there is a C 1smooth bump function b with supp b C B such that/3 = ot - inf8 (g + b) > 0. We define ~p as follows:

~ ( x ) = g + b(x)

if ilx - x01l ~< 6,

~p(x) = + o o

otherwise.

Clearly ~p is 1.s.c. and b o u n d e d below. M o r e o v e r infsc ~p ~> inf8 ~p + / 3 . variational principle (cf., e.g., [44]) provides us with a C 1-smooth bump IIb011~ ~ 0 such that for all x E Sx, lim sup t - l [ l l x -t- thll-+-IIx - t h l l - 2]/> e. t--+O hESx

The computations in E x a m p l e 1.7(2) show that the natural norm of ll is 2-rough. This also follows from Fact 2.7 below. It is clear that a rough norm is nowhere F - s m o o t h . Consequently, such a norm cannot exist on spaces with separable dual. But conversely, we have: PROPOSITION 2.6. Let X be a separable Banach space. The following assertions are equivalent: (i) The dual X* is not separable. (ii) There exists an equivalent rough norm on X. PROOF. (ii) implies (i) is clear by Corollary 2.4. Conversely assume (i). We will consider a "Cantor derivation" on the set of weak* closed subsets of X*. For any e > 0 and any weak* closed subset F of X*, we define the following notation: F e' = {x 6 F; II 9II-diam(V n F ) > e for all weak* open neighbourhoods V of x }. If ot is any countable ordinal, we d e f i n e F f f +1 - - ( F ~ot) e ! and if 13 is a limit ordinal, we set F [ -- n ~ 0 a countable ordinal or(e) such that K~(e)+l = K~(e). Since X* is nonseparable, there exists some e > 0 such that K~ (~) -- D ~ 0. Hence we have found a weak* compact subset D of X* such that every weak* open n o n e m p t y subset of D has II 9 II-diameter at least e. If C denotes the weak* closed convex hull of D, it follows that every weak* open slice of C has II 9II-diameter at least e. Note that D and C are both symmetric with respect to 0. If we define now B0 = K + C, one checks easily that every weak* open slice of B0 has II 9II-diameter at least e. Moreover, B0 is the unit ball of some equivalent dual norm on X*. Our conclusion now follows from the following fact. FACT 2.7. Assume that f o r some 8 > O, every weak* open slice of B x , has norm diameter at least 8. Then the norm on X is rough. Indeed, pick any x E Sx. For any positive t and v, we pick f and g in B x , such that m i n ( f ( x ) , g ( x ) ) > 1 - tv8 and [If - gl[ > 8(1 - v). Let h ~ Sx be such that ( f - g)(h) > 8 (1 - v). We have IIx 9 thll 9 IIx - thll ~ f (x -4- th) 4- g(x - th)


797

and an easy computation shows IIx -+- thll + IIx - thll ~ 2 + t6(1 - 3v). Since t and v were arbitrary, this proves that the norm is 6-rough. REMARK 2.8. (1) Proposition 2.6 shows in particular that Corollary 2.4. has a strong converse: if X is separable and every continuous convex function on X has at least one point of F-smoothness, then X* is separable. (2) A modification of the proof of (i) implies (ii) in Proposition 2.6 allows to show that a dual space X* has the weak* dentability property if (and only if) every separable subspace Y of X has a separable dual. Indeed, if X* fails to be weak* dentable, there is a weak* compact convex subset C of X* with no small weak* open slices. Proceeding as above, we construct an equivalent rough norm 11 . l] on X. Now, a separable exhaustion argument provides a separable subspace Y of X such that the restriction of ]J . 11to Y is rough, and then Proposition 2.6 shows that Y* is not separable. Theorem 2.2 and Proposition 2.6 are tightly connected with "harmonic" behaviour of smooth functions on nonsmooth spaces. More precisely, the following holds. PROPOSITION 2.9. Let X be a separable Banach space such that X* is not separable, and let Y be any Banach space. Let U be a bounded open subset of X, and let OU be its boundary. Let f be a continuous function from U to Y whose restriction to U is Fdifferentiable. Then" (i) the set f (U) is contained in the weak closure of f (OU) in Y. In particular one has suPu IIf II -- suP0u IIf Jl. Hence f satisfies the maximum principle. (ii) If the derivative f1(x) is a compact operator for every x ~ U, then f (U) is contained in the norm closure of f (OU) in Y. (iii) If Y* is separable, then f (U) is contained in the norm closure of f (OU) in Y. PROOF. If the weak closure of f ( O U ) does not contain f ( U ) , there exists a linear map T from Y to a finite dimensional space R n such that the closure of (T o f ) ( O U ) in R n does not contain (T o f ) ( U ) . Composing with an appropriate smooth bump function b from R n to R, we construct a smooth bump B = b o T o f on X. This contradiction with Theorem 2.2 shows (i), and (iii) can be shown in a completely similar manner, using this time a smooth bump function b from Y to R provided by Theorem 2.2. We outline the proof of (ii), which requires the use of rough norms provided by Proposition 2.6. A first step consists of showing that the restriction of a rough norm II 9II to any finite codimensional subspace is still rough. Then compactness of the derivative provides at every x E U a finite codimensional subspace Nx such that the norm of the restriction of f ' ( x ) to Nx is very small. Since II 9II is rough on Nx, we can find a direction h ~ Nx along which f does not vary much while II 9 II is significantly increasing. In this way, starting from an arbitrary x0 6 U, we construct a path along which f does not change much and which will eventually cross OU since U is bounded, and (ii) follows. D

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REMARK 2.10. Let X be a separable Banach space. If X* is separable, then Theorem 2.2 and the use of partitions of unity (or of Theorem 6.2) show that if F is any closed subset of X and f is any real valued continuous function on F, then there is a continuous extension of f to X such that f is C 1 on X \ F. On the other hand, if X* is not separable, pick g X*\{0}. Set f = Igl, and F = Sx. If I were a continuous function on Bx extending f and having a C 1-smooth restriction to the open unit ball, then applying Proposition 2.9(i) to the level sets of g would show that 1 = f on Bx. But f is not smooth at 0. Hence continuous functions on a closed subset of a separable space can be extended in a differentiable manner exactly when the dual is separable.

Notes and comments. Theorem 2.2 combines the work of [ 109] and [ 116] (for (i) implies (iii)) and of [51] (for (ii) implies (i)). It should be noted that the implication still holds if the bump function is simply assumed to be differentiable [46, p. 59]. If the bump function is Lipschitz, this can be seen by applying the smooth variational principle (see [44] in this Handbook) to a rough norm constructed in Proposition 2.6. Ekeland's variational principle (see [44]) has been applied to many different problems, for which we refer to [ 10]. Subdifferentiability of 1.s.c. functions on Asplund spaces is demonstrated in [56], using [ 19]. We refer to [98] for a recent survey on this topic. Generic F-smoothness of convex functions on spaces with separable dual is due to Lindenstrauss and Asplund [ 127,8]. Corollary 2.4 can be made more precise, using the notion of porous sets [ 151 ]. Note that the proof of this corollary shows that the set of points where a convex function is F-smooth is topologically simple, namely G~ regardless of the space. On the other hand, the set of G-smooth points can be very complicated in nonseparable spaces [97]. It is shown in [140] through topological games that if a Banach space has an equivalent G-smooth norm, then every convex function is G-smooth on the complement of a meager set. It is not known whether this latter property is equivalent in full generality to saying that every continuous convex function is G-smooth on a dense set. We refer to [58] for a comprehensive survey of G~teaux smoothness of convex functions on Banach spaces. Understanding the set of points where a given norm is F-smooth is an active field of research. The topological situation is quite clear, but the measure theoretic analogue, which would say that "norms on Asplund spaces are almost everywhere differentiable" is much harder to handle, since on one hand it is nontrivial to define properly what "almost everywhere" means, and on the other hand natural conjectures turn out to be false. We refer in particular to [133] and [134] where equivalent norms on the Hilbert space, and on infinite dimensional super-reflexive spaces, whose set of Fr6chet smoothness is Aronszajn null are constructed. The behaviour of the space c0(N) in this respect is unclear. Pioneering works on rough norms include [37] and [119]. Proposition 2.6 is a result of [125], which characterizes non-Asplund spaces by the existence of an equivalent rough norm, regardless of the density character, with essentially the same proof. We refer to [ 158] for quantitative estimates on roughness and related results. Proposition 2.6 leads the way to Cantor-Bendixon derivations and their applications to Banach space theory, which are displayed in Section 3 below. Remark 2.8.2) is in [125]. We refer to [162,163] and [164] for a comprehensive investigation of the duality between Asplund spaces and dual spaces with the Radon-Nikodym property. Note that the class of separable spaces which have an equivalent G~teaux smooth but rough norm lies strictly between the class of spaces


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containing 11 and the class of spaces with nonseparable dual [45]. We refer to [73] and references therein for related weak forms of the Radon-Nikodym property. It should be noted that rough norms exist on every non-Asplund space, while smooth norms fail to exist in general on Asplund spaces; examples are provided by R. Haydon's fundamental work [95]. Let us note that [95] contains also positive renorming results, and one of the original features of Haydon's renormings is that norms can be defined by recursion. This should be compared with the crucial use of recursion techniques in the definition of important examples of Banach spaces [ 171 ]. It is shown in [ 101 ], under the continuum hypothesis, that there are Asplund spaces with no equivalent norm having Mazur's intersection property. No such example has been constructed so far which would dispense with using special axioms of set theory. It is not known whether (not necessarily equivalent) C 1 smooth norms exist on every Asplund space. The behaviour of smooth functions on Banach spaces and Banach manifolds is investigated in the seminal paper [18], where the maximum principle is shown for smooth functions on Banach spaces containing I i. We refer to [46] and references therein for a recent account of the theory. The general principle which lies behind Proposition 2.9 is that functions satisfy the maximum principle when their order of smoothness exceeds the order of smoothness of the space on which they are defined. Approximation of continuous functions by smooth ones is investigated in Section 6 below, where some applications of renormings are displayed. The classical approximation technique, for which we refer to [46, Chapter VIII], consists of showing the existence of smooth partitions of unity. This is difficult in the nonseparable case, and for instance it is still open in full generality whether a Banach space X with an F-smooth norm has C 1-smooth partitions of unity. This is true when the space X is weakly compactly generated (cf., e.g., [46, Chapter VIII]), hence in particular every reflexive space has C 1-smooth partitions of unity [ 169]. Let us note that when a smooth approximation of a Banach space valued function is obtained from locally finite coverings and smooth real-valued partitions of unity, the approximating function has derivatives which are of finite rank at every point. The "harmonic" behaviour of smooth functions on nonsmooth spaces leads naturally to the problem of finding a "Poisson formula" for representing the values inside U by the values on 0 U. Very little seems to be known about this question which goes back to [ 18]. It is shown in [11], among other things, that if X is an infinite dimensional Banach space which has a C 1 smooth norm (nonnecessarily an equivalent one!), then X is C 1 diffeomorphic to X\{0}. One of the motivations for such results is that it easily implies the failure of Rolle's theorem in infinite dimensional spaces: there exists, e.g., on 12 a C~ bump function b such that b' (x) 7~ 0 for every x such that b(x) ~ O. The proof in [ 11 ] relies in part on a renorming technique, namely on Bessaga's "incomplete norm" approach [ 16]. It is natural to "integrate linear theory", that is, to investigate which theorems remain true when linear operators between Banach spaces are replaced by smooth functions. For instance, it is conjectured that any C 1 function with uniformly continuous derivative from c0(N) to a Banach space which does not contain c0(N) maps weakly Cauchy sequences into norm convergent ones (see [91 ]). This would mean that co enjoys a nonlinear form of Pelczyfiski's property (V). This field of research is nearly unexplored.

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3. Renormings of super-reflexive spaces We will make a crucial use in this section of a modification of the "Cantor derivation" which has been used in the proof of Proposition 2.6. Let C be a weak* compact convex subset of a dual space X*, and pick any e > 0. We denote

De(C) = {x ~ C; II 9II-diam(S) > e for every weak* open slice of C with x ~ C }. Note that De(C) is convex, and that we can define by induction D~' for any positive integer n. With this notation, we have LEMMA 3.1. Let X be a Banach space.The space X is super-reflexive if and only if for any e > O, there is an integer n(e) such that D~ (e) (Bx, ) -- 0. PROOF. Let (YTk) be a finite increasing sequence of Boolean algebras generated by partitions of [0, 1] into subintervals, with S0 = {0, [0, 1]}. An X-valued martingale of height n is a sequence (Mk)0~ ~ for all k and all x 6 [0, 1]. A space X is not super-reflexive if and only if its dual X* is not super-reflexive, if and only if there exists some e > 0 such that for every n, there is an e-separated martingale M~ of height n with values in B x , . It is clear that if {Xn } = M~ ([0, 1]), then xn is in D~ (Bx,). Hence if e is as above, D n ( B x , ) ~ 0 for every integer n. For the converse, we first observe that if C is convex and x 6 De (C) then one has by H a h n - B a n a c h theorem that x ~ c o n v ( C \ B ( x , e/2)). If D n ( B x , ) ~ 0, then it follows from this observation and an easy perturbation argument that there is an e/2-separated martingale of height n with values in B x , . The conclusion follows. [3 THEOREM 3.2. A Banach space X is super-reflexive if and only if there is an equivalent uniformly convex norm on X, if and only if there is an equivalent uniformly smooth norm on X. PROOF. By Proposition 1.3 any uniformly convex space is super-reflexive. To prove conversely that any super-reflexive space is uniformly convexifiable and smoothable, it suffices by Corollary 1.11 to show that there exists a uniformly convex norm on X*. By the above, for every e > 0 there is a smallest n(e) 6 N such that D n(e) ( B x , ) = 0. For all k E N, we put Nk = n (2-k). We also denote


801

Let II 9II be the original n o r m on X*; distances d ( . , C) to convex sets are taken with respect to this original norm. We define a convex function f on X* by the following formula: oc

f(x)-

Nk

2_ k

[Ix[] + Zk=, n~l= - ~ k d ( x , K ; ) .

We define the new n o r m I 9 [ as the Minkowski functional of the set C -- {x 6 X*; f (x) ~< 1 }. It is clear that I 9 I is an equivalent n o r m on X*. We show that this n o r m is uniformly convex. This follows from the claim below via an easy computation which is left to the reader. CLAIM 3.3. Let x and y in X* be such that f (x) = f (y) = 1 and [Ix - YI[ >/e. Then

f

( ) x+y 2

O;

xEK~andyEK~}.

Note that n < Ark since [Ix - yl[ ~> e. We let ), = e/4Nk. SUB-CLAIM. For some integer I with 1 0 such that f = l i m n ~ Tnf(x) uniformly on B(x, 3(x)). We say that a norm is C k, C ~ , analytic . . . if it is so outside {0}. EXAMPLES 4.2. (1) If p = 2n is an even integer, the natural norm on L p is analytic, since we can write Ilfllp = ( p ( f ) ) l / p , where P is a monomial of degree p. Note that absolute values are not needed by positivity of even powers. (2) Let us define on c0(N) the function ~((xi)) - - ~_~i~=1X 2 i . Then the set C = {x co; ~b(x) ~< 1} is the unit ball of an equivalent norm on co. It follows from the analytic case of the implicit function theorem that this norm is analytic on co. The results from this section will show that the above classical spaces are the canonical examples of spaces which have C ~ smooth equivalent norms. In order to sketch the proofs, we need a compact variational principle, valid for spaces which do not contain an isomorphic copy of c0(N). We also state and prove a quantitative version of this principle for spaces with a nontrivial cotype.


807

LEMMA 4.3. Let X be a Banach space, let U be a bounded symmetric open subset of X, and let f : U --+ R be a continuous function, such that f (O) O. For any compact subset K of U, we let f K ( x ) = sup{f(x + k); k ~ K} for all x E X such thatx + K C U. (i) If X does not contain an isomorphic copy of c0(N), then there exist a symmetric compact set K and a symmetric neighbourhood V of 0 with K + V C U, such that for every 6 > O, there exists a finite subset K~ of K such that i n f { f K ~ ( x ) - fK~(0); x 6 V,

Ilxll ~ ~} >

0.

(ii) If X has a finite cotype q, there exist a symmetric compact set K, a symmetric neighbourhood V of 0 with K + V C U and A > 0 such that fK (x) ~> fK (0) + A IIx IIq

for all x ~ V. PROOF. We first prove (i). We let xo = O. If xo, x 1. . . . .

Xn

have been constructed, we define

Kn--{~eixi;i=o 8 i E { - 1 , 1 } } and

En--

xEX;

for a l l k 6 K ~ ,

k+xEUandf(k+x)~0 K~. Since K is contained in U, it is bounded. Since X does not contain c0(N), we have by Bessaga-Petczyfiski's theorem that K is compact and limx~ = 0, therefore lim oe~ = 0. Since f is continuous, we have f ( x ) 0, and we choose n such that an < 6. Pick x 6 V such that IIx II/> ~. Since otn < 6, we have fKn (x) > m(1 - 2 - n ) / 2 . On the other hand, fK~ (0) -- max { ft:._, (Xn), fK~_, ( - x ~ ) } -- fK._, (Xn) and since Xn E En-1 this implies fKn (0) ~< m(1 - 2n-1)/2. Therefore fKn (x) > fK~ (0) + 2 - ~ - 1 m and this shows (i). The proof of (ii) is similar. We assume for convenience that m -- 1 and U C Bx. Using the same notation, we set

G. Godefroy

808 and

En - {x 9 X; for all k 9 K . , k + x 9 U and fKn (k .Af_X) ~ fKn (0) .Af_(2C) -1 [ixllq }, where C is the cotype q constant of X. We let as before an = sup{ Ilx II; x e En } and we choose Xn+l 9 En such that IIx~+~ II/> c~/2. Since X has a nontrivial cotype, it does not contain c0(N) and thus we have as before that lim an = 0. We have now

supf (ei)

6iXi

~n for n large enough. Since Kn is an increasing sequence of sets, we have

f K ( X ) - - l i m fKn (x) t> lim fKn (0) + ( 2 c ) - l l l x l l q - fK(O) + ( 2 c ) - l l l x l l q 11 fl This shows (ii).

[]

Our first application of L e m m a 4.3 is the following uniformization result. THEOREM 4.4. Let X be a Banach space which does not contain an isomorphic copy of ,(k) co(N), and let k 9 N. If there is a Ck-smooth bump function bo defined on X such that o 0 is locally Lipschitz, then there is a Ck-smooth bump function b defined on X such that b (k) is globally Lipschitz on X. PROOF. We give the proof for k = 1; trivial modifications provide the general case. We may and do assume that b0(0) = 1, that bo(x) = b o ( - x ) for all x and that bo(x) = 0 if IIx II/> 1. We let f = 1 - b0, and U = {x 9 X; f ( x ) # 1 }. We apply L e m m a 4.3 to f , U and m = 1 to get V and K as in (i). Since f~ is locally Lipschitz, there is fl > 0 such that f l B x C V and f t is Lipschitz on K + f l B x . Pick 0 < 3 < fl/2. L e m m a 4.3 provides K~ and e > 0 such that if ~ < IIx II >f x~ (0) + e. For x 9 fi B x , we define now

~ ( x ) = Z ( f (x + y ) yeKa

f(y))2


809

Then ~ ( 0 ) - 0, the derivative (Tr)' is Lipschitz on f i B x and ~p(x) ~> 6 2 if x E f l B x and [Ix II ~> 8. Composing with an appropriate smooth function from R to R provides the conclusion. D The uniform behaviour of derivatives provides important information on the space, through the construction of uniformly smooth equivalent norms, as sketched in the following result. P R O P O S I T I O N 4.5. Let X be a Banach space. I f there exists a C 1-smooth bump function b defined on X whose derivative b ~ is globally Lipschitz on X, then there is an equivalent norm N ( . ) on X whose derivative is globally Lipschitz. In particular, N is uniformly

smooth and tON(rl) ~ Crl 2. PROOF. We outline the construction of the norm. We may and do assume that b(0) > 0, that b is an even function and that 0 ~< b(x) ~< 1 for all x E X. Let us define c/)(x) --

F

OG

b ( s x ) ds

OG

and gr(x) = 4)(x) -1 for x -r O, with gr(0) --O. Substitution shows that gr(tx) --Itlgr(x) for all x and all t 6 R. It follows that there exist a > 0 and b > 0 such that allxll ~< gr(x) ~< b IIx II for all x 6 X. There is k E R such that for all x with 89~< IIx II ~< 3 one has

4)(x) --

b ( s x ) ds.

k It follows that 4 / i s Lipschitz on the unit sphere. We define now ~(x) = ~ ( x ) 2. Clearly (al[xi]) 2 ~< ~(x) ~< (bllxll) 2. One checks by direct computation that the derivative ~' exists and is globally Lipschitz on X. We now use ~ and a convexification procedure to construct the equivalent norm. Let F be the largest convex function which is less than ~ on X. The epigraph of F is the closed convex hull of the epigraph of ~, and F can be estimated through the formula

where of course the 0ti'S are positive a n d z i n = l o/i - 1. Let C be the Lipschitz constant of ~'. For all x and h, one has F ( x § h) + F ( x - h) - 2 F ( x ) ~< 2CIIhll 2.

(1)

For proving (1), we pick any 6 > 0 and find (Ofi) and (Xi) as above with x -- Z i L 1 olixi and 17

Z i=1

oti~(xi) < F ( x ) + ~.

G. Godefroy

810 Then we have

F ( x + h) + F ( x - h) - 2F(x)

Z

Oti~(Xi @ h) +

i=1

Oti~(Xi -- h) - 2 Z i--1

Oli~(Xi) -+-2e.

i=1

Thus F ( x + h) + F ( x - h) - 2F(x)

n Z Oli(~(Xi + h) + ~(Xi -- h) - 2 ~ ( x i ) ) -k- 2e ~< 2CIIh[I 2 + 2e i=1

by the Mean Value Theorem. Since e is arbitrary, this proves (1). Since F is convex, (1) shows that F is C 1 and that F t is Lipschitz. The norm N is simply the Minkowski functional of K = {x ~ X; F(x) ~< 1}. ff] By the Mean Value Theorem, every C 2 smooth function has a locally Lipschitz derivative. Hence Theorem 4.4 and Proposition 4.5 imply the following corollary. COROLLARY 4.6. Let X be a Banach space. If there exists a C 2 smooth bump function on X, then X contains an isomorphic copy of co(N) or X is superreflexive of type 2. This corollary, when compared with Theorem 2.2, shows that C 2 smoothness is a much more restrictive condition than C 1 smoothness. The next theorem shows that C ~ smoothness provides considerable information on the space. THEOREM 4.7. Let X be a Banach space which does not contain an isomorphic copy of c0(N), and such that there exists a C ~ smooth bump function defined on X. Then: (i) There exist an integer k ~ 1, a 2k-form a(.) on X, c > 0 and d > 0 such that f o r all xEX, cllxll 2k ~ a(x) O, there exists 6 > 0 such that f o r every x ~ K and every h ~ X with ]lh II 0 such that T2(x) /> Iix112/2 whenever Ilxll -- e. It follows that I x l - ~/T2(x) is an equivalent Hilbertian norm on X. [2 Note that the above function f is clearly differentiable at 0 for any Banach space X. Hence this simple example demonstrates that smoothness of order 2 is considerably stronger than smoothness of order 1. This can also be said of the related following statement, since any reflexive space has an equivalent C 1 norm (see Theorem 2.2.). THEOREM 5.2. Let X be a Banach space such that X and X* admit equivalent C2-smooth norms. Then X is isomorphic to a Hilbert space. PROOF. If X* contains an isomorphic copy of co, then X* contains a copy of l ~ and thus X* is not an Asplund space, which contradicts the existence of an equivalent Frechetsmooth norm. By Theorem 4.3 above, it follows that X* is a superreflexive space of type 2. Hence X is superreflexive as well, and by Theorem 4.3. again it is superreflexive of type 2. But since X* has type 2, X has cotype 2, and then Kwapiefi's theorem provides the conclusion. U] REMARK 5.3. Let us outline a simpler proof of this result in the case where the norm II 9 II of s and its dual norm II 9 I1" are both C 2. We define the convex function g by


815

g(x) = IIx 112/2. Recall that the (Fenchel) dual function g* to g is defined on X* by Fenchel duality through the formula g*(x*) = sup{x*(x) - g(x); x E X}. In this case one checks easily that g*(x*) = Iix*112/2. We pick any xo r 0 in X. The convexity of the functions g and g* shows that if f (xo) - x~ then (g*)'(x~)) - xo. Since g* is C 2, there is 6 > 0 such that for some c 6 R and all h* 6 X* with IIh* II < 6, one has

g*(x~) + h*) - g*(x~)) - h*(xo) ~ Iih112/4c for all h c X such that IIh II < 2c3. But then Taylor's expansion of order 2 shows that

g" (xo)(h, h) >~ Iih112/2c for all h 6 X. Hence [g"(xo)(h, h)] 1/2 = N ( h ) is an equivalent Hilbertian n o r m on X. THEOREM 5.4. Let X be a separable Banach space. The following assertions are equiv-

alent: (i) X is reflexive. (ii) There is an equivalent norm II 9 IIM on X such that every bounded sequence (Xn) such that limm limn IlXm + xn IIM = 2 limn IlXn IIM is norm convergent. PROOF. If II 9II denotes some equivalent n o r m on a Banach space X and x ~ X, we define the symmetrized type n o r m II 9 I1~ by

Ily IIx = IIx Ily II + Y II + IIx Ily II - y II. For every x e X, II 9 IIx is an equivalent n o r m on X such that 211yll ~ Ilyll~ ~ (2 + 211xll)llyll for all y. To check the triangle inequality, one uses the fact that for fixed u and v in X, the function s(r) = Ilru + vii + Ilru - vii is convex and even on R and thus is increasing on R +. We fix now a countable dense Q-vector subspace of X, which we denote C, and we choose a sequence (pc)ccC of positive real numbers such that Y~c~c pc(1 + Ilcll) < ~ . We define a map A from the set of equivalent norms on X into itself as follows:

A(II. II)(x) -- ~

P~llxll~.

ccC

We first prove that (i) implies (ii). Let II 9 II be a strictly convex equivalent norm. It turns out that A(A(II . II)) = II 9 IIM satisfies the conclusion of the theorem. This relies on the following crucial fact.

816

G. Godefroy

FACT. Let us denote II 9Ill = A(II. II). Let (Xn) C X be a sequence such that Ilxn II - 1 f o r all n and lim lim Ilxm -+-Xn Ill -- 2 lim Ilxn Ill, m

n

n

then there is a subsequence (x 88 o f (xn) such that f o r all y E X and ~, fl ~ 0 one has

lim lim Ily -+- oeXZm-4-/3XZnII - lim Ily + (~ -4- ~)X'm lie n m To show this fact, we extract a subsequence (x'/7) such that for every c 6 C, y E C and or, fl E Q + , the limits lim lim IIY -4- c~x~ -4-/3x'~ IIc m

n

exist, which is easy through a diagonal argument. The assumptions of the fact and a classical convexity argument imply that for all c E C,

lim IIx'm + X'n IIc -- 2 lim IIx'nlice lim n n If we let c -- 0, we obtain since

(Xn) is normalized that

/

lim lim IIx'm -4- x n II - 2 m

n

and thus for c~, fl ~> 0 one has limlim IIotXtm + m

/7

~x'nll-~ + ~.

(1)

Similarly we have for all c E C that

lim lim IIc~x~m+ ~X'n IIc -- (c~ + ~) lim IIx~mIIc, m /7 m

(2)

Let y E C and or, t3 E Q+. We apply (2) to c = (or + f l ) - l y . Using (1), we obtain that limlim(lly + m

/7

otx'm +

fix'n II -+- Ily - ~X'm - - ~X'n II)

--limm(lly + (~ + 13)X'm II + IlY -- (~ +/~)X'm II). By the triangle inequality, we have for r/6 { - 1 , 1 } that limlim Ily -4- r/(oex'm + ~X'n) II m

n

immlyl ~

0, there is a subsequence (xff) of (Xn) such that for all k E N and all positive real numbers oil, 0/2 . . . . . Otk, one has k

OliX i

(4)

/> (1 - - e ) ~ O t i . i=1

i=1

In particular, it follows from (4) and Mazur's theorem that (x~I) has no weakly null subsequence. We now conclude the proof of (i) implies (ii). Starting from a strictly convex norm ]l 9II, we let II 9Ill - A(II. II) and we denote II 9IIM -- A(II 9Ill). We consider a sequence (Xn) such that lim lim Ilxm -+- x/7 IIM - 2 lim m

/7

/7

Ilxn IIM.

Since the n o r m II 9 IIM is strictly convex, it suffices to show that any such sequence has a norm convergent subsequence. Since X is reflexive, passing to a subsequence we m a y assume that Xn = x -Jr-Yn where yn is weakly null and lim Ily~ II = A exists. If A = 0 we are done. A s s u m e it is not so. Then we m a y also assume that IlYn Ill = 1 for all n. We now apply the Fact to II 9 II1 and II 9 IIM = A(II 9 Ill) to find a further subsequence which we still denote (x/7) such that for all y E X, one has Y lim lim II Y + Xm + X/7 I1~ - 2 lim ~ + X m m

/7

m

Letting y -- - 2 x , it follows with the above notation that (5)

lim lim IIYm + Y/7 I1~ - 2 lim IIYm I1~ - 2. m

n

m

Choose y* E X such that IlY* II - 1 and lim

IIAy*

- yn II - o. It follows from (5) that

lim lim II Y* + Y* Ill - 2 lim II Y* Ill. m

n

m

We m a y now apply again the fact, this time with II 9 II and II 9 Ill = A(II. II), and its consequence to conclude that (y*) has no weakly null subsequence. But this is a contradiction since (Yn) is weakly null. We now prove that (ii) implies (i). Pick f E X* with IlfllM -- 1. Let (x/7) be a sequence such that IIx~ll ~ 1 for all n and lim f ( x n ) = 1. It is clear that the sequence (Xn) satisfies

818

G. Godefroy

the assumption of (ii) and thus it is norm convergent. Its limit x satisfies IIx IIM = f ( x ) = 1. Hence every f E X* attains its norm and thus by James' theorem X is reflexive. Let us mention that a proper use of Goldstine's theorem provides an alternative proof which does not rely on James' theorem. [3 PROPOSITION 5.5. Let X be a Banach space. The following assertions are equivalent: (i) X contains a subspace isomorphic to l 1(N). (ii) There is an equivalentnorm II 9l] on X and t ~ X**\{0} such that Ilx + tll - Ilxll + Iltll f o r every x E X.

Let us mention that when X is separable, these conditions are also equivalent to the following: for every finite dimensional subspace F of X and every e > 0, there is h E X\{0} such that for all x E F and ot E R, one has IIx + ~hll/> (1 - e)(llxll + IIc~hll). It is therefore clear that the above conditions define an "extreme" form of roughness. The canonical norm of 11 (N) provides the simplest example. PROOF. To prove that (ii) implies (i), we observe that the local reflexivity principle permits to construct, using t and a simple inductive argument, a subspace of X isomorphic to 11 (N). We simply outline the proof of (i) implies (ii). Starting from a subspace Y of X which is isomorphic to l l(N), we identify the bidual of Y with a subspace of X**, and we pick any completely singular t E Y**\{0}; in other words, we pick a nontrivial ultrafilter of N, and t is the weak* limit in X** along this ultrafilter of a sequence in X equivalent to the canonical basis of l 1(N). We observe now that a completely singular element t of (l l) **, when considered as a function on the Hilbert cube C (that is, on the unit ball of l ~ equipped with the weak* topology), has at every point of C a weak* oscillation equal to 211tll. Then we lift C to a minimal weak* compact subset K of B x , . Simple geometrical considerations permit to construct a dual unit ball B in X* such that if Osc(t) denotes the oscillation of t : ( B , weak*) --+ R and N = sup8 Itl, then O s c ( t ) ( f ) = 2N for all t E B. The proof is concluded by the following lemma: LEMMA 5.6. Let X be a Banach space and t E X**. The following are equivalent: (i) IIx + t ll--IIx II + Iltll f o r every x E X. (ii) / f Osc(t) denotes the oscillation o f t: ( B x , , weak*) --+ R, then O s c ( t ) ( f ) = 211tll f o r all f E B x , .

PROOF. Let 7" denote the infimum of all weak* continuous functions on B x , which are greater or equal of t. Since t is affine, 7"is concave upper semi-continuous, hence by H a h n Banach theorem, it is the infimum of the weak* continuous affine functions on B x , which are greater or equal to t. This translates into the following formula: ~'(f) -- inf{f(x) + [it - xl[; x E X} for every f E B x , . A straightforward computation now shows that if (i) is true, then 7"(f) Iltl] for every f E B x , , and (ii) follows. Conversely, if (ii) holds, given any e > 0 and any x E X, there is f0 E B x , such that f o ( x ) -- Ilxll, and by (ii) we can pick f E B x , "close to

Renormings of Banach spaces fo" suchthat t ( f ) > and (i) follows.

IltlI-E and f ( x ) >

Ilxll-e. Wehavethen f ( t + x ) >

819

Iltll + Ilxll-2E, []

Lemma 5.6 will be used in Section 6 when roughness of norms will be related with weak sequential completeness of certain spaces. Proposition 5.5 states a characterization of spaces containing l l (N) by an extremal form of roughness. Dually, it turns out that subspaces of c0(N) can be characterized by the existence of equivalent norms enjoying optimal properties of smoothness, as shown by the following THEOREM 5.7. Let X be a separable Banach space. The following are equivalent: (i) X is isomorphic to a subspace of co(N). (ii) There is an equivalent norm [I 911on X and k > 0 such that if (fn) is a sequence in Bx* which isweak* convergentto f andlimllfn - fl] = e > O, then Ilfll ~< 1 - k s . PROOF. (i) implies (ii) simply follows from the fact that the natural norm of c0(N) satisfies (ii) with k -- 1, and the property goes to subspaces with a straightforward proof. Let us mention that the canonical norm of c0(N) (and the equivalence) show that if (ii) can be done with some k > 0, then it can be done with k = 1. The proof of (ii) implies (i) is more involved. We first observe that (ii) clearly implies that the weak* and norm topologies agree on the dual unit sphere, from which it follows that X* is separable. We now outline the proof in the case when X* has a Schauder basis, that is, when X has a shrinking basis. Let (e~) be such a basis. We denote by En,k the linear span of {e j; n 1, if x is in the unit sphere of El,nk and y is in Enk+l,oo with Ilyll ~< t, then Ilx + yll ~< 1 + sk. Define the desired blocking {Fn } of {En} by setting Fk = Enk_l,nk for k = 1,2 . . . . . It is enough to check that {F2~ }nC~=l and {FZn-1 }~~1621 are both co-decompositions, since then X is the direct sum of the spaces (y~,, {Fzn }~--1)c0 and (y~.~ {F2~-1 }~~1621)c0. To check that, for example, {F2n }n~__j is a co-decomposition, it is sufficient to observe that if xk is in F2k and supk Ilxk [ ~< t, then for each m - 1, 2 . . . . for which II ~km_-I xk II ~> 1, the inequality

Xk ~< (1 -+-62m+1) k--1

Xk k=l

is true. This completes the proof when X has a shrinking basis. A straightforward modification of this proof provides the result under the assumption that X has a shrinking finite dimensional decomposition.

820

G. Godefroy

The general case can be deduced from the above argument by using the JohnsonRosenthal theorem which states that any Banach space X with separable dual has a subspace Y such that both Y and X~ Y have shrinking finite dimensional decompositions. The conclusion follows from the fact that (ii) goes to subspaces and quotient spaces, and that "being a subspace of c0(N)" is a three-space property. D Theorem 5.7 will be applied in Section 6 to Lipschitz isomorphisms. Renormings can sometimes be combined with special structures on Banach spaces, such as order structures. The following result characterizes "nice" lattices through the existence of special lattice norms. A Banach lattice is a Banach space X equipped with a lattice structure which is compatible with the vector lattice operations and such that Ilxll :

II Ix l II

(2)

for all x 6 X. A norm which satisfies (2) is called a lattice norm. Recall that a Banach lattice L is said to be order continuous if every downward directed net with infimum 0 converges in norm to 0. In other words, L satisfies a "Lebesgue dominated convergence theorem". Order continuity turns out to be equivalent with weak compactness of the order intervals. They are characterized among Banach lattices by the following result. THEOREM 5.8. Let X be a Banach lattice. Then X is order-continuous if and only if there is an equivalent lattice norm II 9 II on S such that the weak and norm topologies agree on the unit sphere o f II 9 II. PROOF. We only outline the arguments. If X is not order continuous there is a sequence (Yn) of disjoint positive vectors which is equivalent to the unit vector basis of c0(N) and y 6 X with Yn 0, denote

D ? p ( x ) -- { f ~ X*" p ( x + h) >~ p ( x ) + f (h) - s for all h E X ] . The n o n h o m o g e n e o u s version of L e m m a 1.14, which follows from a similar proof, says that p is F - s m o o t h at x if D o P ( X ) is a singleton {f} and if fn E D s with Sn tending to 0 implies that [[f , - f [1 tends to 0. But since p is a Lipschitz function with constant 1 and x r C, we have lim [[f , [[ = 1 and 1[f [[ = 1. M o r e o v e r (fn + f )

cD~p(x)

and thus lim ][fn + f [I = 2. Since the dual norm is 1.u.c. it follows that lim ]lfn - f l[ = 0 and thus p is F - s m o o t h on X \ C . If we consider now the function p2, it is easily seen to be F - s m o o t h on X, and even C 1 since it is convex. We now show that every continuous b o u n d e d function f on Sx is uniform limit on Sx of restrictions of C 1 smooth functions on X. Indeed assume that - 1 < f (x) < 1 for all

G. Godefroy

824

x ~ Sx, pick e > 0 and N E N such that 7 < 2Ne. For 0 ~< i ~< 2 N + 1, we consider the interval

Ii=

i-l-N N

'

and we let Oi - f - l ( I i ) . If Ci - 0 then g(x) - ~

i-N] N

If 1 ~< i ~< 2 N , we let Ci -c-o--fi-9(Bx\(Oi-1 U Oi g Q i + I ) ) . is s-close to f on Sx. Hence we may assume that Ci :/: 13 for

1 ~< i ~< 2N. For these i's, we define ri (x) - dist2(x, Ci). The functions ri create some kind of smooth partition o f u n i t y on Sx. Indeed pick xo ~ Sx, and i0 such that xo ~ Qio. Since f is continuous, there is ~ > 0 such that if IIx - yll < ~ and y E Sx then y ~ (Oi0-1 u Oio u Qio+l)- Since the n o r m of X is 1.u.c. it follows that xo ~ Cio, and thus rio (xo) > 0. Since xo was arbitrary, we have ~ 2 N 1 ri (x) > 0 for all x E Sx. if we let now ro(x) - (1 - IIx 112)2, we have that ro -- 0 on Sx and ro > 0 elsewhere. For 1 ~< i ~< 2 N and x E X, we define

h i ( x ) --

ro(x) + ri (x) 2N

Y~d=o rj (x ) The hi 's are C 1 functions on X and the function f is "nearly constant" on the support of each hi. It follows by an easy computation that if we denote 0/i the midpoint of each interval li, the function g(x) -- E 2 N 1 otihi(x) is C 1 smooth on X and we have I f ( x ) - g(x)l < s for every x ~ Sx. It suffices for concluding the proof to use a stereographic projection. Let us equip the space Y = X G R with the n o r m II (x, r) Ilr - (llxll 2 -+- r2) 1/2. This n o r m II 9 Ilr and its dual n o r m are clearly 1.u.c. We denote H -- {(x, 1); x 6 X} and we define p from Y\{0} to Sy by p ( y ) -- (llyll r) -1 y; let pl be the restriction of p to H . If f is a b o u n d e d continuous function from H to R and B is a closed b o u n d e d subset of H , we extend ( f o p 11), defined on the closed subset pl (B) of Sx, to a b o u n d e d continuous function f l on Sx. For any s > 0, there exists by the above a C 1 smooth function g on Y such that If(Y) - g(Y)l < s for every y E Sy. Hence I f - g o Pl] < s on B, and since g o pl is C 1 smooth on H this concludes the proof of T h e o r e m 6.2. D We now consider another kind of approximation, namely approximation of Lipschitz functions by differences of convex functions which are b o u n d e d on b o u n d e d sets. W h e n such an approximation is possible, one could say that the metric structure of the space is s o m e h o w "close" to the affine structure. This turns out to be a characterization of superreflexive spaces. THEOREM 6.3. Let X be a Banach space. The following assertions are equivalent: (i) X is superreflexive. (ii) Every Lipschitz function is uniform limit on X of a sequence of functions (gn) such

that f o r every n, gn = Cn - d n , where the Cn's and dn's are convex functions which are bounded on bounded sets.


825

PROOF. Assume (i). Let II 9 II be an equivalent uniformly convex norm on X provided by Theorem 3.2. We will show (ii) with uniform convergence on bounded subsets of X and indicate later how to obtain uniform convergence on X. Let f be a Lipschitz function on X. We may and do assume that the Lipschitz constant of f is 1. We define

fn(x)-

inf {f(y)-+- n(211xll 2 -+- 211Yll2 -IIx + yl12)}.

y6X

(1)

If we let cn(x) -- 2nllxll 2 and dn(x) = Cn(X) - f~(x) it is clear that the functions Cn and d~ are convex and that c~ is bounded on bounded sets. Moreover the sequence (fn) is uniformly convergent to f on bounded subsets of X. Indeed if we let x -- y in (1) we see that f (x) ~> f~ (x) for every x 6 X and every n 6 N. On the other hand, for every (x, y) 6 X 2 and every n 6 N one has 211xll 2 -4- 211yll2 - I I x -+- yll 2 ~ ( l l x l l - Ilyll) 2 ~ 0,

(2)

hence the sequence of functions (fn) is increasing. Let now n ~> 3 and y 6 X be such that

f ( Y ) + n(211x II2 -+-2llyll 2 - IIx -+-yll 2) ~ f ( x ) .

(3)

It follows from (2) and (3) that n ( l l x l l - Ilyll) 2 ~ f ( x ) - f ( y ) 2. Such a norm exists by Theorem 3.6. Then the sequence of functions defined by

gn(x) -- inf { f ( y ) + n(2 p-111xl[ p + 2 p-I Ilyl] p - Ilx + yIIP)} y~X

converges to f uniformly on X. The computations are similar to the above but slightly more involved.

G. Godefroy

826

We only sketch the proof of (ii) implies (i). The following is true: if X is not superreflexive there exists a 1-Lipschitz function defined on X such that for every pair {c, d} of continuous convex functions bounded on Bx one has

suplf(x)

- (c - d ) ( x ) I >1 1/4.

Bx

When X = c0(N) the function f can be constructed as follows: let T be the set consisting of all finite sequences with values in {-1, 1}, which we embed into c0(N) in the obvious way. Let E be the subset of T consisting of sequences with even length. Then the function f defined on c0(N) by f ( x ) = dist(x, E) works. The idea of the proof is that T is an infinite dyadic tree, and that f oscillates between 0 and 1 on each branch of T. Assuming that f = c - d with c and d convex and following an appropriate branch contradicts the boundedness of c and d. The general case of nonsuperreflexive spaces is obtained by a similar trick, using the fact that arbitrarily large finite trees grow in the unit ball of such spaces. [] REMARK 6.4. Denote by Convb(X) the cone of convex functions which are bounded on bounded sets, by UCb(X) the space of functions on X which are uniformly continuous on bounded sets, and by rb the topology of uniform convergence on bounded sets. Theorem 6.3 implies that X is superreflexive if and only if the rb-closed linear span of Convb (X) is equal to UCb (X). We continue with more applications of renormings to nonlinear theory. The next result relies on the characterization of subspaces of c0(N) given in Theorem 5.7. We recall that two Banach spaces X and Y are Lipschitz isomorphic if there is a bijective map F from X onto Y such that both F and F - 1 are Lipschitz maps. THEOREM 6.5. Let Y be a Banach space which is Lipschitz isomorphic to a linear subspace of c0(N). Then Y is linearly isomorphic to a subspace of c0(N). If Y is Lipschitz isomorphic to c0(N) then Y is linearly isomorphic to c0(N). PROOF. We only outline the arguments. The main topological tool we need for proving this result is Gorelik's principle, which provides a substitute for the lack of weak continuity of Lipschitz isomorphisms between Banach spaces X and Y. This principle can be stated as follows: Let U be a Lipschitz isomorphism from X onto Y, and let k be the Lipschitz constant of U -1 . Let X0 be a subspace of finite codimension of X. Let d > 0 and b > 0 be l such that d > kb. Then there exists a compact subset K of Y such that bByf C U(2dBxo) + K, where B I denotes the open unit ball. This topological result is an application of the Bartle-Graves selection theorem and of Brouwer's fixed point theorem. Gorelik's theorem permits to show the following "quasi-linearity" result: if (fn) is a weak* null sequence in Y*, then U(Bxo) asymptotically norms, up to a constant independent of X0 and (fn), the sequence (fn). In other words, when n is large enough, suPv(Bx0) Ifnl controls Ilfnll. This simply follows from Gorelik's principle and the fact that (fn) goes uniformly to 0 on K.


827

Recall that Theorem 5.7. provides a characterization of subspaces of c0(N) through the behaviour of weak* null sequences in the dual space. Assuming that X is a linear subspace of c0(N) and U is a Lipschitz isomorphism from X onto Y, it suffices therefore to construct an equivalent norm N on Y which satisfies the assumptions of this Theorem. This equivalent norm is provided by the following "maximal rate of change" formula: if f E Y*, define N(f)

- supJ I f ( U x - U x ' ) l ; (X,X') E X 2 x ~;kxt} I IIx - x' II

This norm N is an equivalent dual norm since it is clearly weak* 1.s.c. Moreover, N satisfies the assumptions of Theorem 5.7. For checking this point, we pick (in the notation of Theorem 5.7.) x and x t in X which nearly realize the supremum in the definition of N ( f ) , and we may assume without loss of generality that x = - x I and U (x) = - U (xl). Then we use the fact that since X is a subspace of c0(N), the set M of metric midpoints between x and x ~ essentially contains a ball of a finite codimensional vector subspace. Moreover the choice of x implies that f (t) is small for every t E M. On the other hand, Gorelik's principle implies that the set U (M) asymptotically norms the weak* null sequence ( f - fn) with vectors U(xn). Computing ( f - ( f - f~), U ( x ) - U(Xn)) shows that the norm N is asymptotically nearly additive on f and ( f - f , ) since the cross products are small, and this shows our claim and the first part of the theorem. The second part of the result follows from the first, since by a theorem of Heinrich and Mankiewicz, being a/Z or is stable under Lipschitz isomorphisms, and by a theorem of Johnson and Zippin every s162 subspace of c0(N) is linearly isomorphic to c0(N). D We conclude our section with a linear result. Rough norms were applied in Section 2 to prove the "harmonic" behaviour of certain smooth functions (see Proposition 2.9). The next statement shows that some extreme form of roughness bears applications in a different direction. PROPOSITION 6.6. Let X be a Banach space such that there exists a projection P f r o m the bidual X** onto X such that Ilull = Ilu - P(u)ll + IlP(u)ll f o r every u E X**. Then X is weakly sequentially complete.

PROOF. Pick u E X * * \ X , and denote t = u - P ( u ) . Our assumption on use L e m m a 5.6., which shows in particular that the restriction of t to B x , weak* continuity. Now Baire's theorem implies that t is not the pointwise a sequence of weak* continuous functions. In particular, it is not limit of sequence from X, and thus weak sequential completeness of X is shown.

P allows us to has no point of limit on B x , of a weak Cauchy D

Of course, weakly sequentially complete spaces cannot in general be renormed so as to satisfy the assumption of Proposition 6.6 since, for instance, they are in general not even complemented in their bidual. However, many natural examples of weakly sequentially complete spaces equipped with their natural norms turn out to satisfy this assumption. For instance, many natural quotient spaces of L 1 satisfy it.

828

G. Godefroy

Notes and comments. Theorem 6.1 is an example of Asplund's averaging technique which originated in [9]. The Baire category approach we follow here comes from [64]. It should be noted that the set of equivalent 1.u.c. norms is topologically quite complicated; in particular, it is not a Borel set for the natural topology on the set E of equivalent norms [22]. Dually, the set of C 1 smooth norms [22] on a separable Asplund space, or of C 1 smooth convex functions on a separable space [174], is not a Borel set. Many natural properties of norms or convex sets lead to residual sets, in other words, these properties are satisfied for almost every norm in the Baire category sense (see, e.g., [72]). Theorem 6.2 is shown in [172], and it has been extended further in [70] and [152]. It is shown in these two works that if X has a equivalent 1.u.c. norm, or alternatively if X has the Radon-Nikodym property, then X has Ck-smooth partitions of unity (where k 6 N or k - e~) provided that every equivalent norm is uniform limit on bounded sets of C ~smooth functions. The proofs rely on particular on the result that if X has the RadonNikodym property or if X has an equivalent 1.u.c. norm, then the norm topology of X has a a-discrete basis made up of convex sets. It is apparently not known whether every Banach space satisfies this conclusion (or the weaker one of existence of such a a-locally finite basis). Such results are important in the nonseparable theory, where a major problem is to know whether every Banach space which has a C 1-smooth equivalent norm has C 1-smooth partitions of unity. Several partial positive answers are available (see [176]) but the general conjecture still holds. It is not even known whether every space with a C 1 smooth norm is such that the set of C 1 smooth norms is dense in the set of all equivalent norms. Finally, no example is known of an Asplund space without C 1-smooth partitions of unity. Theorem 6.3 is a recent result of [28]. Its motivation is to relate as closely as possible convex and Lipschitz functions, and it asserts that this is possible in super-reflexive spaces and only there. So for instance there is no way to reduce analytic approximation of Lipschitz functions to approximation of norms in the nonreflexive case, and other methods have to be used for showing this approximation result in c0(N); this is done in [31 ] and independently in [71 ]. It is an intriguing fact that no simple formula exists so far which would provide uniform approximation by C 2 or analytic functions and which would replace the use of partitions of unity. The existence of distorted norms in minimal super-reflexive Banach spaces (such as I p with 1 < p < oo) follows easily by Theorem 6.3 from the existence of a distorted Lipschitz function. Much more precise results than Theorem 6.3 are actually shown in [28]. For instance, the uniform convergence on bounded sets of the sequence ( f n ) given by Eq. (1) in the proof is actually equivalent to the uniform convexity of the norm, and locally uniformly convex renormings provide with the same formula uniform approximation on compact sets. On the other hand, there is a Lipschitz function on loc (N) which is not even a pointwise limit of a sequence of differences of convex functions. This follows from the fact that continuous convex functions are weakly 1.s.c. and in particular weakly Borel, while the norm and weak Borel structure on loc (N) are distinct [ 166]. More characterizations of super-reflexive spaces are shown by Cepedello [29,30]: for instance, X is super-reflexive if and only if for every closed subset F of X, there is a difference of two convex functions f = c - d such that f attains its strong minimum exactly on F. Remark 6.4 is from [28].


829

Theorem 6.5 is the main result of [78] and the proof is taken from this paper. Gorelik's principle, which is one of the main tools in the proof, originates in [86] and has been expanded in [ 106]. Among the milestones of nonlinear geometry of Banach spaces, one should mention [52,154,96] and [106]. We refer to [14] for an up-to-date account of the whole theory. The maximal rate of change argument which lies behind the definition of the equivalent norm is related with D. Preiss' theorem on the differentiability of real-valued Lipschitz maps on Asplund spaces [150] and its proof. Johnson-Zippin's theorem on s subspaces of c0(N) is in [108]. Theorem 6.5 somehow means that c0(N) is the "smoothest" of all / ~ spaces, and under this form it can be extended in various ways; for instance, c0(N) is the only separable s space which can be equivalently renormed into an M-ideal in its bidual ([82]; see [93]). However natural conjectures in this direction remain unsolved; for instance, it is not known whether c0(N) is the only s separable space which has Petczyfiski's property (u). Nonlinear theory of Banach spaces contains many open questions. To mention a few of them, it is not known whether a Banach space which is Lipschitz isomorphic to 11 (N) is linearly isomorphic to it, nor whether a space which is uniformly homeomorphic to c0(N) is linearly isomorphic to it. Moreover, no example is known of a couple of separable Banach spaces which are Lipschitz isomorphic but not linearly isomorphic. Separable examples for uniform homeomorphisms [ 155,1], and nonseparable examples for Lipschitz isomorphisms [2], are available. Proposition 6.6 is a simple application of Baire's lemma and goes back to [75]. It has been used in a variety of situations; we refer to [93, Chapter IV] where many of these applications are displayed, and for a more recent work to [80]. It is shown in [75] that the quotient of L 1 by a subspace X whose unit ball is closed in L 1 equipped with the topology of convergence in measure is weakly sequentially complete. This applies in particular to the Hardy space X = HI(D), considered as a subspace of L 1(T), and provides a proof of the Mooney-Havin theorem. Many natural generalizations of the Mooney-Havin theorem can be obtained by the same token (see, e.g., [93]). Proposition 6.6 itself can be widely generalized: for instance, any space X such that there exists a projection P from X** onto X such that I I I - 2PII = 1 is weakly sequentially complete; this follows from [135] or from [77]. We refer to [81] for the related notion of u-ideal and its applications. As mentioned before, there are weakly sequentially complete Banach spaces which are not complemented in their bidual (e.g., the kernel of a quotient map from 11 onto co [129]), and thus Proposition 6.6 as stated has no converse. However, it can be shown that under the assumption of Proposition 6.6, if (xn) is a sequence in X and a 6 R are such that for every f ~ X*, one has limn sup{If(x1 - xk)l; I, k ~> n} ~< al[fll, then dist(t, X) ~< a for every weak-star cluster point t to (xn) in (X**, w*). It is not clear which weakly sequentially complete spaces can be renormed to have such a quantitative form of weak sequential completeness.

Acknowledgement I am glad to thank V~iclav Zizler, who introduced me to renorming theory, and Mari~in Fabian for their help in preparing this chapter.

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[ 121] S. Kwapiefi, Isomorphic characterization of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (1972), 583-595. [122] D. Kutzarova and S. Troyanski, On equivalent lattice norms which are uniformly convex or uniformly differentiable in every direction in Banach lattices with a weak unit, Serdica Bulgaricae Math. Publ. 9 (1983), 249-262. [123] G. Lancien, Dentability indices and locally uniformly convex renormings, Rocky Mountain J. Math. 23 (1993), 635-647. [124] G. Lancien, On the Szlenk index and the weak*-dentability index, Quart. J. Math. Oxford (2) 47 (1996), 59-71. [ 125] E.B. Leach and J.H.M. Whitfield, Differentiable functions and rough norms on Banach spaces, Proc. Amer. Math. Soc. 33 (1972), 120-126. [ 126] M. Leduc, Densit~ de certaines familles d'hyperplans tangents, C. R. Acad. Sci. Paris, S6rie A 270 (1970), 326-328. [127] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139-148. [128] J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10 (1963), 241-252. [129] J. Lindenstrauss, On a certain subspace ofl 1, Bull. Acad. Pol. Sci. 12 (1964), 539-542. [130] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I and II, Springer-Verlag (1977 and 1979). [131] A.R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225-238. [132] R. Maleev and S. Troyanski, Smooth norms in Orlicz spaces, Canad. Math. Bull. 34 (1991), 74-82. [133] J. Matou~ek and E. Matou~kov~, A highly nonsmooth norm on Hilbert space, Israel J. Math. 112 (1999), 1-27. [134] E. Matou~kov~, An almost nowhere Fr~chet smooth norm on superreflexive spaces, Studia Math. 133 (1999), 93-99. [135] B. Maurey, Types and ll-subspaces, Longhorn Notes, Texas Functional Analysis Seminar (1982-83), 123-137. [136] B. Maurey and G. Pisier, S~ries de variables al~atoires vectorielles ind~pendantes et propri~t~s g~om~triques des espaces de Banach, Studia Math. 58 (1976), 45-90. [137] D.E Milman, On some criteria for the regularity of spaces of the type (B), Dokl. Akad. Nauk SSSR 20 (1938), 243-246. [138] V.D. Milman, Geometric theory of Banach spaces. 2. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), 6(162), 73-149 (Russian). English translation: Russian Math. Surveys 26 (1971), 6, 79-163. [ 139] A. Molto, J. Orihuela and S. Troyanski, Locally uniformly rotund renorming and fragmentability, Proc. London Math. Soc. 75 (1997), 619-640. [140] I. Namioka, R.R. Phelps and D. Preiss, Smooth Banach spaces, weak Asplund spaces and monotone or usco mappings, Israel J. Math. 72 (1990), 257-279. [ 141 ] A.M. Nemirovski and E.M. Semenov, On polynomial approximation in function spaces, Mat. Sbornik 21 (1973), 255-277. [142] G. N6rdlander, The modulus of convexity in normed linear spaces, Ark. Mat. 4 (1960), 15-17. [143] E. Odell and Th. Schlumprecht, Distorsion, Handbook of the Geometry of Banach Spaces, Vol. 2, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001) (to be published). [144] E. Odell and Th. Schlumprecht, On asymptotic properties of Banach spaces under renormings, J. Amer. Math. Soc. 11 (1998), 175-188. [145] B.J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), 249-253. [ 146] G. Pisier, Un exemple concernant la superr~flexivit~, S6minaire Maurey-Schwartz, Vol. 2 (1974/75). [147] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. [148] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Mathematics, Vol. 60 (1986). [149] G. Pisier, Weak Hilbert spaces, Proc. London Math. Soc. 56 (1988), 547-579. [150] D. Preiss, Differentiability of Lipschitzfunctions on Banach spaces, J. Funct. Anal. 91 (1990), 312-345. [ 151 ] D. Preiss and L. Zajf6ek, Fr~chet differentiation of convex functions in Banach spaces with separable dual, Proc. Amer. Math. Soc. 91 (1984), 202-204. [ 152] M. Raja, Mesurabilit~ de Borel et renormages dans les espaces de Banach, Thbse, Universit6 de Bordeaux I (1998).

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[153] M. Raja, On locally uniformly rotund norms, Mathematika, to appear. [154] M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14 (1976), 237-244. [155] M. Ribe, Existence of separable uniformly homeomorphic nonisomorphic Banach spaces, Israel J. Math. 48 (1984), 139-147. [156] H.E Rosenthal, On subspaces of L p, Ann. of Math. 97 (1973), 344-373. [157] H.E Rosenthal, Embeddings of L 1 into L |, Contemp. Math. 26 (1984), 335-349. [158] W. Schachermayer, A. Sersouri and E. Werner, Moduli ofnon-dentability and the Radon-Nikodym property in Banach spaces, Israel J. Math. 65 (1989), 225-257. [ 159] H.H. Schaefer, Banach Lattices and Positive Operators, Grundlehren der Math. Wiss. 215, Springer-Verlag (1974). [160] J.J. Sch~iffer, Geometry of Spheres in Normed Spaces, Lecture Notes Pure Appl. Math. 20, M. Dekker, New York (1976). [ 161 ] V.L. Smulyan, Sur la ddrivabilitd de la norme dans l'espace de Banach, C. R. Acad. Sci. URSS (Doklady), N. S. 27 (1940), 643-648. [162] C. Stegall, The Radon-Nikodym property in conjugate Banach spaces, Trans. Amer. Math. Soc. 206 (1975), 213-223. [163] C. Stegall, The duality between Asplund spaces and spaces with the Radon-Nikodym property, Israel J. Math. 29 (1978), 408-412. [164] C. Stegall, The Radon-Nikodym property in conjugate Banach spaces, II, Trans. Amer. Math. Soc. 264 (1981), 507-519. [165] W. Szlenk, The nonexistence of a separable reflexive Banach space universal for all separable reflexive Banach spaces, Studia Math. 30 (1968), 53-61. [166] M. Talagrand, Comparaison des bordliens d'un espace de Banach pour les topologies faibles et fortes, Indiana Math. J. 27 (1978), 1001-1004. [167] W.K. Tang, On extension of rotund norms, Note aux C. R. Acad. Sci. Paris 323 (1996), 487-490. [168] N. Tomczak-Jaegermann, The moduli of smoothness and convexity and the Rademacher averages of the trace classes Sp (1 1 see [60, 27.4, 28.6]. The change of density in Theorem 2 is used in Section 2.1 to show that a k dimensional subspace of L p well embeds into ~pn with n not too large. There what is needed is the relation between the L ~ and the Lp(g d/z) norms on Mg,pE. The relevant estimate follows from a trivial observation which we record for later reference.

Zkn=l

LEMMA 4. Let lZ be a probability measure and let E be a k dimensional subspace of Lp(lZ) which has a basis {fl . . . . . fk} which is orthonormal in Lz(/z) and such that 1]fi] 2 = k . Then f o r e a c h f in E, ]]f]]~ ~ 1/2 which satisfies, for arbitrary f in Lp(#),

J'lvii 2g(p-2)/p

d/z ~< 2 K 2 IITII2

f

Ifl2 g fp-2)/p dlz.

This gives Theorem 7 with a slightly better constant when T L p ( # ) has full support and hence also T h e o r e m 7 as stated in the range 1 < p < oc. For 0 < p ~< 1, T h e o r e m 7 follows via interpolation from T h e o r e m 10. THEOREM 10. If 0 < p 1, # is a probability measure, and T is an operator on L p ( # ) , then IlMg,pTMg-,lp " L ~ ( g d # ) -+ L ~ ( g d#)ll ~< alITll for some density g. To prove T h e o r e m 10, it is enough to find a strictly positive function ~l/p in L p ( # ) so that T maps the order interval [_~l/p, ~l/p] into [-al]Tll~, 1/p, allT]l~,l/p]. The main point is that every operator T on L p (#), 0 < p ~< 1, has a modulus ITI which satisfies IllTlll = IITII and ITfl 1 there is a positive constant C = C(p, )~) such that whenever n < )~k, ~kp C-embeds into e nI 9. Some progress on this problem has recently been achieved in [48].

W.B. Johnson and G. Schechtman

846

Next we would like to sketch some of the ideas involved in the proofs of some of the statements of Theorems 12 and 13. As we already indicated above, a common feature of all the proofs we shall sketch is that, using a change of density, we first find an isometric copy of X with some additional good properties. We delay stating theses properties and the actual change of density that ensure them until later and denote the new space by the same notation X. We may also assume without loss of generality that X lies in a finite (but large) dimensional s m space, say L mp (/z), where # is a probability measure on {1 . . . . . m}. We denote #i = # ( { i }). We would like to show that the restriction operator to a set of relatively few of the m coordinates is a good isomorphism on X. We prefer to do it iteratively by first showing how to find a subset of cardinality at most m/2 such that the restriction operator is a (very) good isomorphism on X, provided m is much larger than k. We shall then show how to iterate this procedure. The choice of the subset will be random; for example, it is enough to show that, for an appropriate e(k, m), m

AveA

sup 2 ~ I.Li IXi [P -- ~ #i [Xi Ip x~X, Ilxll~ 1 icA i=1 m

= E

sup Y~ei#ilxil p t) 1, while kl(X, C1) ~ C-llm(/1 + logn/m). Except for the constants involved the results are best possible. Note that for m proportional to n the resulting dimension of the contained g p space is also proportional to n. For p - 1 this case was observed earlier in [15]. Note also that the conclusion of this theorem is "isomorphic" rather than "almost isometric". We do not know if one can replace the constant C p in the left hand side of the inequalities by 1 + e (of course paying by replacing the constant in the right hand side by one depending on 8). There is also a version of Theorem 19 for p -- cx~" If m ~> n ~, with 3 > 0, then every mdimensional subspace X of ~ contains a well isomorphic copy of ~ with k >~c(S)m 1/2. This was proved in [20] for m proportional to n, and in [5] in general. When m is very large there is also a version of Theorem 19 for p = 1. It was proved in [25] that for every m-dimensional subspace X of ~ , k, (X, K) ~> c min{ (n/(n - m)) log(n/(n - m)), n }. K and c are universal constants.

3. Finite dimensional subspaces of

Lp

with special structure

3.1. Subspaces with symmetric basis In this section we treat the classification of the finite symmetric basic sequences in L p, 1 ~< p < c~ and to some extent also the classification of the finite unconditional basic sequences in Lp, 1 ~< p < 2. Recall that a sequence x l . . . . , xn in a quasi normed space X (over R) is said to be K-symmetric if for all scalars {ai }, all sequences of signs {8i } and all permutations Jr of {1 . . . . . n}

• i=l

ai xi

~ 2 so we only state the result (from [29]; see [61, Theorem 4.4]). THEOREM 21. For every 2 < p < ec and every constant K there is a constant D such that any normalized K-symmetric basic sequence in L p is D equivalent to the unit vector basis o f N n with the norm

]]{ai} 11--max[ ( Z

]ailP) ' / p , w ( Z

]ai 12) ./2 }

(18)

f o r some w ~ (0, 1).

Of course, since g~ isometrically embeds in L p, any norm of the form (18) embeds, with constant 2, into L p. For 1 ~< p < 2 the structure of the symmetric sequences in L p is more involved. Let M be a Orlicz function (see [28, Section 5]) and g i the associated Orlicz sequence space. It turns out that the space ~M embeds isomorphically into L p if and only if the unit vector basis of g i is p-convex and 2-concave and this happens if and only if M(]t] I/p) is equivalent to a convex function and M ( t |/2) is equivalent to a concave function on [0, ec) (see [14]). Recall that two functions M1, M2 :R --~ [0, co) are equivalent (at 0) if there exist constants K l, K2, )~, # and xo > 0 such that for all Ix l < xo KIM2()~x) /an2

>/O. Then

1/p 1

}n 2

II a/ /-lll

l/nai ~ ~-~mi>l/n(pai O so that if l < r < 2 a n d S is an operator on L 2n (n ~ n(e, r)) with ]ISII2 ~ c(p) max{kp/2, kp*/2}, where p* = p / ( p - 1). It turns out that, except for logarithmic terms, one can achieve this bound. THEOREM 33 ([8]).

Pp(k, K, e) 2 and k - 2 m. We build a path Yo . . . . . Ym of spaces connecting Y0 - ~ with Ym - e kp with d (Y i, Yi-1) ~< 2. This can be done in several ways but, since we also want Yi to be well isomorphic to a well complemented subspace of e~ with si as small as possible, we take Yi to be the e p sum of 2 i copies of e 2m-i .

_2m-i

Given any 8 > 0, g-e embeds as a complemented subspace, with constants depending only on p and S, into e~ with ri - - [ ~ 2 ( m - i ) p / 2 ] . The fact that this holds for some 8 < oe is

Finite dimensional subspaces o f L p

867

exposed for example in [24]. To get it for all 6 > 0 represent s as the ep sum of u copies of ~2v (introducing of course a constant depending on u and p), embed each summand complementably in an appropriate s space, and take the s sum of these u spaces as the containing gp space. It follows that Yi well embeds complementably into s for si - [62 i2(m-i)p/2]. Lemma 37 implies then that, for s - Ziml si, ~k (~ e Sp ~ ~,p+k. The assumption that s is well complemented in s implies that n -- y k p/2 for some ), bounded away from zero (this is a result of [2]). It is now easy to see that, with the right choice of 6, s -- n - k and s G s ~ s It remains to show that if also s 2 @ X ,~ s np then X ~ s n-k . This follows from the following simple "uniqueness of complement" result of [9] in the form proved in [33]. In the statement " + " denotes a direct sum of subspaces and the isomorphism constants implicit in the notation " ~ " of the conclusions depend only on the constants for the " ~ " and the projections in the hypotheses. PROPOSITION 38. Assume Z = Y + X = H + G with H C X, and assume H ~ Y 9 W. Then X ~ G @ W. Inparticular, if f o r i = 1,2, Z = Yi + X i = Hi-+-Gi with Y ~ Yi, G ~ Gi, H ~ Hi C Xi and H ~ Y G W. Then X1 ~ X2. To end the proof of Theorem 35 we only have to show that X, the complement of Y ~ s in gp, contains a subspace well isomorphic to gk2 and well complemented. Since n is of order at least k p/2, the general theory of Euclidean sections as exposed in [24] implies that, for some 6 > 0, X contains a subspace U1 well isomorphic to s k. U1 is automatically well complemented (see [44] or [18, p. 46]). Now find another copy of s of U1 in X and iterate.

in the complement

PROOF OF PROPOSITION 38 ([33]). It is very easy: Put F = X N G. Then Z = Y + X = Y+F-+-HandconsequentlyG~YGF. ThusX=F+H~FGYGW~GOW. D PROOF OF THEOREM 36. It follows the same outline as that of the proof of Theorem 35. Given a k-dimensional well complemented subspace Y0 of L p, 2 < p < cx~, we find a path Yo, Y1 . . . . . Ym of well complemented subspaces of Lp with d(Yi, Yi-1) < 4, Ym well isomorphic to ~ , and m -- [log 2 k]. We then use Theorem 33 to embed each of Yi in a well complemented fashion in a low dimensional g p space and continue in a way similar to the proof of Theorem 35. To build the path of spaces Yi, apply first the change of density of Theorem 2 and then that of Theorem 7 to get that without loss of generality IlYllp ~< 2nl/2-1/Pllyllz for each y 6 Y0 and the projection P from L p onto Y0 is also well bounded with respect to the L2 norm. Put Yi--{(y,2iy);

Y6Yo}CLpGpL2,

i - - 1 , 2 . . . . . m.

868

W.B. Johnson and G. Schechtman

T h e n c l e a r l y d ( Y i , Y i - 1 ) < 4, i = 1 , 2 . . . . . m, Ym is w e l l i s o m o r p h i c to ~ a n d it o n l y r e m a i n s to s h o w that e a c h o f Yi is w e l l i s o m o r p h i c to a w e l l c o m p l e m e n t e d s u b s p a c e of Lp. S i n c e P is b o u n d e d in b o t h L p a n d L2, Yi is w e l l c o m p l e m e n t e d in Zi - {(z, 2iz); y E L p } C L p O p L2. Finally, b y a t h e o r e m o f R o s e n t h a l [52] (see also [1]) Zi is w e l l i s o m o r p h i c to a w e l l c o m p l e m e n t e d s u b s p a c e o f L p. D

References [ 1] D.E. Alspach and E. Odell, L p spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 123-159. [2] G. Bennett, L.E. Dor, V. Goodman, W.B. Johnson and C.M. Newman, On uncomplemented subspaces of Lp, 1 < p < 2, Israel J. Math. 26 (2) (1977), 178-187. [3] K. Berman, H. Halpem, V. Kaftal and G. Weiss, Matrix norm inequalities and the relative Dixmier property, Integral Equations Operator Theory 11 (1) (1988), 28-48. [4] J. Bourgain, New Classes of s Lecture Notes in Math. 889, Springer-Verlag, Berlin (1981). [5] J. Bourgain, Subspaces ofl N, arithmetical diameter and Sidon sets, Probability in Banach Spaces, V, Medford, MA, 1984, Lecture Notes in Math. 1153, Springer, Berlin (1985), 96-127. [6] J. Bourgain, Bounded orthogonal systems and the A(p)-set problem, Acta Math. 162 (3-4) (1989), 227245. [7] J. Bourgain, N.J. Kalton and L. Tzafriri, Geometry offinite-dimensional subspaces and quotients of L p, Geometric Aspects of Functional Analysis (1987-88), Lecture Notes in Math. 1376, Springer, Berlin (1989), 138-175. [8] J. Bourgain, J. Lindenstrauss and V.D. Milman, Approximation of zonoids by zonotopes, Acta Math. 162 (1989), 73-141. [9] J. Bourgain and L. Tzafriri, Complements of subspaces of l~, p >>,1, which are uniquely determined, Geometrical Aspects of Functional Analysis (1985/86), Lecture Notes in Math. 1267, Springer, Berlin (1987), 39-52. [ 10] J. Bourgain and L. Tzafriri, Invertibility of "large ""submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math. 57 (2) (1987), 137-224. [11] J. Bourgain and L. Tzafriri, Restricted invertibility of matrices and applications, Analysis at Urbana, Vol. II (Urbana, IL, 1986-1987), London Math. Soc. Lecture Note 138, Cambridge Univ. Press, Cambridge (1989), 61-107. [12] J. Bourgain and L. Tzafriri, Embedding lkp in subspaces of Lp for p > 2, Israel J. Math. 72 (3) (1990), 321-340. [13] J. Bourgain and L. Tzafriri, On a problem of Kadison and Singer, J. Reine Angew. Math. 420 (1991), 1-43. [14] J. Bretagnolle and D. Dacunha-Castelle, Application de l'dtude de certaines formes lindaires aldatoires au plongement d'espaces de Banach dans des espaces LP, Ann. Sci. t~cole Norm. Sup. (4) 2 (1969), 437-480 (French). [15] B. Carl and A. Pajor, Gelfand numbers of operators with values in a Hilbert space, Invent. Math. 94 (3) (1988), 479-504. [ 16] K.R. Davidson and S.J. Szarek, Banach space theory and "local" operator theory, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 317-366. [17] W.J. Davis, V.D. Milman and N. Tomczak-Jaegermann, The distance between certain n-dimensional Banach spaces, Israel J. Math. 39 (1-2) (1981), 1-15. [18] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Math., Vol. 43, Cambridge University Press, Cambridge (1995). [19] L.E. Dor, On projections in L1, Ann. of Math. 102 (3) (1975), 463-474. [20] T. Figiel and W.B. Johnson, Large subspaces of 1n and estimates of the Gordon-Lewis constant, Israel J. Math. 37 (1-2) (1980), 92-112.

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869

[21] T. Figiel, W.B. Johnson and G. Schechtman, Factorizations of natural embeddings of l~ into Lr, I, Studia Math. 89 (1) (1988), 79-103. [22] T. Figiel, W.B. Johnson and G. Schechtman, Factorizations of natural embeddings of l~ into Lr, H, Pacific J. Math. 150 (2) (1991), 261-277. [23] T. Figiel, S. Kwapiefi and A. Petczyfiski, Sharp estimates for the constants of local unconditional structure of Minkowski spaces, Bull. Acad. Polon. Sci. Srr. Sci. Math. Astronom. Phys. 25 (12) (1977), 1221-1226. [24] A.A. Giannopoulos and V.D. Milman, Euclidean structure infinite dimensional normed spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 707-779. [25] E.D. Gluskin, N. Tomczak-Jaegermann and L. Tzafriri, Subspaces of l N of small codimension, Israel J. Math. 79 (2-3) (1992), 173-192. [26] Y. Gordon and D.R. Lewis, Absolutely summing operators and local unconditional structures, Acta Math. 133 (1974), 27-48. [27] W.B. Johnson and L. Jones, Every Lp operator is an L 2 operator, Proc. Amer. Math. Soc. 72 (2) (1978), 309-312. [281 W.B. Johnson and J. Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the Geometry of Banach Spaces, Vol. 1, W.B. Johnson and J. Lindenstrauss, eds, Elsevier, Amsterdam (2001), 1-84. [29] W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (217) (1979). [3O] W.B. Johnson and G. Schechtman, On subspaces of L 1 with maximal distances to Euclidean space, Proceedings of Research Workshop on Banach Space Theory (Iowa City, Iowa, 1981), Univ. Iowa, Iowa City, IA (1982), 83-96. n , Acta Math. 149 (1982), 71-85. [311 W.B. Johnson and G. Schechtman, Embedding lpm into 11 [32] W.B. Johnson and G. Schechtman, Remarks on Talagrand's deviation inequality for Rademacher functions, Functional Analysis (Austin, TX, 1987/1989), Lecture Notes in Math. 1470, Springer, Berlin (1991), 72-77. [33] W.B. Johnson and G. Schechtman, On the distance of subspaces of l~ to Ikp, Trans. Amer. Math. Soc. 324 (1) (1991), 319-329. [341 W.B. Johnson and G. Schechtman, Computing p-summing norms with few vectors, Israel J. Math. 87 (1-3) (1994), 19-31. [351 R.V. Kadison and I.M. Singer, Extensions ofpure states, Amer. J. Math. 81 (1959), 383-400. [36] N.J. Kalton, The endomorphisms of Lp (0 L2(G),

where fi is again convolution with V. Since id is 1-integral and fi maps to a reflexive space, v is nuclear. D

11. A counterexample Returning to the beginning of Section 8, let us agree to say that a space X verifies the Grothendieck theorem if every operator from 12 to X has 1-summing adjoint. From the discussion in "Basic concepts" (see [ 10] for more details), it can be seen that this happens if and only if every operator from X to I 1 is 2-summing. From (ii) and (iv) in Section 8 we see that the spaces Cz+ ('IF) and CE (ql"), where E = Z+ U {-2n}n~>0, verify the Grothendieck theorem. However, their injective tensor product does not. Indeed, this injective tensor product is none other than Cz+ xe (~'2), and the latter space contains a complemented copy of 11 (then, surely, the projection to this copy is not 2_2 n summing). This copy is spanned by the functions {zZ"z2 }n~>0.To exhibit a projection, we observe that the characteristic function of {(k, - k ) : k E Z} C Z 2 is the Fourier transform of a measure on qr x "IF.The required projection is given by convolution with this measure. It is known that the dual spaces of Cz+ and C E are of cotype 2 (see, e.g., the survey [20]). The above construction can be adapted to show that the projective tensor product of two cotype 2 spaces may fail to be of cotype 2, moreover, it may contain a complemented copy of co. The conjectures disproved by the above were perceived as natural for some time in the past. A "more radical" answer to these and many other questions is given by Pisier's celebrated counterexamples (see, e.g., [31 ]). I have included the above material (first published in [20]) because of its transparence and relevance to harmonic analysis.

Banach spaces and classical harmonic analysis

883

12. Stein theorem Seemingly, in the proof of Theorem 12 we did not use the full-scale assumption that E is a quasi-Marcinkiewicz set (we needed "only" the (L 1 _ L 1/2) continuity of the corresponding multiplier). The result of Stein stated below shows that this is not so. We denote by S(G) the space of all measurable functions on G endowed with the (metrizable) topology of convergence in measure. THEOREM 13. Let A C F, and let T'L1A (G) --+ S(G) be a continuous linear operator commuting with translations. Then T is of weak type (1,1). PROOF. The point is that something similar can be said if T does not necessarily commute with translations. Specifically, by the Nikishin theorem (see [25]), in the latter case for every e > 0 we can find a set X-2eC G with IG \ X-2cI < e such that I{t ~ X2s" I(rf)(t)[ > ~.}l ~< Cs)~-1 ]If Ill,

f ~ Llx, X > 0.

(3)

Now, if T does commute with translations, it only remains to average. We fix, e.g., e = 1/2, and put s = 1-21/2. Substituting fx for f (x 6 G) and using invariance, we rewrite (3) in the form

f xY2-xX{[Tf[>)~} ~ C~.-~ Ilfll~; integrating in x over G, we get

IX21 [{ITfl > 9~}] ~ C)~-I Ilfll~. 13. Ap-sets Let 1 2 there is a A2n-Set not of type A2n+e for any e > 0. The main idea of the construction is that I ~ • a• yl2, expands explicitly as a linear combination of products of the characters in E and their complex conjugates, so that we may play with arithmetic conditions on E to ensure huge cancellation after integration.

S. V. Kislyakov

884

This leads to nontrivial examples of A zn-sets, some of which turn out to be not of type AZn+e, 6 > O. The question as to whether similar examples exist for p ~: 2n, n >~ 2, n 6 Z, was often referred to as the A p-problem. For p < 2, the solution of this problem has turned out to be easy (at least formally). Namely, if p < 2, then every Ap-set is a A p+~-set for some e > 0. As in the preceding section, the reason is in the Rosenthal-Nikishin-Maurey factorization theorem saying, in particular, that if X is any subspace of L P with p < 2 on which the topologies of LP and S are equivalent, then, after a change of density, X becomes a subspace of Lr with some r > p:

(f

x i o)1'

4c

(/

x

lip

xEX,

where a is a positive weight, f a = 1. See [25] for the details. Now, if X is translation invariant, we easily get rid of the density by averaging, which implies our claim. This observation was made in [2]. However (in reality, so-to-say), no specific examples of Ap-sets with p < 2 that are not Az-sets seem to be known (this is the precise meaning of the above remark on the formality of the solution in question). For any p > 2, Bourgain [5] proved the existence of Ap-sets that are not of type Ap+~. (Thus, the Ap-problem remains unresolved for p -- 2 only.) Bourgain's proof is probabilistic and combinatorial in nature. Technically, it has little in common either with harmonic analysis or with Banach space theory, though philosophically the result may be linked with the latter. It is a known fact of the finite-dimensional theory of Banach spaces that if X is a space of dimension n and its cotype p constant is at most C, then a typical subspace of X of dimension [C1 n z/p] is 2-distant from the Hilbert space of the same dimension. Here C1 depends only on C. See [26, Theorem 9.6]. Thus, in any n-dimensional subspace X of L p (p > 2) there are many nearly Hilbertian subspaces of dimension proportional to n z/p. Now, suppose that X is spanned by characters 9/1. . . . . Yn. If we succeed in finding a subspace Y of X also spanned by characters, of dimension roughly n z/p, and such that the LP-norm is equivalent to the LZ-norm on Y with a constant independent of X, we are very close to the desired example. (In other words, we need to know that subspaces with "typical" behavior occur even among quite specific spaces, namely, among those spanned by characters.) To understand why this finite-dimensional statement suffices, we note that in many cases the exponent 2 / p in the above discussion is optimal. If we restrict ourselves only to the spaces X having this property, then on the above Y the best constant of equivalence of the L p- and L P+~-norms must tend to infinity as n ~ c~, for every e > 0. Then we attach such spaces Y to one another to obtain an infinite set that is precisely of type A p. In [5] an infinite set of this sort was constructed by considering the spaces span{z 2n,..., z 2n+l } on ~' in the role of X; the "attachment" procedure consisted in applying the LittlewoodPaley decomposition. This is easy. It is the above finite-dimensional statement that is really difficult. Bourgain showed that, in fact, translation invariance is irrelevant in it.


885

THEOREM 14 (see [5]). Let q91 ..... q3n be mutually orthogonal functions on a probability space(S2, v), let lqgil 2 and l ~> 1. Clearly, it suffices to prove this only for 1 -- 1, k - 2. Then the role of T~0 in Theorem 16 can be played by the Sobolev embedding operator W~ 1) (qI'2) ~ LZ(T2). In Section 9 it was already explained that C(1)(ql"2) is similar to a space of the form CE(G) with E a quasi-Marcinkiewicz set. The adjustment of the details is left to the reader (or see [23]). Passing to applications of Theorem 17, we take E -- {2n" n ~> 0} C Z (this is a Sidon set for qr) and consider its square E l - - { ( 2 k,2l)" k , l ~ > 0 } C Z 2. From condition (4) for E it is easily seen that E1 is a A p-set for •2 for every p < oc. However, it is also easy to observe that in the analog of (4) for E1 the constant grows as p, i.e., faster than ~/-fi. Thus, E~ is not a Sidon set. By Theorem 17, CEI (q,2) fails to have G-L 1.u.st. (we note that CE~ (~2) is isomorphic to the projective tensor product of 11 by itself). The domain of applicability of Theorem 18 is outlined in Section 13. We refer the reader to [23] for more examples.

16. Theorems 16, 17, and 18 are proved by similar methods. For Theorem 16, we consider the following operators:

CE(G) id>

L~(G) T~>L2E(G) ~> CE(G).

Here ~p is an arbitrary function in 12(E). By Theorem 12, the adjoint T~ is 1-summing. If CE(G) has G-L 1.u.st., the identity embedding id factors through L 1 (because id is 1summing; see [10]). So, we have come across a composition of operators belonging to mutually adjoint operator ideals, hence it follows that

vj (T~oid T~) 0 and K - ~/3(y)

~ ~

y~E

(/3(9/) - 02(y)) ~

flip -

q211~ -

Xllq~ I1~

< K.

TeE

Thus, there is a (signed) measure # on G such that

< K

f o r p E U,

d/z(x) ~> K

f o r p 6 W.

Re

fG p(x)d/z(x)

Re

fa p(x)

We show that the measure v defined by v(A) = / z ( - A ) y 6 E, then K y 6 W, hence we see that

has the desired properties. First, if

K Re K (ef,(y)), and ~(g) = 0.

Key ~ U

(beV]

Now we can restate Theorem 16 as follows: if E is a quasi-Marcinkiewicz set but not a quasi-Cohen set, then C E (G) fails to have G-L 1.u.st.


891

18. S i d o n sets and a r i t h m e t i c d i a m e t e r

For a finite set A C F , its arithmetic diameter d(A) is defined as the smallest N such that CA (G) is at most 2-distant (relative to the Banach-Mazur distance) from a subspace of the N-dimensional space l~c. If E is a Sidon set in F , then CA (G) is at most S(E)-distant f r o m / i l l for every finite A C E. Consequently, the arithmetic diameter of A must grow exponentially as a function of IAI. There are many ways to see this. For instance, we may argue as follows. First, recall that for p ~> 2 the type 2 constant of L P is of order c~/-fi. Next, it is easily seen that the BanachMazur distance between l~c and 1~gN is bounded uniformly in N, so the type 2 constant of l~c does not exceed c~/log N. Finally, if el . . . . . en are the coordinate unit vectors in In1, then the Rademacher average f01 II ~ ri(t)ei IIdt is equal to n, so that the type 2 constant of I1 is at least ,c/ft. Thus, if 11 is 2-embeddable in l~c, then ~ ~< C'~/log N, as desired. It is remarkable that the exponential dependence of d (A) on IAI in fact characterizes the Sidon sets. THEOREM 20 (Bourgain [4]). I f l o g d ( A ) >~6lAIfor everyfinite A C E, then E is a Sidon set. Moreover, we have S(E) >~c6 -11, where c is a universal constant. The proof of this Banach-geometric characterization of Sidon sets can hardly be called "geometric"; rather, it is combinatorial, because the only geometric notion involved is that of entropy. In general, in a metric space with metric p, the e-entropy of a set F is the logarithm of the smallest cardinality Np (e) = Np (F, e) of an e-net for F. The idea of using entropy in the theory of Sidon sets was originally exploited by Pisier (see, e.g., [30]). Later, Bourgain revised Pisier's work on Sidon sets, replacing some fine probability methods by an elementary random choice combined with entropy combinatorics and with harmonic analysis arguments. We refer the reader to the Bourgain's survey [4] (and to the references therein) for this. The following statement is [4, Corollary 8]; this is a slight improvement of Pisier's original entropy characterization of Sidon sets (see [30, p. 941 ]). For a subset A of F, we define a metric pA o n G as follows:

pA(x, y) = sup Iv(x) -- y ( y ) [ . yEA

THEOREM 21. Let E C F. l f for every finite set A C E and some r > 0 we have

NpA (G, r) > 2 ~IAI, then E is a Sidon set and S(E) ~O

+ 1 ) - l l f ( n ) l ~< CIIflll,

f E H 1(qF),

as well as some related questions were discussed. In spite of a quite considerable bulk of information presented in [3], no complete description of the corresponding classes of Banach spaces X is available in these cases as yet.

21. B-convexity and K-convexity Here our basic multiplier is the orthogonal projection of L2(D) (D is the dyadic group, see Section 8 for the definition) onto the subspace generated by the coordinate functions of D. (Identifying D with [0, 1] in the usual way, we arrive at the Rademacher projection.) The spaces X for which this projection extends in a natural way to LZ(D; X) are said to be K-convex. The notion of K-convexity is quite useful in the theory of Banach spaces, primarily due to the fact that it is intimately related to the duality between type and cotype. Remarkably, K-convex spaces admit a complete characterization. THEOREM 24 (Pisier; see [32]). A Banach space X is K-convex if and only if X does not contain the spaces lln uniformly. The spaces with the latter property are called B-convex. A space is B-convex if and only if it is of nontrivial type; see "Basic concepts".

22. UMD-spaces Here our basic multiplier is the Hilbert transformation 7-[ on L 2 (T). We have 7-[ = Tm, where m(k) = - i sgnk, k E Z. It is really quite useful to know for which Banach spaces X the Hilbert transformation (more generally, an arbitrary Calder6n-Zygmund singular integral operator) acts on the LP-space of X-valued functions. The general theory reduces the case of any p E (1, cx~) to the case of p = 2; see, e.g., [35]. Surprisingly, the class of spaces X for which 7-/acts on L2(T, X) admits a complete description. THEOREM 25 (Burkholder, McConnel, Bourgain). A Banach space X has the above property if and only if there is a biconvex function ~ : X x X --+ IR such that ~ (0, O) > 0 and (x, y) ~< IIx + y II if IIx II - IlY II - 1.


895

The same condition is equivalent to the continuity of standard martingale transformations on LZ(x). That is why the spaces X such that 7-/acts on L2(X) are called UMDspaces (UMD is for "unconditionality of martingale differences"). The proof of Theorem 25 is indirect and passes via this statement on martingales. We refer the reader to Burkholder's survey [9] and to the references in it for the details. It should be noted that the condition formulated in Theorem 25 is difficult to work with. Basically, the only way to prove the existence of the above ~ on a Banach space X is to verify the continuity of 7-/(or of the martingale transformations) on L 2 ( X ) directly. The Hilbert space seems to be the only one presenting a simple possibility of exhibiting ~ (for instance, we may put ((x, y) = 1 + Re(x, y)). Direct verification of the continuity of 7-/shows that the spaces L P with 1 < p < cx~ and the Shatten-von Neuman classes Cp (again with 1 < p < oc) are UMD-spaces. Next, the property of being a UMD-space is inherited by the subspaces and quotient spaces. This is a superproperty (if X is finitely representable in Y and Y is UMD, then so is X). A UMDspace must be superreflexive, but not all superreflexive spaces are UMD. As before, we refer the reader to [9] and the references therein. Again, the proofs of the above statements do not use Theorem 25.

23. The above discussion should be supplemented with the following result due to Bourgain. THEOREM 26. I f tx is a finite measure and 0 < r < 1, then the Hilbert transformation 7-[ is bounded f r o m L2('1F; L 1(Ix)) to L2(~I'; L r (Ix)). We refer the reader to the survey [20] for the proof and related material. The result can be used to verify statement (ii) in Section 8. This idea is Bourgain's; the details of this verification can also be found in [20].

24. Returning once again to statements (i)-(v) in Section 8, we note that we may ask any Banach space theory question about any specific space arising in harmonic analysis. This will yield an incontestable point of contact of the two fields, but rarely will this show a real interplay between them. In many cases, a pure problem of hard analysis (even without the adjective "harmonic") arises in this way, as is described in Section 8 after statements

(i)-(v). Some exceptions of these "rule" were, however, discussed above. Another one is presented by the still mysterious space U of uniformly convergent Fourier series on the unit circle. Bourgain was the first to prove that U* is weakly sequentially complete, and his proof involved difficult techniques of hard analysis; see [6]. But later it was discovered that the statement can be verified almost entirely within the ("soft") methods of functional analysis. We refer the reader to the paper [11 ] in this collection for a discussion and references.

S. V. Kislyakov

896

Having mentioned the space U, we probably cannot avoid considering the spaces of trigonometric polynomials on the circle. Let 7Jnp -- span{l, z

. . . . .

s

with the metric of L p (Tf), 1 Lp.~S-22, ITllPdo'2) defined by T f T f / T 1 is a linear isometry satisfying the condition T 1 -- 1. Denote by # and v the measures on R ~, which are the joint distributions from the statement of lemma. By the Equimeasurability theorem, for every b - (bl . . . . . bk) E R ~, the functions ~ bi fi and ~ bi Tfi / T1 have equal distributions with respect to o'1 and IT 11p do'z, respectively. Therefore, the Fourier transforms of the measures # and v are equal at the point b. Since b is arbitrary, the result follows. D THEOREM 4 (Extension theorem). Under the conditions of the Equimeasurability theorem, except f o r T 1 -- 1 which is no longer required, there exists an injective linear isometry T f" Lp(S-21, .fit, o'l) ~ Lp(S-22, o2) so that TtII4 - T, where ..4 is the smallest o'-algebra of subsets o f 1-21 making all functions from H measurable. PROOF. It is easily seen that the space Lp(S21, A, o'l) is the closure in Lp(S'21, o'l) of the set of functions that can be represented in the form B (fl . . . . . f~), where k 6 N, B is a Borel function on R k and fl . . . . . fk belong to H. Define an operator T ~" Lp (~21, A, o'l) w-~ Lp (S-22, o'2) by

T ' ( B ( f l . . . . . f~)) -- T 1 . B ( T f l / T 1 . . . . . T f k / T 1 ) for every choice of k, B, f l . . . . . fk. The operator T ' is well-defined. In fact, suppose that B1 (fl . . . . . fk) - - B2 (gl . . . . . gm) for k, m E N, Borel functions B1, B2 o n R k and R m, and functions f l . . . . . f~:, g l . . . . . gm ~ H. By Lemma 3 applied to the functions

f l . . . . . fk, gl . . . . . gm,

fs?

l B l ( f l . . . . . f ~ ) - B2(gl . . . . . gm)] p do'1

1

=

]BI(Tfl/T1 ..... Tf~/T1)-

B 2 ( T g l / T 1 . . . . . Tgm/T1)IPITIIPdo'2.

2

Therefore,

T 1 . B I ( T f l / T 1 . . . . . T f ~ / T 1 ) -- T 1 . B 2 ( T g l / T 1 . . . . . T g m / T 1 ) as elements of Lp(.Q2, o'2). The same argument (with B2 - - 0 ) shows that T' is a linear isometry, and its restriction to H is equal to T. D The Extension theorem applies only to subspaces of L p containing constant functions (unital subspaces). However, the general case can be reduced to the unital case by changing the density, since every separable subspace of L p contains a function with maximal support (see, for example, [3]).

Aspects of the isometric theory of Banach spaces

905

Let us show a typical application of the Extension theorem. Let E be a domain in R n. Denote by Lph (E) the subspace of L p ( E ) (with Lebesgue measure) which consists of harmonic functions. The following result was proved in [108]. THEOREM 5. Let p > 0, p r 2N, n/> 2, E1 and E2 be two domains in R n. The spaces L hp(El) and Lph (E2) can be isometric only in one of the following situations: (i) the domains E1 and E2 are similar; (ii) p = 2 n / ( n - 2) and the domains E1 and E2 coincide up to the composition of an inversion and a similarity. PROOF (Sketch). The smallest o--algebra A, making all functions from L hp (El) measurable, is the a-algebra of all Borel subsets of El. By the Extension theorem, every linear isometry T from Lph (El) to Lph (E2) can be extended to an isometry T I from the whole space Lp(E1) to Lp(E2). Using the classical characterization of the isometries between L p-spaces (see [72]), one can show that T' is a composition operator, i.e., T'f(co) = T l ( c o ) f ( r ( w ) ) , where r is a measurable mapping from E2 to the closure of El. Now our problem is reduced to characterization of all composition operators mapping harmonic functions to harmonic functions. Direct differentiation shows that this happens only if r is a conformal mapping. Now one can apply Liouville's theorem that every conformal mapping in R n , n ~> 3, is the composition of a similarity and an inversion. It can be shown that an inversion can be included only when p - 2 n / ( n - 2). In the case n = 2, the conformal mapping r is a holomorphic or an antiholomorphic function, and the fact that r is a similarity follows from a calculation based on the fact that T 1 -- Ir'l 1/p must be a harmonic function. D The Uniqueness theorem was generalized in [37,80] to the case of measures on certain finite and infinite dimensional normed spaces. Suppose that p > 0, E is a separable Banach space, and #, v are finite Borel measures on E so that, for every a c E,

g(a) -- fE IIx -- allP d # ( x ) -- fE IIx - allP dv(x) < cx3.

(3)

We say that p is an exceptional exponent for E if (3) does not necessarily imply that # = v. The exceptional exponents for the space Lq, q > 0 are those for which p / q is an integer, and, in the case of the n-dimensional space l q, n besides the condition p / q E N one of the following must be satisfied: (i) p / q < n, (ii) q is an even integer, (iii) q and p / q - n are both odd integers. If E -- C (K), where K is a compact without isolated points, then there are no exceptional exponents. For the complex space l ~ exceptional exponents are even integers, and in the real case p is exceptional if and only if n + p is an odd integer. Other results in this direction include formulae for calculating the measure # out of the potential g (see [53]), uniqueness theorems for Gaussian measures (see [75,81,82]). Applications and generalizations of the extension method also include [2,18] (see Section 3), [30,41,42,52,56,78,79,85-87,89,95,113,116,122,124,134].

A. Koldobsky and H. KSnig

906

3. Positive definite functions and isometric embedding of normed spaces in Lp-spaces The problem of how to check whether a given Banach space is isometric to a subspace of Lp was posed by Lgvy [74] in 1937. A well-known fact is that a Banach space embeds isometrically in a Hilbert space if and only if its norm satisfies the parallelogram law [26, 47]. Neyman [103] (see also [126] and [15]) proved that subspaces of Lp with p ~ 2 cannot be characterized by a finite number of equations or inequalities. There is a close connection between isometric embeddings in L p and positive definite functions. Recall that a complex valued function f defined on R n is called positive definite if, for every finite sequence {xi }im_1 in R n and every choice of complex numbers {ci }iml, m

E

m

ECiCjf(xi

-- Xj) ~/O.

i=1 j--1 By Bochner's theorem, every continuous positive definite function on R n is the Fourier transform of a positive finite measure. It was known already to L6vy [74] that if B = (R n, I[. II) embeds isometrically in Lp, 0 < p ~< 2, then exp(-l[xl[ p) is a positive definite function, and, hence, is the Fourier transform of a symmetric p-stable random vector. The actual equivalence of the two notions was discovered by Bretagnolle, Dacunha-Castelle and Krivine [13] who proved that, for 0 < p ~< 2, a Banach space B is isometric to a subspace of Lp if and only if the function exp(-llxll p) is positive definite. For different proofs and generalizations see [13,70,1,101,46,127] and [7, Chapter 11]. Bretagnolle, Dacunha-Castelle and Krivine used their result to prove that the space Lq embeds isometrically in L p when 0 < p < q ~< 2. However, it is more difficult to check whether exp(-[Ix I[p) is positive definite for other norms. For example, the following problem posed by Schoenberg in 1938 (see [121]) was open for more than fifty years: for which p e (0, 2) is the function exp(-llx I[p) positive definite, where [Ix [[q is the norm of the space l q,n 2 < q ~< oo .9 An equivalent formulation asks whether the spaces l qn embed in Lp with 0 3 the spaces l q,n 2 < q O, n 9 N, - n < # < qn, #/q ~ NU {0}, ~ 1 3, p 6 (--n + 3, 0), or n -- 3, p 6 (0, 2), or n = 2, p E (1, 2), then the Fourier transform of the distribution IIx IIqp is a sign-changing function on Sn-1. PROOF. By Lemma 7 and properties of the moments Sq (O/), the integral I(O/1 . . . . . O/n-l) = .fl/ [~l [Oil " " " [~n-I [Otn-1 ([[X [[P) A (~1 . . . . . ~ n - l , 1) d~l""" d~n-I = Sq(O/1)'' " S q ( O / n - 1 ) S q ( - O / 1

.....

O/n-1 - P)

converges absolutely if the numbers O/1 . . . . . O/n-l,--19/1 . . . . . O/n-1 + p belong to the interval ( - 1, q). Choosing O/~ 6 ( - 1, 0) for every k = 1 . . . . . n - 1, we have the moments Sq (o/h), k = 1 . . . . . n - 1, positive, and we can make -o/1 . . . . . O/n-1 + P equal to any number from (p, n - 1 + p) A ( - 1, q). Because of our conditions on p, this interval contains a neighborhood of 2, and, since the moment function Sq changes its sign at 2, we can make the integral I (O/1. . . . . O/n-l) positive for one choice of O/'s and negative for another choice. This means that the (homogeneous of degree - n - p) function (llx IlqP)A is sign-changing. []


909

Theorem 6 and the cases n -- 3 and n - 2 of L e m m a 8 imply that the spaces l qn

with

n ~> 3, 2 < q < ec cannot embed in L p, 0 < p ~< 2, and that the spaces l q2 with 2 < q < cx~ can not embed in Lp with 1 < p < 2, which answers Schoenberg's question. Soon after the paper [51] appeared, Zastavnyi [128,129] and Lisitsky [83] independently proved a more general result that there are no non-trivial positive definite functions of the form f([] 9Ilq), where q > 2, n ~> 3. Note that for q = oc a similar result was established earlier in [99]. Zastavnyi's result is even stronger: THEOREM 9. Let E be a three-dimensional normed space with a basis el, e2, e3 such that the function x w-> Ilxel + ye2 + ze31] is differentiable at the point x = 1 for almost all (y, z) 9 R 2, and suppose that the function (y,z) ~

]]xel + ye2 +ze3llxt (1 , y, z)/llel -4- ye2 + ze31]

y, z 9 R,

belongs to the space L1 (R2). Assume that, for some function f : R ~-+ R, the function (x, y, z) ~ f (]lxel Jr- ye2 + ze3 I1) is positive definite on R 3 . Then f is constant. PROOF (Sketch). First, one can assume without loss of generality that f is differentiable on (0, ec), limt--,otf'(t) = O, and there exists a constant C > 0 so that Itft(t)l < C for every t > O. In fact, if this is not the case, one can replace f by a function F(t) = f o f ( t s ) g ( s ) d s , where g is an infinitely differentiable non-negative function with compact support in R. It is easy to check that the function

h(y,z)--

(1 -]y])(1 -Izl),

O,

max{lYl, Izl} ~ 1, otherwise

is positive definite on R 2. Let us prove that

~p(t) - fR2 f (lltel + ye2 nt- ze311)h(y,z)dydz is positive definite on R. In fact, for every e > 0

~/e(t, y , z ) -

f ( l l t e l + ye2 + ze311)h(y,z)exp(-elt])

is integrable and positive definite on R 3 (as the product of positive definite functions). Therefore, for every s E R

~ ( s , 0, 0) - fR exp(--ist) e x p ( - - e l t l ) ~ ( t ) dt >>.O, which means that the function e x p ( - e l t l ) g r ( t ) is positive definite on R. Since e is arbitrary, 7z is positive definite on R.

910

A. Koldobsky and H. K6nig

Using the fact that the derivative of the norm by the first coordinate is a homogeneous function of degree 0 and making the change of variables y = tu, z = tv, we compute the derivative of the function 7t at a point t ~ 0: /,

~r'(t) - I f ' ( l l t e l + ye2 + ze311)lltel + ye2 + ze311't(t, y , z ) h ( y , z ) d y d z JR 2 t2sign(t)

f.2

f'(Itlllel + ue2 + ve311)lltel + ue2 + re3 II't(1 , u, v)

• h(tu, tv) du dr. Multiplying and dividing the expression under the latter integral by Ilel + ue2 -+- re3 II, we can use the condition of the theorem and the properties of the function f mentioned in the beginning of the proof to show that limt~0 ~P!( t ) / t = 0. (Note that this is possible because we have t 2 in front of the integral, which is due to the fact that the dimension of the space is three. This is the point of the proof that fails in dimension 2.) This condition on the derivative of ap implies that ~p!(0) = ~p!!(0) = 0. It follows now from [87, Theorem 4.1.1] that ~p is a constant function. The same is true if we replace the function h(y, z) by the function n2h(ny, nz), where n 6 N. As n --+ cx~, the fact that the corresponding functions ~ are constant immediately implies that f is constant. [] Clearly, every normed space of dimension ~> 3, which is the q-sum with q > 2 of two non-trivial normed spaces, satisfies the conditions of Theorem 9. Another necessary condition for isometric embedding in L p, 0 < p ~< 2, was given in [62]. The proof in [62] is a modification of Zastavnyi's argument. THEOREM 10. Let X be a three-dimensional normed space with a normalized basis {el, e2, e3 } so that: For every fixed (x2, x3) E R2\{0}, the function Xl --+ Ilxlel + x2e2 + x3e311 has a continuous second derivative everywhere on R, and f!

Ilxll' (0 x2, x3)= IlXllx2(O, x2, x 3 ) - 0 X 1

'

' and IIx II'x'2 stand f o r thefirst and second derivatives by Xl, respectively. where Ilxllxl 1

There exists a constant K so that f o r every Xl E R and every (x2, x3) E R 2 with IIx2e2 + " (Xl , x2, x3) (3) Any embedding of l~ into 1N has the form x w-~ ((x, z k ) ) ~ with z~ 6 K n. Let x~ - - z~/llz~ll2 and #k "-IlzkllZs/~-~'~Ul IIz~II2s. Then (3) holds. (3) ==>(2) Let K - R. For x E R n , we denote by x | 6 R nzs the (2s)-fold tensor product of copies of x. Then (x | y| _ (x, y)2S. Consider N

"-- Z

[J"kx~2S

-- fs n-1 x|

dcr (x) E Rn2s .

k=l

Then Sidelnikov's inequality holds,

N o

-

2s

m Crl,s ~ O~

k,l=l hom which implies that ~ = 0: all monomials of degree 2s and hence all p E ,, D 2s,n are integrated exactly as indicated in (5). (2) =~ (1) Let x 6 K n. Apply (2) to p ( y ) - I(Y, x)12s. Then N

E. l(x,xs l

I(x, y)l 2' d o ' ( y ) - Cn,sllXll2, n-1

k=l

i.e., x ~ ((lZk/Cn,s)l/2S(x,xs))~=l defines an isometry of l~ into l~.

917


Condition (3) is useful to prove the existence of certain explicit embeddings since only one e q u a l i t y - (5) - has to be checked. The lower bound is exact, L K ( n , p) -- H K ( n , p) --: N, if and only if the spherical design (Xs)sU__l is tight, i.e., the set of scalar products C = {l(xk, xl)l: k ~ 1} is a very small set, and all #k'S are equal to 1 / N . For p = 4, I C I - 1, and the vectors (xk)~_-i span e q u i a n g u l a r lines. Known cases of equality are n(n + 1) HR (n , 4) -- - , 2

, n2 Hc(n 4)-,

n--2,3,7,23,

n--2,3,8.

For p = 4 it is not known whether there are more values of n such that this holds. For p >i 6, there are only very few cases of equality, the most spectacular being HR (24, 10) -- (28), cf. [115,93,64]. For n = 2 and p E 2N, Proposition 17 and Gaussian quadrature show that HR(2, p) -- p / 2 + 1. Thus, for 2 voln(L).

PROOF. Since (llxllL~) ~ is a continuous sign-changing function o n S n-l, there exists an open subset 12 in S n-1 on which (llxllL-k) A is negative. Let f ~ C c~ (S n - 1) be a nonnegative (and not identically zero) function supported in S2. One can prove (see [59,


923

L e m m a 5]) that the function f(O)r -k is the Fourier transform of a function g(O)r -n+k where g ~ C ~ (S n- 1). Define a body K by -n+k

IIx II~n+~ _ IIx IIL

e

(27r)n g(x)'

where e > 0 is small enough so that the body K is convex (a standard perturbation argument is that, given an infinitely differentiable function on S n-1 , one can choose a small enough e so that the differential properties of the norm I1" IIt-nq-p equivalent to convexity of L are preserved after adding an e-multiple of the ( - n + k)-homogeneous extension of this function). We have

(IIXIILn+P) A - (IIxlI-Kn+P) A - e f (O)r -p,

(lO)

so the distribution IIx IIZ ~+p - IIx II~;~+P is positive definite. On the other hand, by (10) and L e m m a 25

~

(llxllZk)A(o)f(O)dO=(27r)n( voln(L) -- f s n-I

eF/

IIOIIZPlIOII~ n+p ) . n-1

Since the quantity in the left-hand side of the latter formula is negative, we use H61der's inequality (as in Theorem 1) to see that voln (K) > voln (L). D PROOF OF THEOREM 24. Putting k --- 2 in part (a) of Theorem 22 and using the fact that the central section of a convex symmetric body is maximal among sections perpendicular to a given direction, we conclude that the function IIx II~n+3 is a positive definite distribution for every symmetric convex body K. Similarly, IIx I1~-n+2 and IIx I1~-n+l are positive definite (put k = 1 and k = 0 in Theorem 22). Now part (i) of Theorem 24 immediately follows from Theorems 26 and 22. To show (ii), let L be the unit ball of the space with the norm Ilxllz = Ilxl14+ ~llxl12, where e > 0 and I1" Ilq stands for the norm of the space lqn. By Lemma 8, the distribution IIx 114k is not positive definite, therefore IlxllZ ~ is not positive definite for small enough e. Using this value of e in the definition of L (the perturbation of the/~-norm was made to ensure that L has positive curvature) and using Theorem 27 we get a body K giving the desired example (again use Theorem 22 to connect the Fourier transform with the derivatives of parallel section functions). D The condition that IIx IIK ~ is positive definite, that we use in the proofs, has a clear geometric interpretation. For 1 ~< k < n, let us say that a star body K in R n is a k-intersection body of a star body if there exists a star body L in R n so that, for every (n - k)-dimensional subspace H of R n, YOlk(K A H -L) -- vOln-k(L f-) H). It was proved in [61 ] that an infinitely smooth symmetric star body K is a k-intersection body of a star body if and only if IIx IIZ-~ is a positive definite distribution.

924

A. Koldobsky and H. KOnig

Finally, we would like to mention that the isomorphic version of the Busemann-Petty problem is open and equivalent to the famous hyperplane (or slicing) problem (see [97] for details).

5. Approximation of zonoids by zonotopes A zonotope in R n is a special convex polytope, namely the Minkowski sum of finitely many segments I j, j = 1 . . . . . N, in R n,

Z N -- Z

yj" yj 6 Ij, j - - 1 . . . . . N

I j --

j=l

.

j=l

By a segment we mean a compact one-dimensional convex set. For simplicity, we will assume that 0 is the center of all segments; then 0 is the center of symmetry of ZN. A zonoid B is a convex body which can be approximated arbitrarily well by zonotopes in the Hausdorff metric. For n -- 2, all centrally symmetric convex bodies are zonoids, for n ~> 3, the unit balls B pn o f I pn are zonoids if and only if 2 ~ 0 by a zonotope ZN -- ~ U _ 1 lj given by N segments, i.e., ZN C B C (1 + e ) Z N ? This is of particular interest for the Euclidean ball B -- B~. Before stating the estimates for N ( B , e) known for general B and B -- B~, we explain the dual functional analytic formulation. By definition, a zonotope ZN is a linear image of the unit cube B N. Thus, if the interior of ZN is non-empty, the norm II 9II induced by ZN in R n is a quotient norm of l u . Consequently, the polar of the zonotope, P -- Z~v, is the unit ball of an n-dimensional subspace of lN with norm N

ttxtt,-

jl<x,xj)l,

x E R n.

j=l

Choosing Xj E S n-1 , the value )~j is just 1/2 of the length of Ij. Similarly, the polar B ~ of a zonoid B is the unit ball of a norm

IIx II, - fsn-1 I<x, y) Idlz(y), where # is a (positive) measure o n S n - 1 . In other words, B is a zonoid if and only if B ~ is the unit ball of an n-dimensional subspace of L 1 ( S n-1 , IX); one may also take L 1 ([0, 1], IX), cf. [8]. For B = B~, IX is multiple of the usual surface measure. Thus, using the B a n a c h Mazur distance d, the above problem can be restated as follows: Given an n-dimensional subspace X of L1 = L1 (S n-1 , Ix) and e > 0, what is the minimal number N = N ( X , e) such that there is an n-dimensional subspace Yof l~ with d (X, Y) X)]" T E s

v ( T ) = 1, T ( X ) c__X } .

PROOF. Since for projections P ' Y ~ X onto X, Itr(T" X v-->X)] = I t r ( T P ' Y ~ Y)I ~< v ( T P ) ( 2 2~/2~/2~~-)-1 if K - R and c ~> (2 - 4%-/2) -1 if K = C. Problem: Are the latter inequalities in fact equalities? Inequality (26) is used in [67] also to prove that )~(lnp)/~/-ff --> d for 1 ~< p ~< 2 and n --+ where d = ~/2/zr if K = R and d - ~/~-/2 for K - C, independently of 1 ~< p ~< 2. []

Aspects o f the isometric theory o f Banach spaces

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CHAPTER

22

Eigenvalues of Operators and Applications Hermann K6nig Department of Mathematics, University of Kiel, Kiel, Germany E-mail: hkoenig @math. uni-kiel, de

Contents 1. Compact operators, eigenvalues and s-numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Weyl-type inequalities and s-number ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Eigenvalues of p-summing and nuclear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Application to integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF THE GEOMETRY OF BANACH SPACES, VOL. 1 Edited by William B. Johnson and Joram Lindenstrauss 9 2001 Elsevier Science B.V. All rights reserved 941

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Eigenvalues of operators and applications

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1. Compact operators, eigenvalues and s-numbers Classical results relate the order of decay of the eigenvalues of integral operators in L pspaces to the regularity and integrability properties of the defining kernel: the smoother the kernel is, the faster the eigenvalues tend to zero. These operators are clearly compact. Riesz' theory of compact operators admits an abstract quantitative analogue, as far as the decay of the eigenvalues is concerned, for operator classes like p-summing and nuclear operators or operators with summable s-numbers. These latter results will be presented here. Applications to integral operators will be given which extend the well-known classical theorems. The main ingredients of this theory are Weyl's inequality relating eigenvalues and singular numbers, factorization methods for operator ideals - see also [7] - and interpolation theory. All classes of operators which we consider form operator ideals, i.e., they are stable under compositions with continuous operators. To apply the abstract results, an integral operator T in L p is factored over a standard map, like a Sobolev imbedding map, which is known to belong to the operator ideal considered; then T itself belongs to the ideal. Remarks on the historical development of the theory can be found in Pietsch's book [32]. All Banach spaces X, Y. . . . will be complex. We let Bx = {x E X I [Ixll ~< 1} and denote the linear continuous and compact operators, respectively, between X and Y by s Y) and/C(X, Y), respectively, with/2(X) := s X) and/C(X) :=/C(X, X). A map T Es is power-compact if there is n E N such that T n E/C(X). Let a (T) denote the spectrum and p(T) the resolvent set of T. The classical Riesz' spectral theory asserts for power-compact operators T E s (i) For all )~ E C\{0}, (T - )~I) is a Fredholm operator and has finite and equal ascent and descent, say l, with X -- ker((T - )~I) l) @ im((T - )~I)l). (ii) All non-zero spectral values )~ E a ( T ) are eigenvalues of finite multiplicity. They have no accumulation point except possibly zero. These constitute the properties of a Riesz-operator. We recall that the ascent [descent] of (T - )~I) is the smallest integer 1 E 1Nsuch that [im(T - XI) l -- im(T -/kI)/+l].

ker(T - )~I) l = ker(T - )~i)/+1

The (algebraic) multiplicity m(T, )~) of )~ is the dimension of the space of principal vectors ker((T - )~l)l). Principal vectors associated with different eigenvalues are linearly independent. For integers k E 1N

a ( T k) -- a(T) k, m( Tk' #) -

a(T*) -- a(T), E

re(T, )0,

0 =/=# ~ a(Tk),

#=)k, )~Ea(T)

re(T*, )~) = m(T, ~),

0 :/: )~ E a(T).

H. KSnig

944

See, e.g., [32] or [17]. If T is a Riesz-operator, we denote by ()vn(T))nEl~ the sequence of eigenvalues of T, ordered in the following way: they are non-increasing in absolute value, i.e., ];Vn(T)[/> I)~n+l (T)I and each eigenvalue is repeated as often as its multiplicity indicates. If there are no more than n non-zero eigenvalues in this sense, we let ~n+~ (T) = 0 for k E N. The order could be non-uniquely determined; we choose a fixed order of this form. The following localization result is important for our purposes. PROPOSITION 1 (Localization). Let T E s be power-compact, n E N and )~n(T) # O. Then there is an n-dimensional subspace Xn of X, invariant under T, such that T ix, has precisely )~1(T) . . . . . )~n(T) as its eigenvalues. PROOF. If ( # l . . . . . /,t I =: #) are the different eigenvalues in the sequence (~1 (T) . . . . . )~n(T)) and ]#ll ~>"" ~> ]#l], we have # = )~n(T) and l-1

Z

1

m(T, # j ) < n t}. Then also k

k

fn(t) "-- Z ( f l j

- t ) ll

S

> Y,

(x*(x)/[Ix*11)n er~'

S(On)n~N - ~

(21) D(~n)neN-

(llxn* IIIlYnIlffn)n~r~,

OnYn/llYnll

tl

with Ilell IIDII IISII ~ En~N IlXn*II IlYnII. The infimum of all sums on the right side in (20) is the nuclear norm of T, v(T). Any nuclear operator is compact and 2-summing with yr2(T) ~< v(T).

THEOREM 22. Let T E ./V'(X) be nuclear. Then the eigenvalues o f T are square summable, II0~n(T))ll2 ~< v(T). In general this is best possible. If X is an r-convex and r~-concave Banach lattice, where 1 < r 2. Let e > 0. Assume, T has a representation (20) with Z n E N IIx*ll Ily~ II ~< (1 + e)v(T). Let f i n ' - v/llx* II Ily~ II and define D1 "lee --+ 12 and D2"12 --+ ll both by mapping (~n)nEN ~-+ (Crn~n).

Eigenvaluesof operatorsandapplications

965

Let R, D, S be as in (21). Then D - D2D1 and T -- S D2D1R. Here D1 is 2-summing with 7rz(D1) -- lID111 - IIo ll2. Also D~ -- D1. Thus SD2 E/~(/2, lq) has a dual 2-summing and thus q-summing map,

7rq((SD2)*) 01 ~

IX/7(T)I ~< cvo(T) for all T E ,T'(X) }.

/7

We want to show that c(X) < ~ . If c(Z) < cx~ for some subspace Z __ X of finite codimension, Z "" H1 by (ii) and hence also X ~ H2. Thus if c(X) -- oo, the same is true for any subspace Z of finite codimension, c(Z) - oo. Suppose c(X) - cxz. Choose T1 6 U(X) with Y~n [)~n(T1)[ ~/22v0(T1). Looking at finite representations of T1 almost attaining vo(T1), one finds X1 ___X, dim X1 < oo with

TI(X) ~ X1,

~ I~n(T~)l~ 22v0(T1), /7

where T1 E/~(Xl) is the restriction and astriction of/'1. Choose Z1 ___X, codimZ1 < cx~ such that Ilx + zll ~> 1/2llxll for all x 6 X1, z ~ Z1. Since c(Z1) < oo, there is X2 ~ Z1, dimX2 < oo and T2 6 U(Z1) with

T2(Z1) c X2, Z IZ~(T2)I~ 24v0(T2). /7

In general, choose Zk ___ X, codimZk < oo such that Ilx + zll ~> 1/2llxll for all x E X1 @ . . . | Xk and z E Zk and find Xk+l C_ Zk, dimXk+l < c~ and Tk+l ~ U(Zk) with

I~.(Tk+l)l

Tk+l (Zk) ~ Xk+l,

> 2 2k+2

vo(Tk+l).

/7

By normalization, assume that vo(Tk) -- 1 for all k. Define

kEIN

kEN

kEN

kEN

for Xk E Xk. Then T is nuclear and hence has a nuclear extension T to all of X. But the eigenvalues of T contain those of 2-kTk for all k 6 1N. Thus ~ n ])~n(T)] -- oo, contradicting the assumption (1) in Theorem 23. D

H. K6nig

968

The slightly weaker assumption that nuclear operators X have eigenvalues which are of order [)~n( T ) I - O(n -1) leads to the notion of weak Hilbert spaces, see [10]. PROBLEMS.

(1) Let 0 < p < cx~, 0 < q ~< co. Characterize those Banach spaces X such that all nuclear operators on X have absolutely pth power summable eigenvalues (more generally, that the eigenvalues belong to lp,q). (2) Let 0 0 so that for all operators S: Y ~ X of finite rank

)l/p I)~n( SZ) lp

~ cllSII.

(25)

ncN Let ep(T) "-- inf{c > 0l (25) holds}. Is (C~p,ep) a quasi-Banach ideal? (3) With the same notion as in (2), suppose that A and/3 are operator ideals with A _ Cp and/3 ___Ep. What can be said about A + 13? We end this section with a few comments on traces and determinants for operators in Banach spaces 9 Nuclear operators on Banach spaces X, where X has the approximation property, T E A/'(X), admit a well-defined (matrix) trace given by trm (T) -- ~n~N x*(yn) if T has the form (20) (with X - Y). If X does not have the approximation property, (matrix) traces may not be well-defined in the sense that they depend on the representation (20) of the operator. If some operator ideal A _ A/', like S1, has the property that all operators T E A(X) are Riesz operators with absolutely summable eigenvalues for all Banach spaces X, one may define a (spectral) trace trs "A ~ C by letting trs(T) -- Y~n~r~)~n(T). Trace formulas then claim that trm (T) = trs (T) for all T e ,A(X). For X = H and ,A -- .N', this is Lidskii's formula [20], for A = S1 (with respect to the approximation numbers) see [ 17]. Under the condition on A given above, the trace formula is true in general, as shown recently by White [35]. The main problem is to show that the function Em )~m(')" .A(X) ~ C is linear. In the proof, often determinants of operators of infinite rank are used. If the eigenvalues of T E A(X) are absolutely summable, one introduces

D(~,) "--det(I + ~,T)"-- H (1 + X)~j(T)) jeN

and uses that trs(T) -- D'(O)/D(O). The decay of the eigenvalues ~j(T) is reflected in the growth properties of D and vice-versa. Classically, growth properties of determinants have been used to estimate the asymptotic behaviour of eigenvalues, e.g., by Hille and Tamarkin [ 13]. Even before, determinants were introduced for infinite matrices of the form A - I + T where T - (tij), Y~i supj Itijl < CX),by Hill (1877) and von Koch (1901) as the limit of determinants of finite sections of A. Von Koch also showed that

1

~

~

detA-l+E~Z...Zdet~ nEl~

kl=l

{

tklkl

"""

tklkn )

"

kn---1

tknk 1

...

tknkn

Eigenvalues of operators and applications

969

This approach was transferred by Fredholm (1903) to continuous kernels K defined on the unit square, det(I + T/()

-1+~~

l f0, f01 ...

det

K(tl tl)

""

K(t~,tl)

...

K(tl,tn)) 9

dt| .. 9dtn.

K(tn, tn)

Historical remarks concerning traces and determinants can be found in [32].

4. Applicationto integraloperators We apply Theorems 13 and 18 for Weyl number ideals and p-summing operators to estimate the order of decay of the eigenvalues of certain types of integral operators where the kernel satisfies summability and/or regularity assumptions. There is, of course, a vast literature on the asymptotic distribution of the eigenvalues of integral operators 9 The results usually rely on the estimation of singular numbers in Hilbert spaces 9For a recent study, see Edmunds and Evans [8]. For the distribution of the eigenvalues of random Gaussian matrices see the article of Davidson and Szarek [6]. We present results where the Banach space methods of the previous chapters are essential. We start with the Hille-Tamarkin operators 9 PROPOSITION 24. Let (S2, lZ) be a a-finite measure space, 1 < p < ec and k : S2 x S-2 -+ C be a measurable kernel such that

1/p ]]kllp,p'

9~

]k(s, t)]

p!

d/z(t)

d/z(s)

< (X).

Then Tk f (x) 9 fs? k(x, y) f (y) dlz(y)

(26)

defines a power-compact operator Tk : Lp(S2, lZ) --+ Lp(S-2, lZ) with (Xn(Tk)) E lq, where q = max(p, 2) and II)~n(Tk)llq ~ Ilkllp,p,. PROOF. The fact that Ilk!lp,p, < ~ means for a-finite measure spaces that f :$2 --+ X := Lp,(S-2, lZ) given by f ( s ) ( t ) := k(s, t) belongs to the space of Bochner integrable functions Lp(S-2, lZ; X). Lemma 25 below shows that Tk E 1-Ip(Lp(S-2,/z)). Apply Theorem 18. [] LEMMA 25. Let (s

#) be a measure space, 1 0, the vector-valued Besov-spaces are defined by

Bp,q(n; X)"--

I

f E w p ( n ; X)

i

Iflp,q,,~;x "- sup Ic~l=r

(fo'(~

7

N ( 1 / p + i / u - 1) and ,.(-2 C R u be sufficiently regular. Define t by 1/t "-- O~ + cr)/N + 1 / m a x ( 2 , u'). Let k E B p,q )~ ( n ; Bu~v ( n ) ). Then the eigenvalues of the integral operator Tk defined by (26) belong

to the Lorentz sequence space lt,q with

The constant c depends only on the indices and n . The space lt,q is the best possible, i.e., smallest among the Lorentz sequence spaces.

H. KOnig

972

REMARK. The integral operator T~ is well-defined, e.g., in any space Ls (I2) where s satisfies )~ > N ( 1 / p - 1/s), cr > N ( 1 / s - 1/ut), as follows from the Sobolev-Besov imbedding theorems, or in the spaces Bp,q (I2) or Bu~, (s It is a consequence of the principle of related operators that the eigenvalue sequence is independent of the choice of these spaces 9 It is interesting to note that the space It,q does not depend on the indices p and v. v

9

PROOF (Ideas). The proof relies on the estimate of the Weyl numbers of Tk in suitable function spaces and is fairly technical. We only illustrate the basic techniques in a simple case of indices, namely if u I = p. Any f ~ B p,q (S2", X) defines an operator Tf" X* ~ Bp,q (S-2) by mapping x* ~ X * o f . Here Tk "~ -- Tf " Bu~,v(S2) * --+ Bp,q ~ (S2) . If )~ + cr > N ( 1 / p + 1 / u inclusion map I" Bp,q ( n ) ~

1), as assumed, the

Bu,~ v ( n ) *

exists, 9 actually, I = I~ o 11 where I i ' B p ,~q (f2)--+ Ls(f2) and /2" BuOy(n)--+ Ls,(f2) where s is determined as in the above remark. Hence T~'Bp,q (~2) --+ Bp,q z (f2) factors as T~ -- T~ I. A more complicated analogue of L e m m a 21 derived by interpolation using (27) shows that T~ is actually p-summing if p -- q. In this case, )~ ) ~ Clrt-1/1 rcp(Tl~) "~

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