Tensor Norms and Operator Ideals

TENSOR NORMS AND OPERATOR IDEALS

NORTH-HOLLAND MATHEMATICS STUDIES 176 (Continuation of t h e Notas de Matematica)

Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester N e w York, U.S.A.

NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

TENSOR NORMS AND OPERATOR IDEALS

Andreas DEFANT Fachbereich Mathematik Universitat Oldenburg Oldenburg, Germany

Klaus FLORET Fachbereich Mathematik Universitat Oldenburg Oldenburg, Germany and I ME CC/Un icamp Campinads.F, Brasil

1993 NORTH-HOLLAND -AMSTERDAM

LONDON

NEW YORK

TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211,1000 AE Amsterdam, The Netherlands

ISBN: 0 444 89091 2

0 1993 ELSEVIER SCIENCE PUBLISHERS B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U S A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. 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. This book is printed on acid-free paper. Printed in The Netherlands

Contents

V

Contents

Introduction

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

Chapter I: Basic Concepts . . . . . . . . . . . . . . . . . . . . . .

1

7

1. Bilinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Continuous bilinear mappings, 1.5. non-validity of Hahn-Banach theorem, 1.7. non-validity of open-mapping theorem, 1.8. canonical extension to the bidual.

7

2. The Algebraic Theory of Tensor Products . . . . . . . . . . . . . . 2.2. Universal property and construction of tensor products, 2.4. examples, 2.5. trace, 2.6. trace duality, 2.7. tensor product of operators.

15

3. The Projective Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Minkowski gauge functional, 3.2. basic properties, 3.3. Bochner i n t e grable functions, theorem of Dunford-Pettis, 3.4. compact sets, 3.6. nuclear operators, 3.7. trace, 3.9. A does not respect subspaces, 3.10. extension property, 3.11. Grothendieck’s characterization of L1, 3.12. lifting problems, 3.13. t;-spaces, Ex 3.24. Radon-Nikodfm theorem for operator valued measures.

26

4. The Injective Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Basic properties, 4.2. examples, 4.3. E does not respect quotients, 4.5. lifting of vector-valued continuous functions, compact extension property, 4.6. integral bilinear forms, Ex 4.3. Fourier matrices.

46

5. The Approximation Property . . . . . . . . . . . . . . . . . . . . . . 5.2. Survey about counterexamples, 5.3. compact operators, 5.4. characterization with nuclear operators and the trace, 5.5. injectivity of completions of tensor products of operators, 5.6. operators with nuclear dual, Ex 5.17. compactb approximable operators.

58

6. Duality of the Projective and Injective Norm . . . . . . . . . . . . 6.3. Dense sequences of finite dimensional Banach spaces, Johnson spaces C,,6.4. embedding theorem, 6.5. weak principle of local reflexivity, 6.6. principle of local reflexivity, 6.7. extension lemma for integral bilinear forms, Ex 6.4. Lindenstrauss’ compactness argument.

70

vi

Contents

7. The Natural Norm on the p-Integrable Functions . . . . . . . . . 7.1. Bochner pintegrable functions, A,, 7.2. continuous triangle inequality, 7.3. positive operators, density lemma, 7.4. A, respects subspaces and quotients, quotient lemma, 7.5. Fourier transform, 7.6. Hilbert transform, 7.7. type and cotype, 7.9. a Beckner-like result, Ex 7.1. averaging operator in L,.

77

8. Absolutely and Weakly p-Summable Series and Averaging Tech-

90

9. Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Quasinorms, 9.4. criterion, 9.6. examples, 9.7. injective ideals and the injective hull, 9.8. surjective ideals and the surjective hull, 9.9. dual ideals, 9.10. composition ideals, Ex 9.8. space ideals, Ex 9.13. quasinuclear operators, Ex 9.16. K-convex operators.

108

10. Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Characterization with the trace, 10.4. examples, 10.5. factorization, 10.7. characterization with T @ id^, 10.8. and 10.9. summability of the diagonal of infinite matrices, Ex 10.4. extension and lifting properties of integral operators.

118

11. Absolutely p-Summing Operators . . . . . . . . . . . . . . . . . . . 11.1. Basic characterizations, 11.2. positive operators, 11.3. GrothendieckPietsch domination theorem and factorization, 11.4. Dvoretzky-Rogers theorem, 11.5. composition, 11.6. Hilbert-Schmidt operators, 11.7. Pietsch lemma, 11.8. Kwapieii’s test, 11.9. absolutely 2-summing norm of i d E , 11.10. absolutely p-summing norm of the identity of finite dimensional Hilbert spaces, 11.11. little Grothendieck theorem, 11.12. operators with absolutely 2-summing duals and a characterization of Hilbert spaces, Ex 11.10. extension property of absolutely 2-summing operators, Ex 11.13. the ideal of Hilbert-Schmidt operators, Ex 11.16. Banach-Mazur distances between finite dimensional Banach spaces and the Kadec-Snobar result about projections, Ex 11.18. factorization of Hilbert-Schmidt operators.

127

niques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Absolutely p u m m a b l e and weakly p-summable sequences, 8.2. representations of operators on or into l,, 8.3. unconditional summability, 8.4. general scheme of averaging, 8.5. Rademacher functions, Khintchine inequality, 8.6. type and cotype of L,, 8.7. Gauss functions, 8.9. Orlicz property, Ex 8.9. Rademacher versus Gauss averaging, Ex 8.12. absolutely ( T , s)summing operators.

vii

Contents

Chapter 11: Tensor Norms. . . . . . . . . . . . . . . . . . . . . . . .

146

12. Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Reasonable norms and the metric mapping property, 12.2. criterion, 12.4. finite and cofinite hull, 12.5. Laprestb’s tensor norms 12.6. com12.8. the diagram of Laprestk’s tensor norms, pletion with respect to 12.9. tensor product representation of weakly psummable sequences, Ex 12.7. tensors of finite rank.

146

13. The Five Basic Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. Approximation lemma, 13.2. extension lemma, 13.3. embedding lemma, 13.4. density lemma, 13.5. 2,p-local technique lemma.

159

14. Grothendieck’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . 14.1. Idea of proof, 14.4. proof of Grothendieck’s inequality in tensor form, 14.5. matrix form, 14.7. estimates for the Grothendieck constant K G , Ex 14.1. the original proof - more or less.

166

15. Dual Tensor Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1. Trace duality, 15.2. dual norms, 15.5. duality theorem, 15.6. accessibility of tensor norms, 15.7. conditions for the good behaviour of duality, 15.9. tensor norms and their duals on “symmetric” finite dimensional spaces, 15.10. duality of A,, the Chevet-Persson-Saphar inequalities, 15.11. tensor norms closest to Ap, 15.12. another proof of the Beckner result, Ex 15.10. weakly conditionally compact subsets of tensor products.

177

16. The Bounded Approximation Property . . . . . . . . . . . . . . . 16.1. Topologies on 2 ( E , F), 16.2. characterization with the cofinite hull f of the projective norm, 16.4. results involving the Radon-Nikodjm property, 16.5. and 16.6. duality of 6 and T , 16.7. duality of the operator ideals ff, %, and 2, 16.8. non-nuclear operators with nuclear dual, Ex 16.4. - Ex 16.8. reflexivity of 2 ( E , F) for special spaces.

190

17. The Representation Theorem for Maximal Operator Ideals

...

17.2. Maximal operator ideals, 17.3. tensor norms and operator ideals which are associated with each other, 17.4. right-tensor norms and a general way of constructing maximal operator ideals, 17.5. the representation theorem, 17.6. the embedding theorem, 17.7. the transfer argument, 17.8. the dual ideal, 17.9. the adjoint ideal, 17.10. - 17.13. examples, 17.14. Grothendieck’s theorem, 17.15. and 17.16. characterization with tensor product operators T 8 id^, 17.19. ideal norms of identity operators in symmetric finite dimensional spaces, 17.20. injective embedding of E& F into the space of operators, 17.21. unit ball of % ( E ,F), Ex 17.16. continuity of S 8 2 ” ‘ , Ex 17.17. density lemma for maximal normed operator ideals.

200

viii

Con tents

18. ( p , q)-Factorable Operators . . . . . . . . . . . . . . . . . . . . . . . 223 18.2. The norm of the integrating functional, 18.4. ultraproducts, 18.5. factorization through positive functionals, 18.6. p-factorable operators, 18.7. p-integral operators, 18.9. Maurey’s factorization theorem, 18.11. the factorization theorem, Ex 18.4. - Ex 18.11. properties of ultraproducts. 19. ( p ,q)-Dominated Operators . . . . . . . . . . . . . . . . . . . . . . . 241 19.2. The basic estimates, 19.3. Kwapieli’s factorization theorem, 19.4. composition of factorable and dominated operators, 19.6. Grothendieck inequality for C*-algebras. 20. Projective and Injective Tensor Norms . . . . . . . . . . . . . . . . 20.3. Duality relations, 20.4. examples, 20.6. the projective associate, 20.7. the injective associate, 20.9. finite dimensional characterization, 20.10. general rules for associates, 20.11. - 20.13. relations with operator ideals, 20.14, results about gp, 20.15. table for w ~ , d 2 , g 2and their adjoints, 20.17. Grothendieck’s inequality in its original formulation (operator version), finite dimensional Grothendieck constants, 20.18. Banach spaces satisfying Grothendieck’s theorem, 20.19. a result of Saphar and the best constants in the little Grothendieck theorem.

250

21. Accessible Tensor Norms and Operator Ideals . . . . . . . . . . . 275 21.2. Accessible operator ideals, 21.4. total accessibility of certain composition ideals, 21.5. accessibility of Laprestk’s tensor norms, 21.6. a result about the bounded approximation property, 2 1.7. the a-approximation properties, Reinov’s results in the case Q = gp, 21.11. some results of Kisljakov and Bourgain-Reinov, H w , Ex 21.3. principle of local reflexivity for operator ideals. 22. Minimal Operator Ideals . . . . . . . . . . . . . . . . . . . . . . . . 22.1. The minimal kernel of an operator ideal, 22.2. the representation theorem for minimal operator ideals, 22.3. examples, 22.6. the dual of amin(E, F), 22.7. weak-*-continuous operators in Qmin(F‘, E), 22.8. the dual ideal of the minimal kernel, 22.9. a counterexample, Ex 22.7. extension and lifting properties of minimal operator ideals.

287

23. l!;-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.1. Local techniques, 23.2. various characterizations, 23.3. relations with the Cp-spaces of Lindenstrauss-Peiczyriski, 23.4. dual characterization, 23.5. 25-and Ci-spaces, 23.6. the projection constant, 23.7. quasinuclear operators, 23.8. characterization of space ideals with integral operators, 23.9. coincidence of absolutely l-summing and nuclear operators, Hardy spaces, 23.10. Grothendieck’s theorem for l!;-spaces, a characterization of Hilbert spaces, Ex 23.8. the extension norm of an operator.

300

ix

Contents

24. Stable Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 24.1. The linear dimension of e, and L,, 24.2. positive definite functions and Bochner’s theorem, 24.3. moments of stable measures, 24.4. LCvy’s theorem about the embedding of e, into L, (LCvy embeddings), 24.5. embeddings into L,, 24.6. - 24.7. results due to Saphar, Kwapied and Maurey about absolutely q-summing operators with values in t, or spaces with cotype, 24.8. stable type and Rademacher type, Ex 24.1. Schur product, Ex 24.5. stable type. 25. Composition of Accessible Operator Ideals . . . . . . . . . . . . . 327 25.1. Representation of the minimal kernel of accessible operator ideals, 25.4. cyclic composition theorem, 25.5. Persson-Pietsch multiplication table, 25.6. quotient ideals, 25.7. quotient formula, 25.8. the adjoint of composition ideals, 25.9. the regular hull and characterizations of the associates of ‘cp,q and 2,,in particular, of the associated space ideals - results of Kwapieri, 25.10. isomorphic characterization of subspaces, quotients, etc. of L,, 25.11. the minimal kernel of the injective resp. surjective hull of an operator ideal. 26. More About L, and Hilbert Spaces . . . . . . . . . . . . . . . . . . 26.1. Inequalities about a,,( coming from the Khintchine and Grothendieck inequalities, 26.2. factorization through Hilbert spaces of the identity mapping + 26.3. continuity of operators between spaces of Bochner pintegrable functions, complexification of operators, 26.5. Kwapied’s result about the factorization of operators L, + L, through L p , 26.6. tensor norms and ideals on Hilbert spaces, 26.7. the Hilbert-Schmidt tensor norm u,26.8. Schatten’s result about self-adjoint, symmetric extensions of u to Banach spaces, 26.10. limit orders of tensor norms, Puhl’s result, 26.11. unEx 26.6. unconditionally summable sequences conditional bases in e&,&, in L p ( p 8 v).

344

27. Grothendieck’s Fourteen Natural Norms . . . . . . . 27.2. Grothendieck’s diagram, 27.3. the original notations.

361

C,

........

Chapter 111: Special Topics . . . . . . . . . . . . . . . . . . . . . . .

365

28. More Tensor Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 28.1. Three new classes of tensor norms, 28.3. description of the projective associate of 28.4. characterization of operators in ‘ ; 2 28.5. a characterization of operators factoring through a Hilbert space, 28.6. - 28.8. description of the composition ideals 2, o Zq and its adjoints, 28.9. table of results, Ex 28.14. complexification of operators.

X

Contents

29. The Calculus of Traced Tensor Norms . . . . . . . . . . . . . . . . 378 29.1. The tensor contraction, 29.4. the associated operator ideal of a traced tensor norm, 29.5. characterization of p-dominated operators, Tp-spaces, 29.6. the calculus of traced tensor norms, 29.7. the tensor product of two tensor norms and of maximal normed operator ideals, 29.8. properties of 4 %, 29.9. - 29.11. tensor products of special tensor norms, Ex 29.8. ultrastable ideals. 30. The Vector Valued Fourier Transform . . . . . . . . . . . . . . . . 30.1. Fourier operators, 30.3. their characterization, 30.5. Rademacher and Gauss type and cotype, Kwapieh’s type/cotype theorem, characterization of Hilbert spaces, 30.6. the main theorem, 30.8. type and cotype with respect to orthonormal bases.

394

31. Pisier’s Factorization Theorem . . . . . . . . . . . . . . . . . . . . . 407 31.1. K-convex operators, 31.2. duality of type and cotype, 31.4. Pisier’s factorization theorem, 31.5. factorization of compactly approximable operators through Hilbert spaces, 31.6. non-accessible tensor norms/operator ideals, 31.7. “abstract” proof of Grothendieck’s inequality.

32. Mixing Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 32.2. Reformulation of former results, 32.3. tensor product characterization, continuity of tensor product operators between spaces of Bochner p-integrable functions, 32.4. a domination theorem, 32.5. Maurey-Pisier extrapolation theorem, 32.6. a characterization of 32.7. Maurey’s splitting theorem, 32.10. and 32.11. relation with absolutely ( r ,s)-summing operators, 32.12. a finite dimensional result.

ME’,

33. The Radon-Nikodfm Property for Tensor Norms and Reflexivity 33.1. Duality of E and ?r revisited, 33.3. Lewis’ theorem, 33.4. permanence properties, 33.5. LaprestC’s tensor norms, 33.6. coincidence of pnuclear and p-integral operators, 33.7. p-strong operators, 33.8. - 33.11. reflexivity of tensor products and components of operator ideals.

430

34. Tensorstable Operator Ideals . . . . . . . . . . . . . . . . . . . . . . 445 34.1. P-tensorstable and metrically P-tensorstable operator ideals, 34.2. strongly P-tensorstable operator ideais, 34.3. examples, 34.4. permanence properties, 34.5. factorization arguments, 34.6. projection constant of the injective tensor product, 34.7.stability of space ideals, 34.8. double Khintchine inequality, stability of tensor products of Hilbert spaces, 34.9. 6- and s-stability of (p, q)-factorable operators and their relatives, 34.10. distribution of eigenvalues, Pietsch’s tensor product trick, 34.11. a result of Kwapieli unifying Orlicz’s, Littlewood’s and Grothendieck’s inequalities, 34.12. improving weak inequalities with tensor products, mixing operators tl -+ tp.

Contents

xi

35. Tensor Norm Techniques for Locally Convex Spaces . . . . . . . 469 35.2. Tensor norm topologies, 35.3. traced tensor norms, 35.4. locally convex space ideals, 35.6. injective and projective tensor norms on locally convex spaces, 35.7. tensor product of direct sums, 35.8. lifting of bounded sets, property (BB), 35.9. probkme des topologies, Taskinen’s counterexample, 35.10. injective tensor product of (DF)-spaces.

Appendices: A. Some Structural Properties of Banach Spaces . . . . . . . . . . . A l . Subspaces and quotients, A2. dual systems, A3. lemma of Ky Fan, A4. bases, A5. Banach algebras, A6. lattices, A7. abstract L,-spaces and their representation.

489

B. Integration Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493

Extension procedure and the basic theorems: B1. Daniel1 functionals, B2. the convergence theorems, B3. measurable functions, the fundamental theorem of the Daniell-Stone integration theory, B4. product measures; The Lp-spaces: B5. Holder and continuous triangle inequality, B6. duality, Radon-Nikodim theorem, Segal’s localization theorem, Lebesgue decomposition, B7. strictly localizable measures; Borel-Radon measures and Riesz representation theorem: B8. 7-continuity and Bourbaki’s extension procedure, Kolzow ’s theorem, B9. representation of Borel-Radon measures, B10. L1 is complemented in its bidual; Bochner integration: B11. measurability of Banach space valued functions, B12. .C,(p, E), B14. Pettis integrability, B15. variation lemma. C. Representable Operators . . . . . . . . . . . . . . . . . . . . . . . . 508 Grothendieck’s characterization: C1. Riesz-densities, C3. nuclear operators, C4. factorization through e,; The Dunford-Pettis theorem: C6. a general result about representability, C7. strong version of the DunfordPettis theorem. D. The Radon-NikodQm Property . . . . . . . . . . . . . . . . . . . . 517 Basic properties and examples: D1. reduction to the Lebesgue measure, D3. examples, D4. dual of Lp(p, E); Pietsch integral operators: D5. and D6. relation with other operator ideals; The Radon-Nikod9m property and operator ideals: D7. and D8. characterizations in terms of (Pietsch) integraknuclear, D9. integral operators which are not Pietsch integral. Bibliography

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527

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545

List of Symbols Index

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555

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Introduction

1

Introduction

1. Grothendieck’s “Rhum6 de la thkorie mktrique des produits tensoriels topologiques”, submitted in 1954 and published in 1956 in the Bulletin of the Mathematical Society of SZo Paulo, is one of very few papers which have deeply influenced the course of Functional Analysis: It not only demonstrated the enormous possibilities for the using of tensor products in Banach space theory, but it anticipated the study of Banach spaces in terms of finite dimensional subspaces - the secalled “local” theory, which has 80 much enriched our understanding of Banach spaces. The “R&umk” (as this paper is nowadays called) contained no proofs (with the exception of the main theorem) and apparently few people at the time understood it or even noted its importance. This, in spite of the fact that Grothendieck’s theory of locally convex tensor products and nuclear spaces (also poorly understood when it first appeared) had already won great appreciation. There was simply some reluctance in the Functional Analysis community to accepting the idea of thinking in terms of tensor products. It was only in 1968 that the “R6sumk” received more attention through the famous Studia Mathematica paper of Lindenstrauss and Pelczyliski [173] “Absolutely summing operators in &,-spaces and applications” , which presented Grothendieck’s deep “thkor8me fondamental de la thbrie mktrique des produits tensoriels” (the main result of the “ f i u m k ” ) as an inequality concerning n x n matrices and Hilbert spaces. Without using tensor products, various fascinating applications were given, mainly dealing with the class of absolutely psumming operators. Banach space theory (which in the mid-sixties had been considered to be nearly complete by some people) was reactivated in an incredible way - and many of its important results nowadays trace their roots back to the “Risum6”. It is astonishing to see that many (certainly not all) of the ideas of Banach space theory in the last 20 years are already contained in Grothendieck’s paper, although sometimes in a hidden way. The phrase “this result is implicitly contained in the Risumk” became almost a clichC for a certain time, but was nevertheless quite often true. 2. Tensor products apparently appeared in Functional Analysis for the first time during the late thirties in the work of Murray and John von Neumann on Hilbert spaces. The first systematic study of classes of norms on tensor products of Banach spaces is due to Schatten who in 1943 continued his work in a series of papers (some together with von Neumann). Schatten’s influential monograph “A Theory of CrossSpaces” contains what was known in 1950: the most beautiful applications of the theory dealt with operator ideals on Hilbert spaces [249], the Hilbert-Schmidt operators, the traceclass or more generally the Schatten-von Neumann-classes 6,. Many of the more elementary aspects of Grothendieck’s theory were known to Schatten, but he was

2

Introduction

not aware of the important role of the finite dimensional behaviour of tensor norms. Therefore, he did not succeed in developing - for general Banach spaces - a useful theory of duality; the reader will get a more precise idea of why this happened in section 15. On the other hand, the idea of ideals of operators was always apparent in the study of tensor products - in Schatten’s as well as in Grothendieck’s work. When writing the “R&umB”, Grothendieck was not aware of Schatten’s work. In 1968 Pietsch and his school began a systematic investigation of operator ideals on the class of Banach spaces and, ignoring tensor products, developed a method of thinking in a “categorical” manner which is just as powerful as thinking in terms of tensor products - but has the advantage that it is certainly much easier to learn the basics of operator ideal theory than the basics of the theory of tensor products. This development culminated in 1978 in the publication of Pietsch’s book “Operator Ideals” which contains in a nearly encyclopaedic way everything that was known about operator ideals at the time. Although many of the ideas and results clearly came from the “R&um6”, tensor products were never used in his book.

3. There was a bias against tensor products in Banach space theory, but the projective tensor norm K and the injective norm E proved their usefulness rather quickly through simple examples involving vector valued function spaces. New results about general tensor norms were only obtained around 1970; may be the most important of these “early” papers are those of Saphar and Kwapien’s Bordeaux-paper on operators factoring through L,-spaces. This latter paper, however, was written in the language of operator ideals. Later on (in 1973) Lotz based his investigation of maximal normed operator ideals (he had called them Grothendieck ideals) on tensor products. The rich paper of Gordon-Lewis-Retherford on operator ideals was stimulated by Schatten’s and Grothendieck’s work and uses tensor product methods explicitly. It was Pisier’s work in Banach space theory (starting around 1975) which finally drew new attention to the tensor product approach. A highlight is certainly his solution of the last problem stated in the “Rbumk”: There exist infinite dimensional Banach spaces P such that P @e P = P @ x P holds isomorphically. Pisier’s book “Factorization of Linear Operators and Geometry of Banach Spaces” is centered around the problem of determining conditions under which an operator between Banach spaces factors through a Hilbert space. These investigations finally led to solutions of all six of the problems stated at the end of the “RCsumk” except that the exact constant in the Grothendieck inequality (as the “thCorkme fondamental” is nowadays called) is not yet known (the approximation problem was solved in the negative by Enflo in 1972). Reading Pisier’s book, it becomes strikingly clear that thinking in terms of operator ideals and in terms of tensor products is quite useful. Further support for this way of dual thinking is given by a trick due to Pietsch from 1983 where he uses tensor products of operators in order to give a simple proof of the famous result concerning the distribution of eigenvalues of absolutely psumming operators due to Johnson, Konig, Maurey, and Retherford (see 34.10. for some details).

3

Introduction

4. Unfortunately, the beauty and power of “tensorial” thinking, becomes clear only after one is really accustomed to using it. The “Rksumk” is very difficult to read and for this reason various attempts have been made to present the theory of tensor norms in a clearer fashion (Amemiya-Shiga [3], Lotz [180],Losert-Michor [179], Michor [191], Gilbert-Leih [81] are known to us), but there seems to exist none which is easily accessible and, at the same time, incorporates the wonderful theory of operator ideals as it exists nowadays. We have already expressed our point of view in a sort of prebook in [43]. Our central aim with the present book is to convince the reader that the theory of tensor norms is much less difficult than it sometimes seems and that it gives one a deep and beautiful insight into various questions in Banach space theory - in particular into the theory of operator ideals, which is our main concern apart from the forementioned objective. We hope that we can convince the reader (and the historical development gives clear evidence for this) that both theories, the theory of tensor norms and of normed operator ideals (if we consider them for a moment to be really different), are more easily understood and also richer if one works with both simultaneously. It should become obvious that certain phenomena have their natural framework in tensor products and others in operator ideals. 5. We assume that the reader of this book knows the basics of Banach space theory as they are usually presented in an introductory course on Functional Analysis. Some additional information is collected in the four appendices - in particular, our point of view for the theory of measure and integration. Appendices C and D about the Radon-Nikodim property will not be needed before section 16. The book is divided into three chapters: Basic Concepts, Tensor Norms, and Special Topics. The first chapter may serve as part of an introductory course in Functional Analysis since it already shows the powerful use of the projective and injective tensor norms T and E , as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the “Rhumk” and deals with the relation between tensor norms and operator ideals. This relation is dominated by the following definition (17.3.) which constitutes a one-to-one correspondence between finitely generated tensor norms a and maximal normed operator ideals (%, A) : They are called associated (in symbols a (a,A)) if %(MIN ) = M‘&N

-

holds isometrically for all finite dimensional Banach spaces M and N.We believe that the following theorems are fundamental for the understanding of the interplay between operator ideals !2l and their associated tensor norms a : The representation theorem for maximal operator ideals (17.5.)

!2l(E,F’) A ( E g a 8 F)’

( is o m e t y )

and the representation theorem for minimal operator ideals (22.2.)

~ ‘F 2, 6 a m~i n (El F )

( m e t r i c surjection),

Introduction

4

where E and F are arbitrary Banach spaces and Qmin is the minimal kernel of Q. In view of applications it is natural to first study tensor norms on finite dimensional normed spaces and later extend them to arbitrary normed or Banach spaces. There are two ways to do this - an inductive procedure + a ( z ;E, F) := inf{a(z;

M ,N) I z E M 8 N; M ,N finite dim.}

and a projective procedure %(z; E , F ) := sup{a(Qf 8 Q E ( z ) ;E / L , F / K ) I E / L , F / K finite dim.}

.

Both coincide if (and in some sense only if, see 16.2.) both spaces have the metric approximation property. Grothendieck chose the first approach and this is justified when one looks at examples. However, we have found it very useful for our investigations to also have the “cofinite hull” % at hand and we hope that we can convince the reader that it is helpful for structuring one’s thoughts and is often quite useful for finding and working out proper statements and proofs. For operator ideals the cofinite hull gains importance through the embedding theorem, which states that

E I ~ + FL Q ( E ,F ) a

(metric injection)

if a and Q are associated (see 17.6.). As has already been mentioned, these three theorems describe the relationship between tensor norms and operator ideals. Chapter I11 deals with special questions; for this, as well as a detailed account of the contents of the other two chapters, the reader may consult the table of contents. 6. Each section is accompanied by a series of exercises and the reader is strongly advised to solve them or at least give them a close look. They are of varying degrees of difficulty: if the result is not immediate from what was said in the same section, then there will be hints for the solution, which - we hope - make the solution accessible. Clearly, after a while (more or less after section 23) the reader should be continually mindful of the three fundamental theorems just mentioned as well as the local techniques. We do not claim that a result presented as an exercise is easy (“just an exercise”) - we only say that at this stage of the development of the theory this result can be obtained (together with the hint) without too much difficulty.

7. The book treats only tensor products of two “variables” - and not norms on of more than two Banach spaces. Though there is certainly tensor products El@-. -@En a good part of the theory which easily extends to the case of several variables, there exist new interesting phenomena which we do not study. Moreover, we deal mostly with normed, not quasinormed operator ideals (with a few exceptions: for example the cyclic composition theorem in section 25): this is a consequence of the fact that tensor

Introduction

5

norms and their duality theory cannot be used to study general quasinormed ideals. One may consider this to be a disadvantage, but one should not forget that most of the interesting operator ideals are actually normed and that the theory of operator ideals (as it is presented in Pietsch’s book for example) also deals mainly with normed (maximal or minimal) ideals. The following list contains some topics which we did not include in the book - partly because of lack of space (we never thought that the book would grow so large), partly because we did not feel competent: - Pisier’s example of a space P with P aCP = P P - Geometry of tensor products, stability properties - Tensor products of Banach lattices - Tensor products of C’-algebras - Applications to Harmonic Analysis. 8. A short word about citations: We are very concerned that we state the origin of those results and methods which we consider to be important for the ideas which are presented. Nevertheless, we find it boring to constantly repeat “due to Grothendieck” during most of the first two chapters. Primarily in the first chapter we label many results as now being folklore - a judgement which may be rather subjective; we apologize for any omissions which may occur in this way - although we hope that there are none.

During the preparation of this book we received great support from many colleagues and friends - and we learned very much from them: In particular, for our understanding of the basic ideas of tensor norms and the “Rksumk” we owe much to the work of Harksen, Lotz and Saphar. Our knowledge about operator ideals stems for the most part from Pietsch and his school - his book on operator ideals serves as a permanent reference for us. We profited much from the study of the papers of Gilbert-Leih, Gordon-Lewis-Retherford, Kwapieli , Lewis, Lindenstrauss-Pelczyliski, Maurey and Pelczyfiski. And our vision of the topic was widened when we read in Pisier’s work. We are also grateful to those colleagues who helped us in solving special problems during the preparation of the book - in particular, R. Alencar, B. Carl, M. Defant, P. Harmand, H. Jarchow, A. Pelczyliski ,N. Tomczak-Jaegermann and J . Voigt; we are happy that G. Pisier provided us (shortly before finishing the book) with an example of a non-accessible tensor norm. We thank the universities of Ann Arbor, Campinas, Lecce, Niteroi, Oldenburg, Pretoria, Rio de Janeiro and Zurich for having given us the opportunity to present a series of lectures or seminars on our ideas; we acknowledge the critical assistance of colleagues and students at these universities. Various students from the University of Oldenburg wrote diploma theses related to our work: those of L. Behrens (section 27) and M. Meile (section 33) directly influenced our presentation.

6

Introduction

We gratefully acknowledge the support of the Land Niedersachsen, the Volkswagen foundation, the Gesellschaft f i r Mathematik und Datenverarbeitung GMD, the Brazilian CNPq and the DAAD during the preparation of this book. There were many colleagues and friends who spent a great deal of time and energy helping us in the difficult task of proof-reading: Klaus Bierstedt, Bernd Carl, Jeslis Castillo, Martin Defant, Ute Defant, Peter Harmand, Hans Jarchow, Mario Matos, Jorge Mujica, Peter Stollmann, J o k Prolla and Jurgen Voigt. Bruce Hanson had a close look at our English. We thank all of them wholeheartedly - not only for the errors which they found, but also for their many observations which helped improve the readability of the text; in this respect we are particularly grateful to Hans Jarchow. Nowadays when the publisher’s task is often reduced to copying and selling (we do not say that the latter were easy) the secretarial work of producing a manuscript has gained special importance. We feel that Mrs. 1. Matziwitzki from the University of Oldenburg has done an excellent job - with dedication and a good sense of aesthetics. We are deeply grateful for this - it is always a pleasure to work with her. Our feelings of gratitude go to the first author’s wife Ute for her understanding and patience. It is a particular pleasure for us to have this book published in the series Notas de Matembtica edited by our friend and colleague Leopoldo Nachbin; according to the introduction of the %sum6 Grothendieck and he intended at the time to write a monograph about locally convex spaces including many aspects of the metric theory of tensor products. Unfortunately, this never happened. We thank Leopoldo Nachbin for always having encouraged us to write our book and the North Holland/Elsevier Science Publishers for accepting it for publication.

Campinas and Oldenburg, September 1991 Andreas Defant Klaus Floret

1. Bilinear Mapping8

7

Chapter I Basic Concepts After some remarks on bilinear mappings and the introduction of algebraic tensor products, the projective and injective tensor norms and some of their most important properties are studied in sections 3 - 6. Vector valued pintegrable functions, summability of sequences and averaging methods are treated in section 7 and section 8 - always from the perspective of tensor products. The remaining sections are devoted to the basics of the theory of operator ideals and integral and absolutely psumming operators between Banach spaces, culminating in the so-called little Grothendieck theorem. The reader should be familiar with some fundamental tools from Banach space theory, such as the Hahn-Banach theorem, the Mackey theorem / uniform boundedness principle, the weak- and weak-*-topologies, continuous linear operators and the open mapping and closed graph theorems. This clearly includes some simple knowledge about the classical Banach spaces C ( K ) and L p , as well as the sequence spaces co and l,. In Appendix A some additional information about the structure theory of Banach spaces is collected.

1. Bilinear Mappings This first section treats bilinear mappings and puts special emphasis on the fact that in many respects they behave differently than linear mappings, although they are intimately related with them.

1.1. Let E , F, G be vector spaces over the same scalar field K = R or (Cof the real or complex numbers. A mapping i9: E x F - G

is called bilinear if the mappings

8

1 . Bilinear Mappings

are linear for each z E E and y E F , in symbols : 9 E Bil(E, F; G) if E L(F,G)

and

eYE L(E,G)

for each 2 E E and y E F. For simplicity: Bd(E, F) := Bil(E, F ; K ) . If E , F,G are normed spaces (or more generally: topological vector spaces), the set of continuous bilinearmappings E x F -+ G will be denoted by %il(E, F;G) and %i[(E, F) if G = K. From 9(z, y) - 9(zoryo) = O(z - z o , y

- Yo) + q z -

2 0 ,

yo)

+

@(so,

Y - Yo)

the following result is easily deduced:

PROPOSITION: For 9 E Bil(E, F; G) the following assertions are equivaleni: (a) 9 is continuous. (b) 9 is continuous ut (0,O). ( c ) There is a constant c 2 0 with ll@(z,y)llG 5 cllzllEllyllF for all (z,y) E E x F

.

It is easy to see that

11@11 := min{c 2 0 I c as in (c) } = sup{((@(z,y)ll, I z E B,,y E B,} defines a norm on %i[(E, F; G) which is even a complete norm if 11 ' IIG is. Note that continuous bilinear mappings are not uniformly continuous since, e.g., the restriction R , (2, y) 4zy to the diagonal is the function R 3 z u* z2 E R. of Ra

-

1.2. A bilinear mapping 9 E Bil(E, F;G) is separately continuous if all and 9, : E .--+ C are continuous.

:F

-, G

THEOREM: Let E, F, G be normed spaces and E complete. Every separately continuous bilinear mapping 9 E Bil(E, F;G) is continuous.

PROOF: The set D := { Z ' O @ ~J Z ' E B G I E, ~BF} c E' is n(E', E)-bounded since for each z E E

Mackey's theorem / the uniform boundedness principle shows that D is uniformly bounded, i.e. there is a c 2 0 such that for all t' E BGIand y E BF

9

1. Bilinear Mappings

It follows that llOll 5 c . 0

1.3. Some examples of bilinear mappings: (1) For t’ E E’ and g‘ E F’

L Z’@Y’l(Z,

Y) := (t’,4 ( d ,Y)

defines a continuous bilinear form and unit balls and (A,) E e l , then

llz’&dll= 11z‘11 115/11. If t k

c

and y’, are in the

8

cp(Z,

Y) :=

An[&aY;l(zI

Y)

n=l

is well-defined and IlppIl 5 C,”=llAnl; bilinear forms of this kind are called nuclear bilinear forms. (2) The evaluation map on the space C(E,F) of continuous linear operators

&(E,F)X E (T,z)

--

F Tx

has norm 1 (if E and F are not trivial). (3) If E and F are finite dimensional, then all bilinear mappings E x F continuous (use bases). (4) The convolution mapping

Ll(R) x Ll(R)

-

--*

G are

Ll(R)

(!,! -I f) *s

is bilinear. ( 5 ) Take the continuous functions on a compact space K and E a normed space. Then

C ( K )x E (!I4

--

C ( K ,E ) f(.)Z

is bilinear. 1.4. The mappings

give isomorphisms of vector spaces and are inverse to each other. Since

10

1. Bilinear Mappings

this isomorphism reduces for continuous bilinear forms to an isometry of normed spaces

% s i [ ( E , ~ A) c(E,F')

1 1 ~ ~ =I lIIPII

and

llhll = IlTll.

This relationship is basic for the understanding of the ideas which will be presented in this book: the continuous bilinear forms on E x F are exactly the continuous operators E 3 F'. 1.5. Since there is no Hahn-Banach theorem for operators, there is none for bilinear continuous forms in the following sense: Let G c E be a subspace and cp E %iI(G, F ) ; does there exist an extension $ E %3i((E,F ) of cp ? This would mean, by the identification of bilinear forms and operators, that every T E C(G,F') would have an extension F E C(E,F'):

E

To see some examples of operators which are not extendable take the special case in which G = F' and T = idG the identity-mapping: the extension ? would be a projection of E onto G.

(1) Due to a famous result of Lindenstrauss-Tzafriri [175] every infinite dimensional Banach space which is not isomorphic to a Hilbert space has a non-complemented closed subspace. (2) To see a more concrete example take the Rademacher functions defined on [ 0,1] rn(t) :=(-I)~ if

(they form an orthonormal system in L2[0,1] injection f2

-

tE

;[-,

,,l[

, Lebesgue measure) and

consider the

L1[0,1]

n=l

The Khintchine inequality (8.5. and 8.6.) will show that L1 induces an equivalent norm on &. But f 2 cannot be complemented in L1 : a projection P : L1 -,4 Q L1 would be weakly compact so P = P2 would be compact by the Dunford-Pettis property of L1 (see Appendix C7.), which is not possible since e2 is infinite dimensional. (Another argument for the non-complementation follows from the fact - which will be proved in section 11 - that every operator L1 -,f 2 is absolutely 2-summing.)

11

1 . Bilinear Mappings

1.6. Extension to the completions, fortunately, is no problem. Recall that there is no uniform continuity!

PROPOSITION: Let E , F, G be normed spaces and G complete. Every d E B i I ( E , F ; G ) bas a unique edension 6 E %i[(,@, P;G ) . Moreover, Il0ll = 11611. This follows easily from the isometric relation

B i [ ( E ,F ; G ) = 2 ( E , C ( F ,G ) ) and the extension of linear continuous operators.

1.7. There is also no open mapping theorem for bilinear continuous surjective mappings.

REMARK: If E is a normed space of dimension at least 2, then the scalar multiplication K x E + E is bilinear, continuous, surjective but not open. PROOF: Take an open non-void set V C E and a functional 3/ E E’ with inf l(y‘, V ) (> 0 . If U is the open unit ball in K, then 0 E U . V, but 0 is not an interior point of U . V since U . V n ker y’ = (0). 0 It is also possible to give examples d E B i I ( E , F ; G ) which are surjective and not open in zero, i.e. zero is not an interior point of +(BE,BF):In 1973 Cohen [33] constructed a relatively complicated bilinear mapping 41 x --* 41 of this kind, but two years later Horowitz [118] found the following simple example @ : K3 x K3 + K4

(see also Ex 1.8.). There is, however, a closed graph theorem (Ex 1.11.). 1.8. Another negative property of continuous bilinear mappings is that they do not remain continuous for the weak topologies: the unit vectors en in 4, (real) converge weakly to zero but (enlen)cl = 1 does not (see also Ex 1.2.).

1.9. For normed spaces E and F the following isometries hold: 0 : %iI(E, F ) A 2 ( E ,F‘)

T

-

c ‘ .2 ( F “ , E’) A B i I ( F ” , E ) T’

%it(E,F ” )

,

12

1. Bilinear Mappings

where the last equality is the obvious "transposition" V'(z,y) := V(y,z). For each E %i[(E,F ) define 'p" := @(p)E Bil(E, F"); it satisfies II'p'II = II'pII = llL,ll and

'p

~"(z, Y") = (Lb(Y"),Z ) E ~ , E= (Y", L , ( Z ) ) F ~ ~=, F(Y", ~ ~ ( z. ), ) F ~ ~ , F I for all (2, y") E E x F". Since, by definition ~ ( zy), = ( y ,L,(z))F,F, for all z E E and y E F , the map 'p" extends 'p from E x F to E x F" with equal norm. 'p" is called the canonical (right-) extension of 'p.

-

PROPOSITION: Let E and F be normed spaces and 'p E %i[(E,F ) . Then 'p" is the K unique bilinear separately u(E,E')-u(F", F')-continuous mapping 1c, : E x F" which extends 'p. PROOF: That 'p" is such an extension is clear from the equation following the definition; uniqueness follows from the u(F", F')-density of F in F". 0 Clearly, there is also a canonical left-extension it follows that "'p(Z'Ii

y)

How are the functionals the following is true:

= (zll, (Lb ("'p)"

and

"'p

on E"x F defined by "'p := (('pz)")z;

= (2'1,

IE~)Y),qif,,qi

"('p")

V ( * lY ) ) E i i , E f *

on E" x F" related? Surprisingly enough,

COROLLARY: For 'p E %i[(E,F ) the following three statements are equivalent : (a) The two "canonical" extensions ("'p)" and "(9") of cp to El' x F" coincide. (b) There is a E Bil(E", F") which is separately c(E", E')-u(F'', F')-continuous and extends 'p. (c) L, : E -+ F' is weakly compact. I n this case the II, in (b) is equal to ("'p)" = "('p").

+

PROOF: The proposition implies easily that (a) is equivalent to (b). To see the equivalence of (a) and (c) first note that, by Ex 1.12. and Ex 1.14.

Now recall that L, is weakly compact if and only if L:(E") c F'

.0

1. Bilinear Mappings

13

Exercises:

Ex 1.1. If E and F are finite dimensional, then every p E Bil(E,F ) has the form for some xk E E' and

4 E F' .

k=l

Ex 1.2. (a) If E and F are normed spaces, then every (o E !&[(ElF ) which is continuous for the weak topologies has the form as in Ex 1.1. with continuous x i E E' and 4 E F' . (b) If E and F are vector spaces, then every u(E' , E)-a(F* ,F)-continuous bilinear form (o on E' x F' has the form n (p

=

for some X k E E and

Xk&k

yk

EF

.

k=l

Ex 1.3. Let E be a Banach space and F be normed. If S C %iI(E,F) is a set of separately continuous forms such that {=pI p E S} C F' is equicontinuous for each x E E, then sup{ll(oll I (o E D} is finite. EX 1.4. Using the space cf of "finite" sequences ( a ) (i.e. en E K and
E-G'.

4

T 1

Ex 1.6. Let G c E be a subspace and I : G L* E the isometric injection. Every (o E %iI(G,GI)is extendable to a E Qi[(E,G') if and only if there is a U E C ( E ,G") with U o I = IEG . Ex 1.7. A Banach space F is called injective if for every Banach space E l every subspace G C E and every T E C(G,F) there is an extension E & ( E , F )of T . If there is a constant A 1 1 such that 11fl1 5 AllTll can be achieved, then F is said to have the A-extension properly; F has the metric extension properly if A = 1. (a) tm(I')has the metric extension property for every set l'. (b) F is injective if and only if F is complemented in some &,(I?). If P : &(I?) -+ F 1 is a projection, then F has the IIPII-extension property. Hint: F ~ * t , ( B p ) .

14

I . Bilinear Mappings

Ex 1.8. If @ : K 3 x K 3 + K4 is Horowitz' surjective bilinear mapping which is not open in zero (see 1.7.) and E and F are normed, then

6 :( K 3x K

-

x E ) x ( K 3x F ) ( X I r zzr 23;Y1,Y2)

K4 x F

,I.(@

Yl), 22Y2)

is also continuous, surjective, and not open in zero. In particular, there are such mappings t, x 1, -+ t, for each p E [l1m]. Ex 1.9. Let 3 ( X ) be the space of all scalar-valued functions f : X + K with finite support [f # 01 := {z E X I f(z) # 0). Then

[fa # 01 = [f # 01 x [g # 01 in order to construct a bilinear mapping R2x R2+ R4 the range of which is not a linear subspace; other examples of this

is bilinear, Use

type are easily deduced from 1.3.(5) . Ex 1.10. Under what circumstances is a bilinear mapping injective? Ex 1.11. Let E, F and G be Banach spaces and a, 3 E B d ( E ,F;G). (a) Closed graph theorem: If 0 has a closed graph, then 0 is continuous. (b) Banach-Steinhaus theorem: If 3,(z, y) 4 3(z, y) for all (z,y) E E x F and all 3, are continuous, then 0 is continuous. Hint: In both cases show the separate continuity of @ . Ex 1.12. Show that for the canonical embedding K E : E c* E" of a Banach space and the canonical projection PE' : El" ---+ E' the following relations hold: (a) KL = P E ~ .

(b) P p 0 K$ = idEli. (c) P& = ICE" if and only if E is reflexive. (d) 6% = K E ~ Jif and only if E is reflexive. Ex 1.13. If 9 E %i[(E,E') is defined by ~ ( zz',) := (z',z ) then ~ L, = K E Ex 1.14. Let p E %i[(E,F).Show that L(,A) =

KFI

o

L, : E

-

L(EL,+,) = Pp o L :~ E l 1 -

.

F"' F' .

Hint: Use proposition 1.9. for ' 9 . Ex 1.15. Let @ E %iI(E1,E2; F) and let Ki C E; be compact convex sets; then for the set of extreme points the following holds: ext (conv(a(K1, K z ) ) )

c

QP(ext Kl,ext K z ) .

2. The Algebraic Theory of Tensor Products

15

Hint: Use the fact that the extreme points of conv(L) with L compact are in L and the concept of an extremal set D of a convex set C (i.e. art (1 - a ) y E D with a ~ ] 0 , 1 [ and 2, E C implies 2, E D ;see e.g. f701).

+

2. The Algebraic Theory of Tensor Products With the aid of a certain universal object the study of bilinear mappings can be reduced to the study of linear mappings. The construction of this new vector space E @ F is, at least for an analyst, very simple.

2.1. For an arbitrary set A define T ( A ) to be the set of all functions f : A + K with finite support, i.e. f ( a r ) = 0 except on a finite subset of A. For fixed cr E A the “&-unit vector” is the function e , E T ( A ) defined by e,(P) :=6ap

,

the Kronecker delta. It is clear that each f E F ( A ) has the unique representation

0EA

in other words: (e,),EA is an algebraic basis of F (A ) . Now take two sets A and B and consider, as in Ex 1.9.,the bilinear map q0 : F ( A ) x T ( B )

(fld

--

T ( Ax B)

f(-)s(-*)

-

Since e(,,p) = \ko(ea, e p ) , it is clear that span im8, = F ( A x B )

.

(Note that e , has a double meaning if A n B #jI- but this clearly does not cause any G) define T E L ( F ( A x B), G) by problems.) For ip E B d ( F ( A ) ,F(B); T ( e ( a , p ) ):= we,, e p )

(and linear extension). It is obvious that T is the unique linear map T ( A x B) with To\Eo=@ .

G

16

2. T h e Algebraic Theory of Tensor Products

It follows that

-

L(F(A x B ) ,G ) = BiI(F(A),F(B); G) T To*, is a linear isomorphism of vector spaces. I t is worthwhile to look at G := K :

F ( A x B)* = Bil(F(A),F(B))= L ( F ( A ) ,F(B)*) linear functionals on T ( A x B), the bilinear forms on F(A) x F(B)and the linear mappings F( A) + F(B)*coincide ! This fundamental algebraic relation allows one to consider certain operators T as linear functionals on a new space depending on the domain and range of T - and this idea is essential to the theory presented in this book.

- the

2.2. The universal property of the pair ( F ( Ax B), \ E d ) is expressed in the

DEFINITION: Let E and F be K-vector spaces. A pair ( H Iq,) of a K-vector space H and a bilinear map qo E Bil(E, F ; H ) is called a tensor product of the pair ( E lF ) i f f o r each K-vector space G and each 0 E BiI(E,F ; G ) there is a unique T E L ( H , G ) withO=To*,:

ExF-G

In other words,

L(H,G) T

--

Bil(E,F;G)

To*,

is injective, surjective (and linear) for all G. This easily implies the following properties:

PROPOSITION: (1) If ( H I8,) is a tensor product of (ElF ) , then span im 9,= H . (2) If ( H I ,*:) and ( H 2 , qk,2) are two tensor products of ( E lF ) , then there i s a unique linear isomorphism S : H1 -+ H2 with S o Q: = f&: . ( 3 ) If Ti : Ei -+ Fi are linear isomorphisms (onto) and (HI*,,) is a tensor product of (F1,Fz), then (HIQ,, o (TI x Tz)) is a tensor product of ( E l ,E2) . If E is a vector apace and

(x,),~A

an algebraic basis of El then

2. The Algebraic Theory of Tensor Products

17

is obviously an isomorphism (onto). The construction in 2.1. and the proposition give the Every pair (E, F) of vector spaces has (up to a linear isomorphism as in THEOREM: (2)) a unique tensor product ( H , Q,,).

It will be shown in 2.4.(4) that the space

E @ F : = 93i(((E’,a(E’,E)),(F*,a(F*,F))) with the bilinear mapping

ExF Y)

(2,

--

-

%i(((E’,a(E’,E)),(F*,a(F’, F))) [(C*l

Y’)

(z’,4(Y’, YN

is a tensor product of (E,F). This space has the advantage that it is constructed “naturally”, not with “coordinates” (i.e. with the help of a basis) - let us call it the tensor product of (E,F).Clearly, much of the power of the theory of tensor products stems from the fact that in specific situations other spaces H can serve as a tensor product - it is customary to speak of other “realizations” of the tensor product and to write

H=E@F - the bilinear mapping !Po : E x F + H being either obvious from the specific type of spaces or being given explicitly. In this sense

F ( A x B ) = F(A) @ F(B)

.

Recall other instances in mathematics where we are used to working with a universal object - without always thinking of a specific realization: the completion of a metric or normed space, the inductive or projective limits of spaces - or even the real numbers! The same will be done with E @ F . Before going on to various examples and methods of checking whether or not a space is a / the tensor product, note again the fundamental properties of tensor products:

BiI(E, F ;G)= L(E @ F,G)

(E@ F)’ = BQ(E,F) = L(E,F’) .

-

The tensor product mapping

ExF will be denoted by has the form

(2,y) v*

E@F

z @ y . By (1) of the proposition every element z E E n k=l

@F

18

2. The Algebraic Theory of Tensor Products

elements of the form z €3 y are called elementary tensors. If 0 E Bil(E, F; G) and T E L(E €3 F, G) with q z ,Y) = T(" €3 Y) 1 the then T is called the linearizafion of 0 (sometimes T is denoted by (PL) and associated bilinear mapping of T . From this one-to-one correspondence it is clear that defining a linear T : E 8 F ---+ G is the same as defining a bilinear 0 : E x F ---+ G. So definitions of T very often will look like

-

T:EQF-G c €3 y

and one must check whether

( 2 , y) u+

...

... is bilinear.

2.3. It is clear from the construction in 2.1. and (3) of the last proposition that for ~ B of E@F. bases (z,),€A of E and ( y p ) p E ~of F the family ( Z ~ € ~ Y ~ ) ( , , ~ )is~ aA basis This observation leads to the following test: CRITERION: Let E , F and H be vector spaces and Qo : E x F space (spanQo(E x F ) , Qo) is a tensor product of (ElF ) span im(\k,) = E @ F

(via

H bilinear. Then the

\ko)

i f and only if (QO(Zi,gj))i=1,..., n;i=l,..., m is linearly independent whenever and (yl, ...,ym) are linearly independent in E and F , respectively.

(21,

...,2,)

PROOF:The condition is clearly necessary. Conversely, let (z,),EA and ( y p ) p E ~be bases of E and F. Then, by the condition, (Qo(z,, Y ~ ) ) ( , , ~ ) ~ A ~isBa basis of the linear hull of \k,(E, F ) , and hence this space is isomorphic to F ( A x B) = E @ F and \ko serves as the tensor product map. 0 An easy calculation with the criterion gives

REMARK:If ( e a ) a € A is a basis of F with coeficient functionals e: : F E(A) := e a E ~ isEa tensor product of (El F )

+

lK,then

E @ F = E(*) via E x F 3 (z,y) u+ ((e:,y)2),E~ E E ( A ) .In particular, for every z E E @ F there exists a unique family ( 2 , ) E E(A)with

2. The Algebraic Theory of Tensor Products

Now assume z = ~ ~ =@ eaj l and ~ order j ( t l , ...,zn) such that of span(t1, ...,t n } . Then

19

(ti,

...,t k ) is a basis

hence the

COROLLARY: For every 0 (21, . . . , t n )

and

#z

E E @ F there are an n E

N and linearly independent

( ~ 1 ,...,gn) with n j=1

2.4. Some examples of tensor products (use the criterion or remark 2.3. - readers who are not familiar with tensor products should check the details; in order t o become acquainted with a mathematical object one also has to get used to it ...):

(2) If K and L are compact, then

C ( K )@ C ( L )=

This is why the notation g@h

:= g( .)h( ..)

is used for the product of two functions in different variables. The Stone-Weierstrass theorem shows that C ( K )@ C ( L ) c C(K x L ) is dense with respect to uniform convergence; is there a natural norm on E @ F which, in this case, is the sup-norm? For the polynomials

P ( F )@ P ( P ) = P(CF+") holds and for the entire holomorphic functions

20

2. The Algebmic Theory of Tensor Products

where, by Taylor-expansions, it is easily seen that 'H(P)@'H(P) is dense in 'H(P+") with respect to the topology of uniform convergence on compact sets.

(3) If S ( X ) is any vector space of functions on X of a special type (continuous, differentiable, integrable - or sequences with special properties) and E a vector space, then

Clearly, the notation g @ z for g(.)z will be used. (4) Let E and F be vector spaces. Then, using the criterion and Ex 1.2.(b) it is clear that 'L3i(((E', 4E'l E ) ) ,( F ' , 4 F ' l F ) ) )

is a tensor product of (ElF ) via the map

xBy:

E ' x F'

-

K

This was formally "the" tensor product of (E, F ) , see 2.2.. This means that z&y is t @ y and the notation 3 is not necessary. Nevertheless we shall from time to time use the notation xBy for the bilinear form on E* x F' since it has some advantages in the future; for example, if the mapping bilinear forms/operators/ functions tensor product were extended to larger spaces by density, the injectivity might be lost! This could happen easily in the following dxample as will be seen in section 5 .

-

(5)

E* @ I F= {T E L ( E ,F ) I rank(T)

< m} =: F ( E ,F )

via (x*,y)~.,[z'&y: z ~ * ( x ' x)y] , (still easy with the criterion). If E and F a r e normed spaces, then

E' @ F = {T E 2 ( E , F ) I rank(T) < m} =: $(El F )

,

the space of continuous finite rank operators. This identification will be often written as k=l

k=l

hence T ( x = ) C;=l(zi,t)yk; note, that by the very definition of the tensor product, this is independent from the special representation of z. If E or F is finite dimensional,

E'@F=C(E,F),

2. The Algebraic Theory of Tensor Products

21

in particular, @ Krn= q K " , Krn) = K""

K"

,

the space of matrices as was already seen in example 1. In summary, the basic examples of tensor products, also from the point of view of applications, are vector valued functions functions of two variables finite rank operators

.

-

2.5. If E is a vector space, then the evaluation map

E' x E

K ;

(z*,t)-(z*,~)

has a unique linearization in (E' @ E)' : it is called the trace t r E on E evaluation

E'xE-K F ( E , E ) = E' 8 E

To check that this is the usual trace of matrices for dim E = n, take a basis ( e l , ...,en) of E with coefficient functionals e ; . For T E L ( E , E) with

and representing matrix ((e; ,T e k ) )

n

rn

n

k=l

k l

k=1

hence the trace of T is the sum of the diagonal elements of the representing matrix of the mapping T. 2.6. Take E and F vector spaces and 'p E (E @ F)* = BiI(E, F ) = L ( E , F') with associated operator T = L,. Using the transposition (see Ex 2.1.), the identification

E@F z

--

F @ E=~~~I((F*,u(F',F)),(E',u(E*,E))) -F(F*,E) Zt

2. The Algebraic Theory of Tensor Products

22

is as follows: For z =

xfl

zk @&

the associated h e a r map is n

S : F'

4

E

t

S(Y*) =

C(Y*! / k ) z k 9

k=l

cE!l

or S = &@!k. The duality bracket between E @ F 4 T (F', E ) and its algebraic dual (E@F)' = L ( E , F') turns out to be the so-called trace duality between operators:

TRACE DUALITY: Let p E (E@F)*with associated map T E L ( E , F') and z E E @ F with associated S E T ( F ' , E ) . Then

.

(p,z ) = trE(S o T ) = t r p (T o S )

PROOF:By linearity, it is enough to check this for elementary tensors z = xo €9 yo but this is easy since (cp,

20

€9 Yo) = (Tzo,Yo) S O T = T * y o z o: E

E ToS=yo~Tzo:F*+F' +

.

0

Note that transposition z u* z' means going to the dual of the associated operator (see Ex 2.2.) - more or less: minor difficulties come from the domain and range spaces (see

Ex 2.3.). 2.7. The tensor product of operators is defined by the

PROPOSITION: Let E L(Ei, F,) for i = 1,2. Then there is a unique operator S E L(E18E2, F1@F2) with the p r o p e r t y S ( z l @ x t ) = 2'1+1@T222 for all ( z i , 2 2 ) E Ei xE2. The operator S i s denoted b y TI 8 T2, hence

PROOF: This follows from the diagram

and the universal property of tensor products. 0

PROPERTIES: (1) If

1'7

and

T2

are surjective, then TI@ T2 as Surjective.

2. The Algebraic Theory of Tensor Products

23

(2) If TI and T2 are injective, then TI @ T2 is injective. (3) In particular, if G c E, then G @ F is a subspace of E @ F . (4) The following formula holds: ker(T1 @ T2) = (ker TI) @ E2

+ El @ (ker T2) C El @ E2 .

PROOF: (1) is simple.(2) follows from the criterion 2.3.using a basis of the form zo@yp on El @ E2. To see (4) decompose

,

El = (ker 7'1) @ GI

E2

= (ker Tz) @ G2

and take z = Enz; @ z i E ker(T1 @ Tz) . Then z i = u6 Eng i @ g i = 0 ; therefore z

=

C(Ur,+ g:)

@ (ui

+ g f ) = C u:

+ g i and, by (2) and (3), 2

8 (ui + g f ) + C g k @un

n

n

E (kerT1) @ E2

n

+ El @ (kerT2) .

The other inclusion is obvious. 0 An interesting special case of TI @ T2 is given by zz E E' and T E L(F, G) :

x: @ T :E @F -+ 1K @ F = F

Exercises:

Ex 2.1. Show that the transposition map z

.r)

tt

defined by

is an isomorphism. Ex 2.2. Show that zEEt@F-2(E,F)3

$ 1

zt E

F @ E'

-

T

C(F', El) 3 T'

24

2. The Algebraic Theory of Tensor Producta

is commutative. Ex 2.3. Show that

5E 1

Z*

1 3

E " 8 E' = $(E', El) 3 T ' O K E I

commutes. This means in particular, that t f E l can be considered as being defined on the space S ( E , El'). Ex 2.4. (a) Take two separating dual systems (Ei, Fi). Then the mapping

is bilinear and defines a separating dual system

(b) In the case Fi = E: use E; 8 E,' C (El 8 E2)* to show that the duality bracket is just the restriction of the trace duality. This is why the pairing in (a) is also called trace duality. (c) For normed spaces E and F the dual system (E 8 F, El @ F') is separating. Ex 2.5. Let E and F be finite dimensional normed spaces, z E E 8 F = S(E', F) with associated operator S and p E ( E 8 F)* = 2 ( E ,F') with associated operator T.Then

Ex 2.6. Let E and F be normed. Show that E 8 F = {T E C(E', F ) I T weak-*-continuous,

rank(T)

< XI) .

& ( p ) be the space of p-integrable functions (on a measure space (a,p ) ) and & ( p ) the Banach space of equivalence classes of functions. Show that for every normed space E

Ex 2.7. Let

& ( p ) 8 E = {f : R -+ E I f Bochner integrable, dim span f(R) < m) Li(p) 8 E = & ( p ) 8 E / -

-

where stands for the "almost everywhere equal" equivalence relation. Ex 2.8. Show that for 1 5 p 5 00 and E normed

tp8 E = {(tn) E t P ( E )I dim span{z,} < XI}.

2. The Algebraic Theory of Tensor Products

Describe

co @ E

25

.

Ex 2.9. If E = El

@ E2, then

E €4 F = (El @ F ) @ (E2 €4 F ) ; more generally,

Is something like this true for infinite products? Ex 2.10. (a) If G is a Gvector space, define G to be the Gvector space with the same addition and the new scalar multiplication

Use this construction to show: If E l F,G are Gvector spaces and 0 : E x F + G biantilinear (i.e. biadditive and ~ ( A z ,y) = a(., Ay) = x@(zlY))~ then there is a unique antilinear T : El @ E2 -+ G such that

(b) Recall that the Riesz-identification RH : H H' of a complex Hilbert space with its dual is antilinear. Then there is a unique antilinear map

satisfying (RH€3 R K ) ( z @ y) = RHX@ RKY for all x E H and y E K. Ex 2.11. The formation of tensor products depends upon the underlying field. More precisely: Let Ec and Fc be complex vector spaces and ER, FR the same spaces considered as real ones. Show that there is a natural map ER @R FR -, E c Fc which is R-linear, and surjective but, in general, not injective. Ex 2.12. The mapping E -+ E @E defined by Q(z) := z@z is in some sense a typical quadratic form. (a) Show that Q is nearly injective in the following sense : if z+y # 0 and Q(z) = Q(y), then t = y. Hint: Use symmetric bilinear forms on E @ E , i.e. those for which (9,u @ v) = (p, v @ u), or think in terms of operators. (b) If p E %iI(E,E ) is symmetric, then the two canonical extensions to E" x E" (see 1.9.) coincide on the diagonal: ( A p ) A ( d l , d= ~ ) "(("(d',~'') for all x" E E". Hint: ("(P)~ = p" in this case. EX 2.13. Define a tensor product for n-linear mappings El x . .x En-+ G and denote it by El €4 + + - @ EShow n . that

-

for each 1 5 k

5 n-I

26

9. The Projective Norm

3. The Projective Norm The projective norm A on E 8 F has the following highly desirable universal property: For normed spaces E , F, G each bilinear mapping E x F G is continuous if and only if its linearization E 8=F + G is continuous.

3.1. Before going through the basic calculations, recall that if C c E is an absolutely convex subset of a vector space, its Minkowski gauge functional mc is defined by mc(z) := inf{A 2 0 I z E AC}

E [O,m]

It is easy to check that mc is a seminorm on the linear span of C which coincides with {z E E I mc(z) < 00) ; notation:

[q:= (span C, m c )

.

mc is a norm if C is linearly bounded (i.e. bounded on every 1-dimensional subspace). It follows that (*)

(1 - E)C C&C

c c B[C] c (1 + E ) C

for the open and closed unit balls and each E > 0 . If a Banach disc.

[qis a Banach space, C is called

If A1 C El and Az C EZ are subsets, then

This seems to be confusing since in the case where the Ai are subspaces this is not the tensor product of these two spaces - it is in general not linear! But in practice it will always be clear what is meant. 3.2. Take iI, E Bil(E,F; G) for normed spaces E, F, G and denote its linearization for the moment by OL E L(E 8 F, G). Then

27

3. The Projective Norm

DEFINITION: The projective norm A on the tensor product E 8 F of two normed spaces 0 0 E and F i s the Minkowski gauge functional of r ( B E 8 BF). The name “projective” will become clear in 3.8.. Notations: for z E E 8 F

E & F := ( E 8 F,*) and E&%Ffor the completion. Actually A is a norm and has an alternative and more convenient description (which the reader should always keep in mind):

PROPOSITION: A is a norm on E 8 F and can be calculated

c N

n(z;E , F ) = inf{

as follows:

N

lltnll llYnll

I N E JV , z = C zn 8 Yn}

=

n=l

n=l

Moreover,

% i [ ( E ,F ; G ) = 2 ( E

(1)

F, G )

holds isometrically - and A is the unique seminorm on E 8 F which has this property . In particular,

for G = K

a(%;E , F ) = m={I(S,

.>I

I s E 2 ( E ,F / ) ,IlSll I11

1

the duality bracket given by

(E

F)’ = %i[(E,F ) = 2 ( E ,F’)

,

which (by 2.6.) can be calculated b y trace duality. 0

0

(2) The open unit ball in E 8=F is r ( B E 8 BF).

(3) (4)

4t 8 Y;El F ) = 1141IlYll A

for all (2,Y) E E x F satisfies the metric mapping property: If Z E 2(Eil Fi)l then llT18 7” : El 8 r E2

(5)

A

-

FI 8=F2ll 5 111;11 llT’ll

is finitely generated, i.e. for all z E E 8 F

E , F ) = inf{*(z; M , N) I z E M 8 N ;M C E and N C F finite dimensional}.

A(%;

28

3. The Projective Norm 0

0

By definition r is a seminorm on E @ F = span r ( B E PROOF:

B F ) . The formula (**) for the norm in %iI(E, F; G) and property (*) of 3.l.for the unit balls of a Minkowski gauge functional imply that for each CP E BiI(E, F ; G ) @

in particular: %iK(E, F) = ( E @= F)' isometrically. To see that A is a norm note that by 2.4.(4) and the Hahn-Banach theorem the following holds: for every 0 # z E E @ F there exists an 2' @ g' E E' @ F' c %iK(E,F) with (2' @ g',z) = 1 (this was Ex 2.4.). Hence (E @ F, %iI(E,F))is a separating dual system, which implies that A is a norm. Clearly, every seminorm 11 on E @ F for which (E @ F, a)' = %iK(E, F) holds isometrically, has to be equal to A. So (1) has been proven. To see (2), observe first that, again by (*) of 3.1.,

Conversely, if z

= cE=, a k z k @ yk E&('I (1

for k = 1,...,n . Hence (1

+

b ~ )then , there is an E > o with

+ E)M E ~ F E r ( i E @ b ~ )and , ~ ( zE;, F) 5 (1 +

E ) Z ~ &E

+ &)'z

63

and (1

0

E ) - ~

< 1.

0

Now x has to be calculated more explicitly: For z E B E and y EBF property (2) implies that x ( z @ y; E, F) 5 1, hence for all (z,y) E E x F

But there are

t'

E BE' and y' E B p with (z',z) = llzll and

11~11llyll =I

(2' @

(y',y)

= llyll, hence

Y', 2 @ ar) 15 112' @ y ' l l ~ ( z8 Y; E , F ) 5 x ( c @ y; E , F ) .

This is (3) - and everything is ready for proving the initial formula for triangle inequality (and an obvious notation)

r ( z ;E, F) 5 infoo(z)5 infj(z) On the other hand, take r ( z ;E, F) < A, i.e.

k=l

which implies

A:

By the

for z E E 8 F.

9. The Projective Norm

29

and therefore inff(z) 5 n(z;El F ) . Clearly, the inf-formula implies ( 5 ) , and it will also be used for the proof of (4) since it is handier than the original definition: Take E C(Ei,Fi) and n

z=

C

X: QD Z:

E El

QD E2

.

k=1

Then

For brevity, the notation TI &, T2, and Tl&T2 for its unique extension to the completions E1&E2 4 Fl&F2, will be used. By considering elementary tensors, it even follows that llT1 8% T211 = 11T111 llT211 holds.

3.3. Let ( 0 , p ) be a measure space and E a Banach space. Then

Cl(f4 8 E c C l h E ) - the

space of Bochner integrable functions and J u P )8 E

c Ll(P7E)

for the equivalence classes. It is clear from the definition of Bochner integrable functions (see Appendix B12.) that the vector space e ( p ) 8 E of E-valued step functions n

EX,,@xk

Ak p-integrable,

Xk

EE

k=l

is dense in & ( p , E ) and hence the space S ( p ) €3 E of classes is dense in & ( p , E). It follows that

are dense inclusions. PROPOSITION: Let p be an arbitrary measure and E a Banach space. Then

3. The Projective Norm

30

is an isometric dense embedding; in other words,

f o r each f E L1(p) 8 E , and L1(p)6&E = L1(p,E ) holds isometrically.

PROOF: Obviously L1 x E

L1(E) has norm

5 1, hence

Since it is easy to see that S(p) 8 E is ?r-dense in L1(p) 8 E (see 3.9. or Ex 3.3.), it is enough to check, by Ex 3.4., that

holds for step functions N

n=l

Clearly, the An can be assumed to be pairwise disjoint, therefore

This example is quite important; for 1 < p see sections 7 and 12 for a discussion if p

5 00 it is not true that L,&E = L,(E), < 00 and 7.1. and Ex 6.8. for p = co .

What is the dual of L1aT F ? In other words, is there a good description of (L1BTF)’ = (Li6LF)’ = & ( p , F)’ ?

THEOREM OF DUNFORD-PETTIS (weak version): If p i s a strictly localizable measure F’ a continuous linear operator into the dual of a Banach on R and T : Ll(R,p) space F, then there is a function g :0 4 F’ such that --+

(1) g is weak-*-measurable (see Appendiz B11.) and g(R) C T B L , ~ ular, Ilg(w)llFI 5 IlTll f o r all w E R . (2) g is a weak*-densiiy of T , i.e.: for each z E F and f E L1

(F’lF)

s

I n other words, T f = f g d p as a a(F‘, F)-Pettis integral.

. In partic-

9. The Projective Norm

31

PROOF:Call 'p E ?ZW(L1, F ) = (-L1 ~3~F)' 3 ~ ( L F') I , the bilinear form associated with T and define ('pz,f) := 'p(f,z) = ( T f , z )for each 2 E F and f E L1. Then pt E L: = L , . Since p is strictly localizable, there is a "lifting" p : L , + 2 , (see Appendix B7.), i.e. p is linear, positive, p ( i ) = 1 and p(f) f j.It follows that Ip(f)(w)l

I IIflL~.

For each w E R the functional g(w) E F' is defined by ( g ( 4

for all z E F ,

4 := P(cPZ)(W)

hence ( g ( - ) ,2 ) = p(p,) is measurable for all z E F and I(S(414I

= IP('pz)(4l 5 Il'pzllt, = sup{l(Tj, 41 I f E

BLJ

which means that g(w) E ( T B L ~ ) " = T B L I 0( F ' r F ) by the bipolar theorem. Clearly, the following representation holds:

( T f ,4 = ('pZ1 f) =

Jnp(cp,)(w)f(w)p(dw)= J,f ( w ) ( s ( w ) M, d w )

0

I f T E 2(L1, F ) is weakly compact, then the set T B L ~c F is weak-*-closed in F" and therefore:

COROLLARY: If p is strictly localizable, F a Banach space and T : L l ( R , p ) --* F weakly compact, then there is a weakly measurable function g : R + F with values in TBL~ such that

(Zt,

for all ( x ' , j ) E F' x L1

Tf)=

J,

f(W)(ZI,

s(w))p(dw)

. In other words, T f = S fgdp

as a Pettis integral.

In Appendix C it will be shown that the function g in this corollary can be chosen to be Bochner integrable if p is a finite measure. Appendix D4.demonstrates conditions under which the dual of L l ( E ) can be described by bounded measurable El-valued functions. 3.4. The following theorem - as most of them in this part of the theory: due to Grothendieck - describes the compact subsets of E& F .

THEOREM: Let E and F be nonned spaces and K c E& F relatively compact. Then there are zero sequences (2,) and (yn) in E and F, respectively, and a compact subset H C L1 such that for each z E K there i s a (A,) f H with

n=l

3. The Projective Norm

32

PROOF:The idea (following Pietsch) is to expand each z E E&F convergent series z = C zn and to choose a representation

in an absolutely

- and all of this somehow simultaneously in K, with the aid of its precompactness. For this, take eo := 1 and En > 0 with Cr., en = 1+ e and define

Assume K C B, ; then there is a finite Do C B, with

K C Do

+ 81

and, by induction, finite sets Dn c Bn with

There are (pairwise disjoint) finite indexing sets A, and BT with 0, := {zY I j E An} and representations zj” = a:kxy,,t 8 $,k kEB?

with llzjlkll

= ll$,kll

=En

and

c

5 En

+

k€B?

The set IIYzkll

-+

r

:= { ( n , j , k ) I n E N U { O } , j E An,k E BY} is countable and I l z y k l l = 0 under some enumeration of I’. Take as the desired compact set in Ll(I’)

For z E K there are jnE An with z

- zj”, E ( K - 0,) n B1= F~

- z ; ~- ~ j ’ ,E ( z - z~”O - ... - z;,,

z

F~ ol)n B2 E (Fn- D,) n Bn+l

...

3. The Projective Norm

33

It follows that

which is the desired representation. 0 3.5. Bounded subsets C c E&F A c E and B C F such that

are “liftable” in the sense that there are bounded

ELF

C c r ( A @B )

(why?) - the theorem implies that the same holds for relatively compact sets:

COROLLARY 1: Let E and F be normed and C c E&F relatively compact. Then there are compact sets A C E and B C F such that E&F

C C r ( A8 B ) The fact that

E = E&#

gives another important special case:

COROLLARY 2: Every relatively compact subset K in the completion of a normed space E is contained in the closed absolutely convex hull of a zero sequence in E . Moreover, for every E > 0 the z, can be chosen to be in [(1+ &)supzEK11z11]B~. Since for every z,, -, 0 there is A, 00 such that Anz, -+ 0 , it follows that for every compact subset K of a Banach space (K c I?{ z,}) there is a compact Banach disc L (take L := I”{Anzn})such that K is compact even in [t]. Another consequence of the theorem is that every element z E E&* F has an absolutely convergent series expansion of the form z = Cr.l z, €9 y, with 2, E E and y, E F .

COROLLARY 3: For z E E& F

PROOF:By the triangle inequality for ?r the right hand side defines a norm r m 3 ?r on E&F which , by 3.2., coincides with ?r on the ?r,-dense subspace E €9 F of E&F; this implies the result - e.g. by Ex 3.4.. An application: Every f E L l ( p , E ) has an expansion m

f = CL.zfl fl=l

34

3. The Projective Norm

00

Clearly, T is continuous in this case and

defines a norm on the vector space %(E,F) of all nuclear operators; it is easy to see that %(E, F) becomes a Banach space with this norm. Recall from 2.4.(5) the injective map E' @ F S ( E ,F ) C % ( E ,F ) C 2 ( E , F ) *'@!I zQy

--

with (z'sy)(x) := (x', z)y. With this notation nuclear operators have an expansion m

n=l

being absolutely convergent in %(E,F) and hence in 2 ( E , F), which implies that they are compact. The definition of %(E,F), its norm and corollary 3 give the

PROPOSITION: The map

is a metric sujection.

Note that the shorthand definition for J waa used: first on elementary tensors, then finite sums, and finally extension to the completion by continuity; it is therefore clear that for x E E [J(Z)I

(XI

=

00

00

n=l

n=l

[CX L B Y ~ ] (x)= C ( x k z)Yn

for each representation z = C,"==, zh @ yn. Unfortunately, it will turn out that J is not in general injective - and hence is not an isometry on E' @= F. This has to do with

35

9. The Projective Norm

the approximation property and will be investigated carefully in section 5 (for N on $(E, F ) = G @ F see Ex 5.16.). Clearly, if E or F is finite dimensional, then

E'&F = El@=F = % ( E ,F )

(via J)

and J is an isometry.

-

3.7. Since I(d,z)I 5 Ilz'l11z11, the trace defined in 2.5. trE : E' @= E

K

has norm 1 (if E # (0)) and hence extends to the completion t r E E (El& E)'

, lltr~ll= 1

with (by continuity) n=1

-

for any convergent representation z = C,"==l z', 8 zn.If the mapping

J : E'&E

% ( E ,E )

is injective, then the trace is defined for nuclear operators - but it will be seen in section 5 that this is not always possible. The trace is a very useful tool for many calculations. To see an example:

REMARK: N ( i d E ) = dim E for all finite dimensional Banach spaces E PROOF: Take an Auerbach basis (el,

a * ,

.

en) of E with coefficient functionals e i

Ilekll= I l e i l l = 1 and

(see Appendix A4.). Then idE = CrZle&eb = and

, i.e.

(eS,et) = 6k,f

= J(C;=le i @ e b ) , hence N ( i d ~ )5

C;=Ille611 llekll

since J is an isometry. 0 For many calculations the following result will be helpful:

3. The Projective Norm

36

PROOF: By linearity and continuity it is enough to check this for elementary tensors z=z@y:

In particular,

3.8. For % E 2 ( E j , Fi) the operator T I ~ & T: ~ E1G3,E2 -+ Fl&Fz is well-defined; is it a homomorphism if the are?

PROPOSITION: If Qi : Ei

1

+ Fi

tare two metric surjections, then

is a metric surjection. In particular,

is a metric surjection as well.

0

0

PROOF: Qj(BEi) =BF, and the description 3.2.(2) of the open unit ball in the projective tensor product imply

which is the claim. 0 This property of

?r

is the reason for calling it projective.

3.9. Let G c E be a subspace. If G@., F were isomorphically a subspace of E@* F, the Hahn-Banach theorem would imply that every linear operator in E(G, F’) = ( G a rF)’ could be extended to E -+ F’. There have been given various examples in 1.5. showing that this is not true; therefore: T does not respect subspaces, not even isomorphically ! Later on (5.8.) it will even be shown that Tl&&T2 may fail to be injective if the T i are. But A respects complemented subspaces, dense subspaces and the embedding into the bidual:

PROPOSITION: Let E and F be normed. (1) If G C E i s complemented with a projection P : E

4

G , then

37

3. The Projective Norm

for all z E G @ F and P @ idF i s a projection of E @, F onto G @= F . (2) If G c E is dense, then G @= F is an isometric dense subspace of E @= F and, consequently, of B B~P. ( 3 ) I respects the natural embedding into the bidual isometrically:

PROOF:(1) follows from the mapping property and (2) from the unique extension of a continuous bilinear form to its completion (1.6.). To see (3), observe first that IlidE @ ~ ~ 5 11, hence 1 n(z;E , F") 5 I(%;E, F) for each z E E m F , Take a p E ( E @ =F)' of norm

5 1 with

and its canonicaI right-extension p" E (E @= F")' (see 1.9.). Then

The result (3) also follows from Ex 3.19. on weak retractions. The following example (taken from Jameson's book [119]) shows in a simple way that x does not respect subspaces isometrically: Take for n 2 2

G = %(G,G) Q %(G, ti') = GI & li' and zo corresponding to i d c . and consider G' Then, by remark 3.7., x(z,;G',G)=n-l .

c,

hence zo = corresponds to the embedding G + canonical basis. It is easily checked that IleilJGt = 1/2 and therefore

zo E GI @=

x ( z o ;GI,

since q

@=

e i 8 ek for the

q )= n / 2 ,

GI = G(G') holds isometrically by 3.3..

3.10. The following characterization holds:

PROPOSITION: For every Banach space E the following assertions are equivalent: (a) E @= respects subspaces isomorphically .

3. The Projective Norm

38

(b) There is a A 2 1 such that: Whenever G c F is a subspace, then ~ ( zE; , G)5 AT(%; E , F ) for all z

E E 8G .

(c) E’ has the A-extension property for some A 2 1. (d) E’ is an injective Banach space. Moreover, the constants A in (b) and (c) can be taken to be the same.

PROOF: (a) w (d) and (b)

-

(c) follow from the Hahn-Banach theorem applied to

C(F,E’) = ( F @r E)’ (d)

TC

restriction

(G @, E)’ = E(G, E’)

.

(c) was shown by the reader in Ex 1.7..

If E = LI(p), then for every

f E L1(p) 8 G

hence L l ( p ) @ ~respects ~* subspaces isometrically. For a localizable measure (i.e. L,(p) is order complete, see Appendix B6.) Lm(p) = (Ll(p))’ holds, hence:

COROLLARY: If p is localizable, then L,(p)

has the metric extension property.

A theorem of Nachbin-Goodner-Hasumi-Kelley states that the spaces with the metric extension property are exactly the C ( K ) where K is compact and extremally disconnected (=Stonean), see Day [38], p.123 or Lacey [164], p.92 . This fits with the Gelfand representation of the C*-algebra L, . Moreover, due to a theorem of Kakutani (see Appendix A7.), abstract L-spaces L are isometric to some L1(p) (where p is a Borel-Radon measure on a locally compact space and, in particular, is strictly localizable). This implies that:

(1) L respects subspaces isometrically. (2) L‘ has the metric extension propertg. The duality between abstract M - and L-spaces has various consequences; first observe that every C ( K ) is an abstract M-space, hence C(K)” is the dual of an abstract L : (3) C(K)” has the metric extension property for each compact space K ; in particular, For G C F every operator T E C ( G , C ( K ) ) can be extended t o some operator T E C ( F ,C ( K ) ” )with the same norm Il5?((= llTll - but there is in general no ettension T :F 4 C ( K ) .

e)

39

3. The Projective Norm

The last statement follows from the fact that co = C(N U (00)) is not complemented in 1,. So, in general, C ( K ) is not an injective Banach space - but it will be seen in 4.5. that compact operators with values in C ( K ) can be extended.

3.11. The following characterization is due to Grothendieck [91]

THEOREM: Let E be a Banach space. Then E @, and only if E is isometric to some L l ( p ) .

respects subspaces isometrically if

PROOF: If E @=I respects subspaces isometrically (the other direction was shown before), then, by the last proposition, El has the metric extension property and therefore El = C ( K ) by the theorem of Nachbin et al.. Grothendieck showed (see Appendix A7.) that this forces E to be isometric to some L 1 ( p ) . 0 3.12. Dual to the extension problem is the lifting problem: Given a surjective mapping + G between Banach spaces and T E C ( E ,G). Is there a ? E C ( E ,F) with

Q :F

T=QOF?

P

It is easy to see (lift the T e , ) that for E = !1(r) and E > 0 there is always a lifting with IjFll (1 e)llTII - but, in general, the same norm cannot be achieved (Ex 3.21.). Conversely, a result of Kothe [151] (with the aid of Ex 3.22.) implies that each Banach space E such that all T's are liftable (with arbitrary norm) is isomorphic to some el (r).

+

-

PROPOSITION: Let E be a Banach space such that E ~3~ respects subspaces with a constant X (as in proposition 3.10.(b)). If Q : F --Y G is a metric surjection between normed spaces, then for every T E C(E,G ) there is an operator E C ( E ,Fii) such that ~ C G o T = Q" o and 11?[1 5 AIITII:

PROOF:Since

40

3. The Projective Norm

the Hahn-Banach theorem shows that there is a p E C ( E ,F”) with 11p1 ;15 AllTll and id^ @ Q’)’(P).It is easy to check that the latter means that ICGoT = Q”oT . 0

T=

COROLLARY 1: Let E be a Banach space such that E @T respects subspaces with a constant A and F , G normed spaces. If for Q E C ( F , G ) and T E C ( E , G ) there is a bounded and absolutely convex subset C c F with Q(C)3 T ( B E ) ,then there is a “lifling” E C ( E ,F”) with ICG o T = Q” o p and BE) c ACoo.

PROOF: Obviously, the mapping [q-+ [&(C)]is a metric surjection; moreover, with I : [q Q F, it is clear that I”(B[c])c Cooc F“. So it is enough to apply the proposition. 0

(See also Ex 3.23..) Clearly, any space L1(p) satisfies the assumption with A = 1 : COROLLARY 2: Let Q : F -W G be a metric surjection between Banach spaces. (1) For every measure p and every T E 2 ( L 1 ( p ) , G )there is a p E C ( L l ( p ) ,F”) with llTll= IlTll and K G o T = Q” of’ . (2) For every measure p , every compact operator T E C ( L l ( p ) , G ) and every E > 0 there as a compact lifting p E C ( L l ( p ) ,F ) with Ilrl’ll 5 (1 &)llTll.

+

PROOF:(1) is clear. For (2) observe that T ( B L ~C) I’{yn} with

(y,,) a zero sequence

+ &)IITIIBc(see corollary 2 in 3.5.). For z,,E F with llznll 5 (1 + ~)IIy,,ll and Qz, = y,, take the compact set C := I’{zn}, which clearly satisfies Q(C)3 T(BL,). Since Coo = C C (1 + ~ ) ~ 1 1 T l holds, l B ~ corollary 1 implies that im(T) c F and in (1

1 1 ~ 1 15 (1 + 4211Tll.0

See also Ex

3.31..

3.13. The metric results on extensions and liftings connected with the spaces C ( K ) ,L , and L1 have natural isomorphic counterparts for spaces which “look locally like” Pw or 6.

DEFINITION: For 1 5 p 5 00 and 1 5 A < 00 a normed space E i s called an C:px-space, if for every finite dimensional subspace M c E and E > 0 there are R E C ( M ,q )and S E C ( q ,E ) for some rn E N such that SRz = 2 for all z E M and IlS((IlRll 5 A+& :

41

3. The Projectiue Norm

E is called an 2;-space if it is an 2;,A for some A. In section 4 (Ex 4.7.) it will be shown that each Lp(c() is an A!:,l-space. The 2;-spaces are very closely related to the $-spaces of Lindenstrauss-Pelczyriski [173] ; they will be studied to some extent in Chapter 11, section 23 (where the exact relationship between 2,- and Zg-spaces will also be clarified). A starter for the “local techniques” (i.e. the method of investigating a Banach space in terms of its finite dimensional subspaces) is the following PROPOSITION: If E is A!T,A-space and G C F is

a subspace, ihen

?r(z;El F ) 5 ~ ( zE ;, G)5 An(%;El F )

for all z E E 8 G ; in other words, E

@x

-

respects subspaces up to the constant A.

PROOF:Take z E E 8 G and E > 0. Since ?r is finitely generated (proposition 3.2.(5) and Ex 3.30.), there is a finite dimensional subspace M c E with z E M 8 G and ~ ( zM, ; F ) 5 (1

+ e ) r ( z ;E , F ) .

For a factorization I E = SR of the embedding I E : M llSll llRll5 A E the mapping property gives

+

4

E through some

with

It follows that if E is a 2{,A-space its dual E’ has the &extension property (3.10.) and E has the lifting property expressed in 3.12. . Grothendieck’s result 3.11. (and EX4.7.) says that the 2:,l-spaces are exactly the Ll(p)-spaces. It will be shown in section 23 that the C:-spaces E are those such that E @ x respects subspaces isomorphically.

Exercises: Ex 3.1. Show that ?r(.; E, F) is the Minkowski gauge functional of I’(BE 63 BF). Ex 3.2. Show that 112’1 Bx 2’211 = 11T111 llT”ll . Ex 3.3. Prove directly, by using the infimum-description of the ?r-norm, that

42

3. The Projective Norm

is a dense isometric injection if E, c E and F, C F are dense. Ex 3.4. Let G be a vector space with two norms 11 111 and II.112 which coincide on a 11 Ill-dense subspace Go c G. (a) The norms are equal on G under each of the following three conditions:

-

-

-

(1) II * 112 I All 111 for some A. (2) There is a norm 11 . 113 on G with 11 Ils-dense.

-

11 - Ili 5

Ail1 . 113 for some A i and Go is

(3) (G, 11 112) is a Banach space and there is a separated topological space X and an injective 0 : G + X which is continuous for both norms. (b) Give an example of two non-equivalent Banach space norms which coincide on a subspace which is dense for both norms. Hint: 11~112:= llSzll1 defines a norm for each injective S E L(G, G) - not necessarily continuous. Ex 3.5. A subset K C E g 3 , F is compact if and only if there are a compact H C el, zero sequences (2,) in E and (yn) in F such that

If K C Eg,, F is absolutely convex or convex, then the set H C of the same type.

el

can be chosen to be

E x 3.6. Let T E C(Ll(p),Ll(v)) and S E C ( E ,F ) . Then there is a unique continuous operator U E C(Ll(p, E),Ll(v, F)) with U ( f 8 z)= Tf@sz

.

E x 3.7. Let T E C(Ll(p),Ll(v)) and (fn) a sequence in L l ( p ) . Then

for all 1 5 p < 00

. Hint: Consider T 8 idt, .

Ex 3.8. If j E L1(p, F) and f E L w ( p ) , then fj is Bochner integrable. Show that T :L , ( p )

-

F

is nuclear and N(T) 5 SIlglldp. Actually, there is equality of norms; see Appendix c3.. Ex 3.9. Show that there is a metric surjection t 2 G 3 , t 2 ---+ el; in particular, ep$,t, is not reflexive.

43

9. The Projective Norm

Ex 3.10. Closed graph theorem: Let E , F and G be Banach spaces. Show that every linear operator T : E @= F + G with closed graph is continuous. Hint: Ex 1.11.. Ex 3.11. Does ti 8% ti = hold isometrically? Ex 3.12. If G C E has codimension n, then

for all z E G 8 F . Can this inequality be improved in general? Ex 3.13. (a) Show that t1&F -+ 2(co, F) is injective (F a Banach space). (b) If I : G 4F is a metric injection, then: T E %(co, G ) if and only if I o T E %(co, F ) ; in this case, N ( T ) = N ( I o T ) = C,"=lI(Te,II. Ex 3.14. N ( T ) 5 min{dim E , dim F } IlTll for all T E 2 ( E , F ) . Ex 3.15. Let E c and Fc be complex Banach spaces and ER and FR the same spaces considered as real spaces. (a) For every @linear T E C ( E ,F )

holds. Is it true that N d T ) = N R ( T ) ? Hint: 3.7.. (b) Denote by QDK the tensor product over K. Show that the natural map

from Ex 2.11. is a metric surjection. Ex 3.16. For U E C(Eo,E ) and S E C(F,F,,) (all spaces are complete) the diagram

W'6-S

I

.

1

z

commutes. Deduce N ( S T U ) I IlS((N(T)[lUll.Clearly, this also follows directly from the definition of nuclear operators. Ex 3.17. If G is a closed subspace of a Banach space E with %(F,G) = F'&G for some F and F' @% G L-, F' @, E is not an isomorphic embedding, then there is a non-nuclear T E C ( t P ( F ) , t p ( G )which ) is nuclear as an operator t p ( F )--+ t P ( E ) .Hint: The operator F + G associated with z E F' @ G has a norm I T(Z;F', E ) . Ex 3.18. Let E and F be Banach spaces such that F' c E isometrically. (a) Then there is a projection P : E -W F' with norm 5 X if and only if ~ ( zF', ; F) 5 AT(%; E , F ) for all z E F' @ F .

44

3. The Projective Norm

(b) If p(F‘, E ) := inf{ llPll I P : E + F‘ projection } < 00, then there is a projection with norm p(F’, E) . Ex 3.19. For a metric injection I : G 4 E of normed spaces the following statements are equivalent: (a) I @ idF : G Br F + E @r F is an isomorphic [resp. metric] injection for every normed space F. (b) I @ idGn : G @ , ,G’+ E a,, G‘ is an isomorphic [resp. metric] injection. ( c ) There is U E C(E,G”) with U o I = ICG [and llUll 5 11 . (d) There is S E C ( C ,El) with I’ o S = id^ [and llSll 5 11 . Hint: Ex 1.6.. In this case, properties (c) and (d) justify calling I a weak retraction. Ex 3.20. Let M be an abstract M-space and G C F. Then every operator T : G -+ M can be extended to an operator f : E -+ M” with 115?11 = llTll . Ex 3.21. Let H be a closed subspace of a Banach space F and Q : E -+ F/H the metric surjection. (a) If T E C(tl(I’2, F / H ) and E > 0, then there is a “lifting” 5? E C(t,(I’), F ) of T (this means: T = Q o T ) with I l f l l 5 (1 ~)llTll. (b) Every T E i!(tl,F / H ) is liftable with the same norm if and only if H is proximinal in F (i.e. every element in F has a best-approximation in H ) . (c) If F is a dual space and H weak-*-closed, then every T E C(tl(I’), F/H) is liftable to F with the same norm. Ex 3.22. Assume that the canonical surjection t l ( B ~ -W) E has a right inverse S E C ( E , ~ ~ ~ BThen E ) ) for . every metric surjection Q : F -W G and T E C ( E ,G) there is a lifting T E C ( E ,F) with I l f l l 5 (1 ~)llSllIITll. Ex 3.23. Show that in 3.12., corollary 1, the assumption Q(C) 3 T ( B E )may be replaced by Q(C)3 T ( B E ) .Hint: Use E g r (F‘/ker mace). Ex 3.24. Radon-Nikodim theorem for operator valued measures: (a) Let p be a strictly localizable measure, E and F normed spaces and T : Ll(R,p) -+ C ( E ,F ) continuous and linear. Use the weak version of the Dunford-Pettis-theorem 3.3. to show that there is a a(C(E,F”), E&F’)-density D : fl-+ C ( E ,F”) such that

+

+

(as a weak-*-Pettis integral in (E&F‘)’ = C(E,F”)). (b) Let A be a a-algebra of subsets of R and p a (non-negative) measure on A. If E and F are Banach spaces and M :A --t C ( E , F) is a-additive, then there is an operator T E C ( L l ( p ) ,C ( E ,F ) ) with T ~ =AM ( A ) if and only if for each z E E and y‘ E F’ the signed measure (M(.)z, y’) is p-absolutely continuous with density in C m ( p ) . Ex 3.25. A special case of the foregoing Radon-Nikodfm theorem is E = K. Show for the Fourier transform

45

3. The Projective Norm

that the values of the density cannot be chosen in F = co (see also Appendix C). Ex 3.26. Every space L m ( p ) has the compact extension property: If F C E and T E f f ( F , L m ( p )(the ) space of compact operators), then there is, for every E > 0, an , with 11?;11 5 (1 ~)llTll. Hint: Dualize 3.12., corollary 2. extension ? E f f ( E L,(p)) Ex 3.27. Show that for arbitrary measure spaces (nilpi)

+

Ll(Pl@P2)

A ~ 1 ( P l I L 1 ( P 2 ) A) Ll(Pl)&Ll(P2)

*

1

c

@ p2) by Fubini-Tonelli. Hint: Ll(p1) @r Ll(p2) Ex 3.28. (a) Let ( e n ) be an orthonormal system in 12 and (An) E 11 with non-negative An. Using the trace, show that r(C,"==, Anen @ en;12,12) = CZzIAn . (b) If (en) and (fn) are two orthonormal systems and (A,) E 1 1 , then m

m

*(CAnen@fn;P;r,12) = n=l

C1Anr.

*

n=l

Hint: Use U e , := fn. Recall the spectral decomposition theorem: Every compact operator 12 -+ 12 has the form CAnen&fn with (A,) E c, and (en) and (fn) orthonormal systems . (c) Take E = 1, (for 1 5 p 5 00) or c, and T := C:=l A n e n e n E 2 ( E ) for (An) E co. Then T is nuclear if and only if (An) E 11. Hint: Use l t r ( q r-, lP%lp -W C)I 5 N(T) and adjust the signs. Ex 3.29. If G C F is a subspace and E @* F does not induce an equivalent norm on E @, G, then there are MnE FIN(E),Nn E FIN(F) and zn E Mn@ (N,, fl G) with r(zn;Mnr Nnfl G)> nn(zn;Mn, Nn).

Hint: Use the fact that r is finitely generated . EX 3.30. r ( z ;E l F) = inf{r(z; M ,F) I z E M @ F, M c E finite dimensional}. Ex 3.31. If E is an 2:,A-space and Q : F -W G a metric surjection, then there is for every compact operator T E 2 ( E ,G ) a compact operator ? E 2 ( E ,F ) with Q? = T and Ili'll 5 (A ~)llTll. Hint: As in corollary 2 in 3.12. use a lifting of compact sets. EX 3.32. If G C E is a subspace and T E %(G,F), then for every E > 0 there is an extension ? E %(ElF ) of T with N(F) 5 (1 & ) N ( T ). EX 3.33. If F C E is a complemented subspace (with projection P) and E an 2:,A-space, then F is an .i2;,A,,p,,-space. EX 3.34. Let E and F be Banach spaces, E complemented in its bidual E" with projection P . If T E 'X(E',F ) is u(E', E)-u(F, F')-continuous, then for every E > 0 there are xn E E and yn E F such that

+

+

c m

T=

m

2n&Yn

and

n=l

Hint: Look at T' : F'

4

C llxnll

n=l

E

L,

E"

P

+ E.

IlYnII

5 IPI")

+

&

*

46

4 . The Injective Norm

4. The Injective Norm The tensor product E @ F of two normed spaces is canonically embedded into the normed space %il(E‘, F‘) = ( E ’ @ r F‘)’of continuous bilinear forms on E’ x F‘. This space induces another natural norm on E @ F, the injective norm e, which is the norm of uniform convergence on the Cartesian product BE’ x BF’ of dual unit balls.

4.1. Let E and F be normed spaces. Then for x’ E BE’ and y‘ E B p the linear form x’@y‘ : E @ F -P K @ K = K is defined. For z E E @ F

gives exactly the norm which is induced on E @ F by the embedding into %i[(E’,F‘) =

(E’ @r F’)’since ~ E ’ o - F ’= r ( h E ’ @ E& F.

ip).Notation: E aeF and for the completion

PROPOSITION: Let E and F be n o m e d spaces. Then

- and even E Be F A (E‘ @= F’)‘ if both spaces are finite dimensional. ( 2 ) For norming subsets A c BE’ and B C B p

(3) E ( X @ y; E , F ) = 1 1 ~ 1 1llyll for all x E E and y E F

.

( 4 ) e S n o n E@F.

( 5 ) e saiisfies the metric mapping property: If ?;. E Z(Ej, Fi), then

Recall that a subset A C BE’ is called norming if l l z l l ~= sup (I(x‘,x)J I x’ E A} for all x E E . The set e x t B p of the extreme points is norming since every linear weak-*-continuous functional attains its supremum on the weak-*-compact unit ball BE! at an extreme point.

4. The Injective Norm

47

PROOF:(1) was the definition; for (2) take any representation of z and calculate: m

E(Z;

b,C

E,F ) =~~p{l(z’@

2 ,

I

@ Yn)l

2’

E BE',^ E B F I }=

n=l rn

(3) is clear from the definition, (3) implies (4) and ( 5 ) follows from (2: @

z;, (Tl @ TZ)(Zl @ z2)) = (T:z: @ T;z;, 2 1 8 z2)

.

0

Since BE c BE&* is norming, it follows that

E ( Z ; E ’ , F ’ ) = S U P { I ( Z @ Y , ZE) IB~EZ , Y € BF}= = SUP{ [ ( w ,z)IIw E r(BE 8 BF)}=

= SUP{l(W, .)11.<w,

E , F ) I 1)

1

in other words,

E‘ BCF’

c ‘ . (E

@w

F)’

A C ( E ,F’) .

This shows that even though T and E coincide for elementary tensors they are far from being equal : E is connected with the operator norm, T with the nuclear norm of operators! For explicit examples see Ex 4.2. and Ex 4.3.. 4.2. Examples:

(1) Since E @ F 4 $(El, F ) L C(E’, F )

E @c F

(E’ @r F’)’, it follows that C(E’, F )

- and similarly F = $ ( E , F)

C ( E ,F ) ,

and therefore (for Banach spaces E and F)

~ ‘F 6Z(E, ~ F) , the space of all approximable operaiors T E C(E,F), i.e. those for which there exists a sequence (Tn) of finite rank operators with [IT- T’ll + 0 ; the space Z ( E , F) is equipped with the operator norm.

4. The Injectiue Norm

48

An element z E E @ F gives a u(E’, E)-a(F, F’)-continuous operator in e(E’, E). It follows that

{TE e(E’, E ) I a(E’,,!?)-u(p,F’)-continuous}

E&F

since the latter space is closed in the Banach space g(E‘,E);see also EX 22.2.. Note that E‘& F operators

-

is injective - a fact which was not true for r! (2) C ( K )@ E is the space of all continuous functions K 4 E the image of which is contained in a finite dimensional subspace. If K is compact, then the Dirac functionals

& (for i E K) form a norming subset for C ( K ) ,hence

&(f;C(K),E)=sup{l(61~b,Cf, @.,)I I t E K , d E BE‘} =

I

= s u P { l ( d , ~ f n ( t ) ~ n ) lt E K , d E B E ’ }

= sup{llf(t)llE

I

E K } = IlfllC(K,E)

>

where C ( K ,E) is the space of continuous E-valued functions. Hence

C ( K )BCE

C ( K ,E)

.

A simple argument using uniform continuity and a partition of unity shows that this space is even dense, hence

C(K ) & E A C(K , E ) for each Banach space E.

(3) If K and L are compact topological spaces, the same reasoning with norming subsets shows that C ( K ) C& C ( L ) c C ( K x L ) holds isometrically. The Stone-Weierstrass theorem gives density, hence C (K )& C (L )

C(K x L)

C(K,C(L))

- where

the latter follows from example (2) (check that the identifications are the “natural” ones!) or by direct verification of C(K x L ) = C ( K ,C(L ) ) .

(4) It is impressive to see how the natural mappings usually fit together and give new information. This is one reason for the power of tensor product methods. Consider the following mappings for the space R(E, C ( K ) )of compact operators:

Z ( E , C ( K ) )c R ( E , C ( K ) )

T

rw+

C ( K ,E’) = C(K)&E’ [t * T’&]

4. The Injective Norm - note

(a

49

that 0 is defined on ff. Since

is isometric. For T = xl&f E E’ @ C ( K ) C R ( E , C ( K ) )it follows that

(there was a dualisation /transposition involved!) and hence, by example (l), @ is surjective on T ( E ,C ( K ) )and therefore

Every compact operator with values in C ( K ) is approzimable. 4.3. Concerning subspaces and quotients, E behaves “dually” to n. This is in some sense clear from the definition, but will be made much more precise in the sequel.

PROPOSITION: (1) If Gi c Ei are subspaces, then

i.e.

(2)

E

respects subspaces isometrically. In particular, E is finitely generated, i.e.

If Ti E C(Ej,Fi) are injective, El and E2 Banach spaces, then Ti@CTz: Ei&E2

-

FigCF2

is injective. 1

( 3 ) E does not respect quotients - not even isomorphically: If Q : F -U G i s a metric surjection, then, in general, idE QC Q need not be open and its completion idE& Q need not be sujective.

PROOF:(1) The mapping property implies that for z E G1 @ G2

and the Hahn-Banach theorem gives equality. For (2) consider the commutative diagram

4. T h e Injective Norm

50

Since all operators from E1&Ez are u(E{, El)-u(Ez, E4)-continuous (see Example 4.2.(1) - the completeness of El is used here) and Ti has a u(E{,El)-dense range, it is clear that T1&T2 is injective . (3) By the open mapping theorem it is enough to find a metric surjection Q : F G and an E such that

id,rp&Q :

F = Z ( E ,F) S

+.

--

$(E, G) = E‘geG QS

is not surjective - in other words: an approximable operator T : E -+ G which is not of the form Q S An “advanced” example runs as follows: Take E = G = L2 and Q : l l --y l z (every separable Banach space is a quotient of L1). It will be seen later on (Ex 11.18.) that every operator LZ -, LZ which factors through l l is Hilbert-Schmidt. But there are Lz which are not Hilbert-Schmidt. compact=approximable operators l z A more “elementary” example uses “local techniques” and the fact that .n does not respect subspaces: By Ex 3.29. there are finite dimensional spaces En,Gn c Fn and zn E En @ Gn such that

.

r(zn; En, Fn) For the natural quotient map Qn : FA norm of

1

< -T(zn; E n , Gn) n

-, FA/Gt

-

= Gk this implies that the quotient

cannot be surjective, because of the open mapping theorem and the fact that all spaces EL @e FA are l-complemented in lz(EL)&l2(FA) . 4.4. However,

-

respects metric surjections, i e . : If Q : F PROPOSITION: C ( K )Be surjection between normed spaces, then id @c Q : C ( K )@e F

-

C ( K )@e G

1 --Y

G is a metric

4. The Injective Norm

51

is a metric surjection and therefore also

“Philosophically” this is a consequence of the facts that L1 @r * respects subspaces, is dual to T and C ( K ) is in some sense dual to L1 by the Kakutani theorem (at least C(K)” = L l ( p ) for some p ) . Using the “local techniques” it is possible to make this precise and prove the statement of the proposition: First assume K to be finite, then clearly E

e”,@cE=%(E)

-

9

i.e. Enwith the maximum norm; individual lifting of i d @ Q : e”, @c F

(21,

...,z,,)E G” gives that

e”, @e G

is a metric surjection. [ If F and G are Banach spaces, this follows also from id @ Q’: (&(GI)’ = q(G’) r-+ q ( F ’ ) = (e“,(F))’

1. The strategy of the proof is to next show that C(K)looks

being a metric injection! locally like .

LEMMA: For every e

>0

and every finite dimensional subspace M C C ( K ) there are

S E 2 ( M ,em,) and R E 2(e, C ( K ) )for some rn E Bv with llRll llSllI 1 + E and RSf = f

for all f E M ; in other words, C ( K ) is a 2k,1-space (see 3.13. for the definition).

PROOF: (a) Take ( f 1 , ...,fn; zi,...,xi) an Auerbach basis of M (i.e. l f’II = llzill = 1 and (zi, fi) = S i j ) and E < 1/2 . Then there is a partition of unity in C ( K ) such that there are ti E K with (of(ti)= 1 and gk E M, := span { p i ) with

((oi)zl

llfk - g k l l w

5 E/n

k: = 1, -..,12

For S1 := C x f , @ k : M -+ M , it follows that for each f E M

c n

IIf - S l f l l w I

l(xk,f)l

llfk

- stllw

I ~11f11w

1

k=l

hence IlS1ll 5 1 + E . Since M, is isometric to (with the mapping g * ( g ( t i ) ) i ) , the desired factorization is nearly obtained: Only the embedding M, + C(K)needs a small pertubation. (b) To do this, first observe that

llflloo I Ilf - Slfllw + llslfllw I E l l f l l o o + Ilslfllw

I

4. The Injective Norm

52

11s;'

hence : S 1 M + Mil _< ( l - ~ ) - '. Now extend the functionals (SF')'zi E (S'M)' to 3/, E C(K)' with the same norm, note

and define

n

dB(S1fk - fk) : C ( K )

R1 :=

+

C(K)

k=l

which has norm operator

5 ~ l sz ~k IIziII l ~ llgk ~ - f k l l m 5 &(I - E)-'

< 1 . Therefore the

Rz := i d c ( ~-) R1 :C ( K )+ C ( K )

+

satisfies llRzll 5 1 e(1 - &)-' = (1 - &)-' and R & f k = f k , hence R 2 S 1 is the embedding M cr C ( K ) . Since S 1 ( M ) c Mo and Mo is isometric to em,, the desired factorization is found. 0 Note that the operator Rz : C ( K ) + C ( K ) which was constructed in the proof is even I ( l - ~ ~ R 1 ~ ~5) -(1-2e)-'. ' This means that the subspace an isomorphismand 11&-'11 M is contained in a subspace (namely R2M0) which is nearly isometric to em, .

PROOFof the proposition: Take a metric surjection Q : F -W1 G and

zo E C ( K )@ G. Then there is a finite dimensional subspace M c C ( K ) with zo E M @ G. Since E respects subspaces, E(zo;M , G ) = E ( 2 , ; C ( K ) ,C ) .

Now take 6 > 0 and a factorization R o S of M IlRll IlSll I 1+ 6 :

Choose yo E

@c

C ( K ) through some

F which lifts S @ idc(to)and satisfies

Then it is clear that yl := R @ id&,)

which was to be shown. 0

lifts to= RS @ id^(^,) and

em, with

4. The Injective Norm

53

-

It is clear, from the proof, that E QDC respects metric surjections for each Z&,,-space E, and also: that 2&,A-spacesrespect metric surjections up to the constant A. It will be shown in 23.5. that this property charactizes the 2L-spaces. 4.5. The proposition has interesting analytic consequences.

COROLLARY 1: For every metric sujection Q : F --y G between Banach spaces, every continuous function f : K --+ G on a compact space K and every E > 0 there is a continuous function g : K -+ F with Q o g = f and llglloo5 (1 E)llflloo .

+

PROOF: This follows from

The argument can be applied to function spaces other than C ( K ) , see the work of Kaballo [134], [135] and 23.5..

COROLLARY 2: If G c E i s a subspace, T E R(G,C(K)) and edension E R(E, C ( K ) ) of T with 115?11 (1 ~)llTll.

+

E

> 0, then there is an

PROOF:In 4.2.(4) the relation R(F,C(K))= F’&C(K) was shown, hence

For L,(p) this compact extension property was already shown in Ex 3.26.. Lindenstrauss ([171], p.92) has given examples showing that corollary 2 does not hold for e = 0 - not even for finite dimensional E. Consequently (look at the proofs above) also corollary 1 is false for E = 0 and even for finite dimensional F; see also Ex 3.21.(b). 4.6. The dual of ( E @3r F)’ is the space of continuous biIinear forms. Which bilinear forms cp are continuous even with respect to E , i.e. cp E (E C& F)’ ?

THEOREM: Let cp E ( E @ F ) * .Then ‘p E (E&F)’ if and only if there ezists a (positive) Borel-Radon measure p on BEIx B p (equipped with the weaklc-topologies) such that for all z E E €31 F (cp,

.>J =

B E /x BFt

(2’

€3I y‘, .)Cl(d(z’, y‘))

*

The measure p can be chosen such that 11pl1 = ~ ( B EX IB F I )= I / ’ ~ ~ ~ ( E B . F ) )

.

This is why bilinear forms in ( E B CF)’c (EmDr F)‘ = %il(E,F ) = Z ( E , F’) are called integral. The operators E -+ F’ associated with integral forms are called “integral” as well; they will be studied in detail in section 10.

54

4. The Injectiue Norm

PROOF: (a) By the very definition of the injective norm

is an isometric injection. Then if cp has such a representation, it follows that

F)’ a signed and, vice-versa, the Hahn-Banach theorem gives for every cp E (E measure p E C ( B J x~ B p ) ’ =: M ( B p x B p ) which represents cp as indicated. (b) But p may not be positive. To achieve this, the metric surjection I’ : M ( B p x B p ) ++ ( E @c F)’ has to be examined; denote by M;’ the convex, weak-*-compact set of probability measures on BE’ x B p and D C M t the set of Dirac measures. By the definition of the injective norm

I’(o)O = B E ~ . F holds; moreover M ( D ) c I‘(D) for all 1x1 5 1 and hence the convex hull and the absolutely convex hull of I’(D) coincide. The bipolar theorem gives (closure with respect to the weak-*-topology in (E @c F)’):

It follows that for every cp E (E @, F)‘ there is a p E M;’ with cp = IIcpllI’(p) - this was to be shown. 0 4.7. If p is a positive Borel-Radon measure on a compact set K, then

pI : C ( K ) & C ( K ) = C ( K x K )

-

-

K

Jh(w,w)p(dw)

is clearly continuous, i.e. integral on C ( K ) x C ( K ) ,and of norm p ( K ) . The theorem says that this is somehow the typical integral bilinear form (the details will be given in a moment). REMARK:If p i s a finite measure, then the “integrating functional”

4 . The Injective Norm

is continuous and

55

= p(Q)

PROOF: Choose a lifting p : L,(Q, p ) -, 2,(Q, p ) (Maharam’s theorem, see Appendix B7.). Hence, a f ) := P ( f K 4 defines an element in Lb, of norm 1 for every w E Q. The claim follows from

Note that for these two bilinear forms the norm in ( E @< F)’ and ( E @r F)’ coincide.

COROLLARY: For ‘p E (El @I E2)* the following statements are equivalent: (a) (o i s integral. (b) There exists a compact set Q and there exist a positive Borel-Radon measure p and

PROOF:Clearly, if /?I is the “integrating” form on C(S2)8 C ( Q )or L,(p) @ L,(p), then, as noted before, CP = PI 0 (R1 @c Rz) is integral. It remains to show (a) f i (b) : The representation theorem gives a p on the compact set BE‘ x B p - and the operators [ R l z l l ( 4 , 4 ) :=

[R24(4 4 ) := f

give the factorization. 0

,

(2’121)

(4.2)

4 . The Injective Norm

56

the norms T and E are not equivalent on E’ 8 F whenever there is an approximable operator which is not nuclear. For a long time it was an open problem whether or not there are infinite dimensional Banach spaces E such that

holds isomorphically! Pisier [224] succeeded in 1981 to construct such Banach spaces PI thus solving one of the toughest problems Grothendieck left behind for the functional analysts; P can be chosen to be separable and even to satisfy

P‘& P

-+.

%(PI P ) L $(PIP ) = P’& P

and this mapping is not injective, hence ?r and E are not equivalent on P‘ 8 P. (For the very first step of Pisier’s construction see Ex 4.15..) John [128] showed that every compact operator T : P -, P’ ist nuclear.

Exercises: Ex 4.1. Does t?j8ct?j= q hold isometrically? Check that ?r # E on 4: 8 tg Ex 4.2. Show that E and T are not equivalent on t 2 8 22 . Hint: Ex 3.28.. Ex 4.3. Consider the “Fourier matrix”

.

Show that (a) T ( w n ; G , G ) = n3I2 . (b) 1 S E ( w n ; q , G ) I n (c) llid : Q 4 c i6 ~1 12 n1I2 . (d) E and ?r are not equivalent on t1 8 t1 . Hint: For (b) use the fact that wn represents a unitary operator 4!j --..* 4!j . A result essentially due to Hardy and Littlewood [loo], [178] (see also Dineen-Timoney [57]) says that there is a constant c > 0 such that for all n E N c n ~ ( pI)

e ( w n ; q ,$1 5 nT@)

) l/p - 1/2 if 2 5 p 5 00 and y(p) = 2/p - 1 if 1 _< p I 2 (for p where ~ ( p = c = 1 works) . For (c) and (d) see also proposition 11.1..

2 2 even

4. The Injective Norm

57

Ex 4.4. For Q = E or a = A show that

Ex 4.5. Let E be a real normed space and define a complex multiplication on the real tensor product Ec := E fT by a o z := id^ @ Eiidc)(z) . Show that for a continuous R-linear E + F the operator

.

is @linear and llTc11 = llTl/ Ex 4.6. Show that every z E & & l 2

has a representation

with orthonormal systems (en) and (fn) and (An) E co . In this case the norm of z can be calculated as follows: E ( z ; & , & ) = max IAn[ . Hint: Expansion of compact operators. Ex 4.7. (a) Use the arguments from the proof of lemma 4.4. to show: If E is a normed 2:,A-spacel then the completion @ is an 2 ;,p p a c e . For the converse see Ex 6.6.. (b) For 1 5 p 5 00 every L p ( p ) is an 2:,,-space. Hint: Step functions, and (a). (c) For 1 < p < 00 the Hardy spaces HP are 2z-spaces. Hint: By the M. Riesz theorem they are complemented in Lp[O,27r] , see Duren [60], p.54 and 67. (d) If E is a Banach space which has an increasing family (E,) of subspaces which are X:,,-spaces and the union of which is dense in E, then E is a 2;,A-space. (e) Just for fun: If a (real or complex) vector space E is the union of a (not necessarily increasing) sequence of subspaces En,then every finite dimensional M c E is contained in some En . Ex 4.8. (a) Show that too @c E is isometrically contained in

{(z~) E l,(E)

1 {Zn} c E relatively compact}

and deduce from this that l,&l, c l , ( N x N ) is a proper subspace. (b) If PBv denotes the Stone-Cech compactification of N equipped with the discrete topology, then C(PBv) = t, (note that PnV is the Gelfand space of the C*-algebra 4,). Show that the natural mapping P(Bv x Bv) + PN x PN is not bijective. Ex 4.9. Show that for every measure p and Banach space E

holds isometrically, but the spaces are different in general. Hint: B L is ~ norming for L , ; both norms are dominated by T , hence it is enough to take step functions.

58

5. The Approzimation Property

E x 4.10. Show that A C E& F is bounded if and only if sup l(x'@ 3 / , A ) [ < 00 for all

x' E E' and d E F' . E x 4.11. A sequence (z,,) in E& F is a weak Cauchy sequence if and only if for all x' E E' and d E F' the sequence ((z'@d, zn))converges. Hint: Lebesgue's dominated convergence theorem. Ex 4.12. EBCF = E a r F holds isomorphically if and only if every continuous bilinear form on E x F is integral. Ex 4.13. Let G c E a subspace and (o an integral bilinear form on G x F . Then there is an integral bilinear form on E x F which extends (o . Ex 4.14. Every nuclear bilinear form (see 1.3.) is integral; the natural mappings E ~ &F' E'&F E& F

---

( E @c F)' ( E @e F')' (El F')'

all have norm 1 whenever E and F are different from (0). Ex 4.15. Let 1, : En L-* En+l be a sequence of isometric injections of normed spaces En such that sup IlI,, @ I,, : En BEEn 4 En+l Br E,,+lll =: C < 00 . Then

E S T L C E on E @ E for the space E :=

u,"==l E,, with its natural norm.

5. The Approximation Property Is every compact operator approximable? Is E'&F = % ( E ,F ) ? Is the trace defined for nuclear operators? Is an operator nuclear if its dual is? Is Tl&Tz injective if TI and Tz are? These questions find an answer with the help of Grothendieck's approximation property.

5.1. Let E and F be normed spaces. C,,(E, F) denotes C ( E ,F) equipped with the topology re, of uniform convergence on all absolutely convex compact subsets of E; it is a separated locally convex space with seminorms

PK(T) := SUP IlTKll

for all T E 2 ( E ,F )

where K C E is absolutely convex and compact; EL, := ZC,,(E,K ) .

5. The Approzimation Property

59

DEFINITION: A normed space E has the (1) approximation property if idE E S ( E , E).=O ; in other words, for every absolutely conuex compact K and E > 0 fhere is a T E 5 ( E ,E ) with llTx - xi1 5 E for all t E K . (2) A-bounded approximation property for X E [1,00[ if idE E AB~(E,EI‘ co ;in other words, fhere is a net (Tv) of finite rank operators with llTv;ll 5 A and Tv + idE in &o(E,E). (3) metric approximation property := 1-bounded approximaiion property. (4) bounded approximation property if it has the X-bounded approximation property for some X 2 1

.

Some consequences are immediately clear: (1) E has the approximation property if and only if for all normed spaces F (or only F = E) the space S(E, F) is r,,-dense in Z ( E , F) . (2) E has the A-bounded approximation property if and only if there is a net (Tv)in S ( E , E) with llTqll 5 A and Tvz -+ t for all t E E . This follows from the fact that on equicontinuous subsets of t ( E ,F ) the pointwise

topology and rcocoincide. (3) If a Banach space E has a basis, then it has the bounded approximation property and an equivalent norm Il.llo such that (E, II.llo) has the metric approximation property. This norm is given by 1 1 ~ 1 1:= ~ sup llPnx[[,where the operators P,,are the expansion operators of the basis. 5.2. For many years it was an open problem whether or not every separable Banach space has a basis - or at least the approximation property. This was solved in the negative by Enflo [63] in 1972; the present “state of the art” is the following:

(1) All .fp (for 1 5 p 5 0 0 , p # 2 ) and co have a closed subspace without the approximation property ( 2 < p 5 00 and co : Enflo; 1 5 p < 2: Szankowski [262] ); Pisier’s spaces P mentioned in 4.8. cannot have the approximation property (by theorem 5.6. below). (2) Szankowski [261] constructed a separable Banach lattice without the approximation property.

(3) The space .C(&,&) does not have the approximation property (Szankowski [263], see also Pisier [221]). That this space does not have the approximation property is scandalous! All the “usual” spaces (which means: not artificially constructed) have the metric approximation property. For C ( K ) this follows from arguments with partition of unity, for L p ( p ) with averaging operators. Exception :

(4) It is not known whether the (also non-separable) space H m ( D ) of all bounded holomorphic functions on the open unit ball of Chas the approximation property (but it has the gp-metric approximation property for all 1 < p < 00, see Bourgain-Reinov

60

5. The Approximation Property

[17] and section 21)' while the disc algebra A ( D ) (the subspace of H w ( D ) of functions with continuous boundary values) has the metric approximation property. All the implications between the various approximation properties and the property of having a basis are either false or trivially true: ( 5 ) There is a Banach space with separable dual and with the approximation property but without the bounded approximation property (Figiel-Johnson [66]). (6) There is a Banach space with separable dual and the bounded approximation property but without the metric one (this follows from theorem 1 of Figiel-Johnson [66] with the aid of a construction due to Lindenstrauss, mentioned in this paper; see also Lindenstrauss-Tzafriri [177], I, l.e.20.). It is not known whether a Banach space with the bounded approximation property can be renormed in order to have the metric one. (7) There is a reflexive separable Banach space with the metric approximation property and without a basis; Szarek [264]. Szarek's method was recently refined by Mankiewicz and Nielsen [181]. A positive result will be proven in section 16 with the Radon-Nikodim property: (8) A reflexive Banach space (or a separable dual space) with the approximation property even has the metric approximation property. 5.3. The first question to be attacked is the approximation of compact operators:

E'&F = T ( E ,F ) C ff(E,F ) . When does equality hold?

PROPOSITION: (1) A Banach space E has the approximation property if and only if F'& E = ff(F,E ) f o r all Banach spaces F . (2) The dual E' of a Banach space E has the approximation property if and only if E'& F = R(E, F ) f o r all Banach spaces F . It is enough to check the conditions for reflexive F (see Ex 5.8.).

PROOF:(1) Since TBF is relatively compact for T E ff(F,E), the condition is easily seen to be necessary. Conversely, take K C E compact. Then there is a compact Banach disc L such that K is a compact subset of [L] (see 3.5., corollary 2). The embedding J : [L]+ E is compact = approximable by assumption, hence there is an operator T = xi 8 y k E EL]' 8 E with "

for all x E K

.

k=l

The dual J' : E' + [L]' has a weak-*-dense = [Ll:,-dense range; therefore the xi 's can be approximated on K by $k E El and 9 := dk8 y k gives the result.

ct=l

61

5. The Approzirnation Property

(2) If T E R(E, F ) , then by the approximation property of E‘ there is an S E E“ @ Et such that

llT‘ - ST’II 5 E and hence also llT” - T”S‘ll 5 E . If S =

x:

@J

a$, then

n

TI’S’ =

E E’

xi @I T”a$

@J

F

k=l

since T”(F”) C F ; it follows that [IT - Czi @ T“zrll 5 E . To see the converse it , is approximable. Since, by assumption must be shown, by ( l ) ,that every T E f f ( F E’) T‘ o ICE E R(E,F’) is approximable, it follows that T=(T’o/c~)’o)cF

is also approximable. 0

COROLLARY: If E’ or F has the approximation property, then

E’&F = R ( E ,F ) ( E ,F Banach spaces): each compact operator E

+F

is approximable.

5.4. Is E’&F = % ( E ,F ) ? Is the trace defined for nuclear operators (see 3.7.) ? To solve these types of questions, observe first that there are natural mappings

E’&F E‘&F E’&F

T ( E ,F ) L-, f f ( E ,F ) 4 Z(E, F ) c* %i[(E,F’) E’&F c* f f ( E F , ) c* E ( E , F )c* %[(E,F’) F’)’ ( E @J=F’)’ = %il(E,F‘) . -+ ( E

-

--y

-+

The last mapping comes from the fact that nuclear bilinear forms are integral, Ex 4.14.; all these mappings have norm 1 (provided E and F are # { O } , of course) and they are all the same in the following sense : just E’&F -+ %iI(E,F’) naturally. So, if one i s injective, all three are injective . The same holds for the three mappings

E@*F -+ T ( E ‘ , F ) R(E‘, F ) c* Z(E’, F ) c-,%i[(E‘,F‘) E&F + E&F L+ R(E’, F ) c* %i[(E‘,F’) E& F (E’ @Jc F’)’ 4 (E’ @ J F’)’ ~ = %i[(E‘,F‘) Q

-+

(all are injective if one is); also the same is true for the natural mappings

El& F’

+

... c* %ir(E, F ) .

The question is under which circumstances

E& F -+

operators

/

bilinear forms

62

5. The Approximation Property

is injective. The main result of this section, due to Grothendieck, answers this question in a very satisfactory manner - a result which is basic for the study of tensor products and operators. 5.5. For this, a description of the dual of &,(E, F) will be needed; the Banach space

E will be fixed. Denote by @F the natural mapping

LEMMA:*F : Ec,(E, F) -+ [(F'&E)', weak-*-topology ] is continuous and therefore

i s defined; if z = C;=,dn@ xn E F'gZ E i s an arbitrary expansion, then

f o r all T E E ( E , F )

PROOF: It has to be shown that for z = C,"==, y,' @ xn E F'& E the mapping

is continuous: Choose C,"==, 11&/,11

< 00 and 2,

+ 0;

then

and the latter is a continuous seminorm. Thus DF is defined and the formula follows from m

C (An,Tzn) =

(trF, idF'&'r~(z)). 0

n=l

If F is reflexive, @F is an isomorphism (onto) and therefore also D F .

PROPOSITION:

DF : F'&.,E

-

(&,(El F))'

i s surjective. If F is reflexive, then DF is even an isomorphism (of vector spaces).

5. The Approximation Property

63

PROOF:Given p E (C,,(E, F))‘ , take a compact set K C I’{zn} C E (where (zn) is a zero sequence, see 3.5.) such that

Consider J! : L ( E , F )

T

--

G(F) (Tzn) .

Then the inequality for p and the Hahn-Banach theorem give a @ = (&) E t , ( F ’ ) = bo (c,(F))’ such that @ o J! = cp . For z := Cn=l y,‘ 8 2, E F’&E it follows that

5.6. For every F there are natural mappings

En

&&zn (it is sometimes good to maintain the notation where Jp(C, & 8 2), = y ‘ z for the operator associated with y‘ 8 z).It follows that

for all (z’,y) E E’ x F and z E F’& E

. In particular,

Now E has the approximation property if and only if

since DE : E’&E + (&,,(El E)‘ is surjective, the Hahn-Banach theorem and (**) imply that this is equivalent to : If z E ker J E ~then ,

in other words, E has the approximation property if and only if ker J E I C ker t r E This means that the trace : E‘& E -, K factors through the nuclear operators.

.

64

5. The Approximation Property

THEOREM: For a Banach space E the following statements are equivalent: (a) E has the approximation property. (b) ker J E ~C ker trE , 1.e. there as a continuous linear functional i f E on %(E, E ) with = it-E o JE, (such a functional is called a trace on the space of nuclear operators).

tPE

(c) The natural mapping

J E ~: E'&E

-

%(E,E )

is injective.

(d) For all Banach spaces F (or only F = El) the natural map

is injective.

Usually the trace i r E on %(E,E ) will also be denoted by t f E - if it exists.

PROOF: The equivalence of the first two statements was already shown, clearly (d)& (c) TC (b) and it remains to show that (b) implies (d) : For this take z E F&E with JF(z) = 0 and 'p E ( F @,I E)' with the associated operator L, E E(F, El); note that

(see 3.7.). A look at the commutative diagram

shows (p, z ) = 0 , hence z = 0 . 0 It will be seen later on (21.9.) that it is enough to check (d) for separable reflexive spaces F only. It follows from 5.4. that E has the approximation property if and only if for all Banach spaces F (or only F = E')

is injective.

65

5. The Approzirnation Property

5.7. Other consequences are immediate.

COROLLARY 1: If E' or F has the approtimation property, then E'g3,F = % ( E , F ) holds isometrically (E and F Banach spaces).

-

If E' has the approximation property, then, by the theorem,

E

E'&E

is injective, hence - again by the theorem - E has the approximation property:

COROLLARY 2: A Banach space E has the approximation property whenever E' has it. The converse is false : It will be seen in Ex 12.9. that the space .&&& has a basis but its dual (ty&&)' = l!(.f2, &) does not even have the approximation property. Since it will be shown in 16.3. that coroilary 2 has an analogue for the metric approximation property, 5.2.(8) implies that the examples in 5.2.(5) and (6) also have duals without the approximation property.

COROLLARY 3: Let E and F be Banach spaces. If F is reflexive or: if F' or E has the approximat ion proper2 y, then

F'&E = ( Z c 0 ( E F))' ,

(via

DF)

.

We do not know whether or not DF is always injective.

PROOF: The reflexive case was already treated in 5.5.. If F' or E has the approximation property, then J p .FI& E -+ F'& F

-

is injective. Since, by the formula (*) in 5.6., ker DF C ker J p the result follows. 0 This is a duality result between E and ?r : if one defines the injective tensor product for locally convex spaces, then Ei0gt F = C,,(E, F ) (if E or F has the approximation property), hence

E&,,F' = (E:,&F)' whenever E or F' has the approximation property. The duality between be studied in sections 6 and 16.

E

and

?r

will

66

5.

The Approximation Property

5.8. If E A.!(Ei,fi) are injective operators and Ei are Banach spaces, then the operator T1&T2 : E1&E2 -+ Fl&F2 is injective (see 4.3.). What about Tl&Tz ? The commutative diagram

and the theorem lead to

COROLLARY 4: (1) If the Banach space El or E2 has the approzimation property, then Tl&Tz is injective whenever both E C(Et,Fi) are injective. (2) Conversely, i f for a Banach space E and its natural metric injection IE into ~ , ( B E I )the mapping idEn&IE : E'&E

---+

E'&e,(Bp)

is injective, then E has the approzimation property.

For (2) use the fact that eoo(BEn) has the approximation property.

n=l

and this shows at the same time that N ( T ' ) 5 N ( T ) .

PROPOSITION: Let E and F be Banach spaces, E' with the approzimation property and T E C ( E ,F ) . IfT' is nuclear, then T is nuclear and N ( T ) = N ( T ' ) holds. PROOF:First recall that F&rE' Q F"&E' is a metric injection (3.9.). For a functional 'p E (F"&E')' the following diagram holds (recall that transposition for tensors = dualization of operators): E t 6 r F L C ( E ,F ) 3 T

I

I. 3

5. The Approzimation Property

67

The three horizontal maps in the middle are injective since E’ has the approximation property. If T’ is nuclear, there is a t E F”& E’ with J ( z ) = T’ . It must be shown that I E E’& F : for this, take, by the Hahn-Banach theorem, a cp E (F”& E’)’ which is zero on E‘&F ; this means precisely that L, : F“ -t EN is zero on F. Since T‘ is nuclear, T is compact and hence T“(E”) c F, which implies that L, o T” = 0 and the diagram shows that (cp, t) = 0 . 0 This result will be improved in 22.8.; but in general there are non-nuclear operators the duals of which are nuclear (see 16.8. and 22.9.). For the nuclear norm it may even happen that N(T’) < N(T) for finite rank operators (see Ex 16.10.)!

Exercises: Ex 5.1. ( 2 ( E ,F ) , r6)‘ = E @ F’, where rs is the topology of pointwise convergence in 2 ( E , F ) . Hint: Use the Hahn-Banach theorem as in proposition 5.5.. Ex 5.2. (a) The completion ?J! of a normed space E has the X-bounded approximation property if E has it. (b) For every normed space E there is a net (Tq)of finite rank operators which converges pointwise to the idE. Hint: Finite dimensional subspaces are complemented. (c) Every normed space of countable dimension has the approximation property. Hint: Banach discs are finite dimensional. (d) The statement (a) is not true for the approximation property.

Ex 5.3. A separable Banach space E has the bounded approximation property if and only if there is a sequence (Tn) in $(El E ) converging pointwise to i d E . Ex 5.4. If E and F are Banach spaces with the approximation property (resp. bounded a.p.; metric a.p.), then E& F has the approximation property (resp. bounded a.p., metric a.p.). Hint: Lift compact sets.

Ex 5.5. The property of Banach spaces having a basis, the approximation property, the bounded or the metric approximation property is not stable under forming subspaces or quotients. The bounded approximation property and the approximation property are inherited by complemented subspaces. The latter is not true for having a basis this follows from a result of Pelczyriski’s [200]together with Szarek’s counterexample mentioned in 5.2.(7). Ex 5.6. (a) Every abstract L-space has the metric approximation property. (b) Using the fact that E has the metric approximation property if E’ has it (this will be proven in 16.3.), show that abstract M-spaces have the metric approximation property.

68

5. The Approzimation Property

Ex 5.7. If E has the A-bounded approximation property for all X the A,-bounded approximation property.

> A,,

then it has

Ex 5.8. Use the Davis-Figiel-Johnson-Pekzyhski factorization theorem (see 9.6.) to show that in proposition 5.3. it is enough to test for reflexive spaces F .

Ex

5.9. Let I : E2

1

1

G be a metric injection, Q : E --y El a metric surjection and T E ff( El ,E2) such that I o T and T o Q are approximable. Is T approximable? Ex 5.10. (a) If F C F” is complemented and T E C ( E ,F ) with TI E %(F’, El), then T is nuclear as well. What about the nuclear norms ? (b) If F is reflexive, then T E C ( E ,F ) is nuclear if and only if its dual is; moreover, N(T) = N(T’). (c) If T’ is nuclear and S weakly compact, then S o T is nuclear. Ex 5.11. The natural map J : E’&F -+ %(El F ) is a metric surjection. Is it possible to find a finite rank operator T E %(El F) such that for each n there exists zn E E‘& F (or El @, F ) with “(Zn; E‘, F ) 2 n and J ( Z n ) = T ? Ex 5.12. Take z E E&F such that for all ‘p E E’ L,

Is it true that z = 0 ? Under which conditions ?

Ex 5.13. Let E and F be Banach spaces, G C E a closed subspace with canonical injection I and assume that F‘ has the approximation property. Then the following two statements are equivalent:

(a) Each T E C(F,G ) is nuclear if I o T is nuclear. (b) F’ @, E induces an equivalent norm on F’

@r

G

.

J

Hint: F’&G L, F‘&E 4 C(F, E ) are injective and M := J-’(C(F,G)) is closed, hence (a)n(b) is simple. For the converse try to use an argument as in the proof of 5.9. and observe that for S E C ( E , F ” ) = (F‘&E)’ with SIC = 0 and z E M the operator U E C(F”, F”) coming from i d p & S ( z ) is just S o J(z)” = 0 .

Ex 5.14. If E is a Banach space with the approximation property, S E C ( E ,F) invertible, then ( t r E , T ) = ( t r F , STS-’) for all T E % ( E , E )

Ex

5.15.

.

If (e,) is an orthonormal basis of a Hilbert space, then

for each T E % ( H , H )

.

69

5. The Approximation Property

Ex 5.16. Denote by J : Et 8 F

4

5 ( E ,F) the canonical isomorphism and define a

new norm on $ ( E , F) by

N"(J(z)) := n(z;El, F )

.

(a) ($(E, F),No)'= C(E', F') holds isometrically with the duality bracket (trace duality)

(S,T)g,r::= t r p ( T o S') = trE'(s' o T ) (the trace is defined for finite rank operators).

(b) For T E 2 ( E , F )

(c) If F is reflexive, S E S ( F , E), then there is a T E C ( E , F ) with IlTll = 1 and

N"(S) = trE(S o T ) . (d) If E' or F has the approximation property, then No= N on $(E, F) (e) N = No on S ( E ,E) if and only if E has the approximation property

.

Ex 5.17. The operators T E $(E, F)"O C C(E,F ) are called compactly approtamable. With the notation of 5.5. and 5.6. observe that ( t r F , T o J p ( z ) ) = ( D ~ ( z ) , T ) f o r a l l T E C ( E , F ) a n d z EF'C3Eand

for all T E C ( E ,F) . Show that for T E C ( E ,F) the following statements are equivalent: (a) There is a linear functional 7 on % ( F ,E) such that for all z E F'6&E

(c) T is compactly approximable. Note that 17 is necessariIy continuous if it exists.

70

6. Duality of the Projective and Injective Norm

6. Duality of the Projective and Injective Norm The injective norm E was defined to be the “dual” of the projective norm 1 in the sense of trace duality; surprisingly enough, z is not dual to E in this sense - but nearly. A consequence of this restricted duality is the (weak) principle of local reflexivity which says that the bidual E“ of a normed space looks “locally” like E - a fact which is fundamental for the understanding of modern Banach space theory. 0

0

0

6.1. The fact that BE’~,,F’= r(BEl €3 B p ) implied, by the very definition of the injective norm E , that 1

c (E’ €3.,

E&F

F’)‘

.

Using the fact that the e-norm on E’ @ F can be calculated on norming subsets of the unit balls instead of BEII€3 B p (e.g. BE c B E I I ) ,it follows in the same way that 1

E’

F

( E @,. F’)‘ 1

F‘

.

(E€3., F)’

C--+

In this sense E is dual to a on all pairs (E, F) of normed spaces. 6.2. Again by the definition of E, each x’ €3 g‘ E BE’ €3 B p defines a functional on E @< F of “integral” norm I 1 (and x‘ @ y on E QDr F‘, ...). This implies that

11 E’@.,F’ 11 El@., F 11 EB., F

c

(E€3cF)’ 1151 ( E @ c F’)’ 11 I 1 ( E t a F’)‘ 11 I1

.

Note that this also follows from dualizing the natural embeddings in 6.1.. Since the space (E’I& F’)‘ is complete and continuously embedded into (El@.,F’)’ = C(E’,F”), the study of the approximation property showed that the extension to the completion

E& F

-

(E’

F’)’

may fail to be injective - in particular, E@.,F (E’BCF‘)‘ may not be an isomorphic embedding. However, it will be shown that these embeddings are even isometries if one of the spaces is finite dimensional. The proof of this needs some preparation. 6.3. The Banach-Mazur distance of two normed spaces is defined by

I

d ( E , F ) := inf {llTll l ~ T - l ~T~E C (E ,F ) bijective 1 E [I, 001

LEMMA:For each finite dimensional normed space M and of some such that d ( M , G ) I 1 + ~.

E

.

> 0 there is a subspace G

6. Duality of the Projective and Injective Norm

Clearly, by dualization, there is also a metric quotient of

71

with this property.

PROOF: Choose an €-net (z;,...,zL) in the dual unit ball B M ~ (i.e. min,1126 for all x' E B M I )and define

-z'II 5 E

Every finite dimensional space M (for short M E FIN) is isometric to K" with some norm. Choosing the functionals z i E (K")'= K" in the preceding proof with rational components, it is clear that there is a sequence (Gn) of normed spaces, each a subspace of some such that for every finite dimensional normed space M and E > 0 there is an n with d ( M , Gn) 5 1 E .

+

If (Gn)nEN is such a sequence, Cp := ep(Gn) @p ep(GL)

+

is said to be a Johnson space (1 5 p 5 m). Every M E FIN is (1 &)-isometric to a 1-complemented subspace of Cp. All the C, are reflexive for 1 < p < 00 and it is easy to show that for 1 5 p < 00 the Cp have the metric approximation property (Ex 6.3.). However, Cm does not even have the approximation property since (by Ex 6.5.) every reflexive separable Banach space is 1-complemented in Cm,hence also those which do not have the approximation property. 6.4. With the lemma the announced duality result can be demonstrated.

(canonical embeddings).

If F is finite dimensional, then the mappings in (2) and (3) are surjective.

72

6. Duality of the Projective and Injective Norm

PROOF: The third isometry follows from (2) and 3.9.(3). For the proof of (1) and (2) assume E to be finite dimensional. Since by 6.1. E'

aCF' = ( E @= F)'

holds isometrically, the isometry (1)follows by dualization. The statement (2) is more difficult to prove - it will be shown in three steps: (a) Assume first E = P ,, then by the examples 3.3. and 4.2.(2)

F' = q ( F ' ) = ( C ( F ) ) ' = (Pm (b) If I : G

F)' isometrically

.

P , is a subspace, then the diagram

commutes. Since ?r respects quotients and E subspaces metrically, the vertical mappings are metric surjections and therefore J is an isometry (clearly, onto as well). (c) Finally, s u m e that E is an arbitrary finite dimensional normed space. Choose, ! and a bijection T : E 4G with IlTll ~ ~5 1T E 1 . The~ by 6.3.,a subspace G c & diagram

+

implies 11J-'11 follows. 0

5 llT-'llllTll 5 1 + by~ the metric mapping property and the conclusion

Since q ( E , F ) = E'&F if E' or F has the approximation property, the following is a simple consequence of (3).

COROLLARY: If E and F are Banach spaces, one of them finite dimensional, then

(V, E))'

%(E, F ) 2+

holds isometrically, the duality given by the trace. Clearly (%(El F))' = (El F)' = C(E', F') in this case. We shall come back to the question of the duality between C,ff, and 9l in section 16.

~

6. Duality of the Projective and Injective Norm

73

6.5. This kind of duality between E and r produces an important tool for the local

techniques in Banach space theory: the "local reflexivity" of all normed spaces; more precisely:

WEAK PRINCIPLE OF LOCAL REFLEXIVITY: If M is a finite dimensional n o m e d space, F an arbitrary normed space, and S E C ( M ,F"), then for every E > 0 and every finite dimensional subspace N of F' there is an R E C ( M , F ) with IlRll 5 (1 ~)llSll such that for all x E M and y' E N

+

PROOF: It follows from the duality relations that

A check on elementary tensors shows that T

*r,

KF o

T

C(M ,F ) 4 C(M , F") = (C(M , F))" is exactly the canonical injection of a normed space into its bidual. The space

L :=

N

c M @F'=

(~(M,F))'

is finite dimensional, hence the result follows from

HELLY'SLEMMA:If E is a normed space, L c El finite dimensional and xt E El', E > 0 there is an z, E E with 1 1 ~ ~ 5 1 1 (1 +~)IIzflland

then for every

(XI,2,)

= (zff,2')

for all x' E L

.

PROOF:If Lo is the polar of L in E" and OL := Lo n E the polar of L in E l then it follows that dim E/OL = dim L < 00 and hence the natural mapping

is a surjective isometry. This implies the result. 0 6.6. Although in most cases the weak principle is sufficient, it is interesting to know

that the operator R can be chosen such that Sx = Rx whenever Sx E F, and injective if S is - with control of the norm of the inverse:

6. Duality of the Projective and Injective Norm

74

PRINCIPLE OF LOCAL REFLEXIVITY:Let M and F be normed, M finite dimensional, S E 2 ( M , F“) and N a finite dimensional subspace of F‘. Then for every E > 0 there is an R E 2 ( M , F ) such that (1) IlRll 5 (1 + 4llSll * (2) (z’,Ry) = (Sy,z’) for all x’ E N and y E M . (3) Ry = S y whenever Sy E F . If S is injective, then R can be chosen to be injective such that additionally (4) llR-lll

I (1 + 4llS-’ll

-

*

-

PROOF:Denote by J the embedding Mo := {y E M I Sy E F} M and define So : M , F by Soy := Sy. Now choose, according .to the weak principle, for every finite dimensional N C L C F’ an operator RL E C ( M , F) with 11R~l15 (l+e)llSll and ( d ,R L ~=) (Sy,z’) for all 2’ E L , y E M ; in particular, ( z ’ , R L J ~=) (.’,Soy) for all 2’ E L, y E Mo . Since (C(Mo,F))’= (ML F)’ = Mo@ F‘, this means that the net ( R L J )converges in 2 ( M o ,F) to So with respect to the weak topology on C ( M o ,F); it follows that So is in the weak=norm closure of the convex set

hence there is for every 6 > 0 a UoJ E C with

-

clearly, (z’,(So U o J ) y ) = 0 for all Q : M + M, and define

2‘

E N and y

E

M,

.

Now take any projection

Then (2) and (3) are obvious and (1) follows from

IlRll 5 l l s o - UJll

IlQll + lluoll I 6llQll+

(1 + ~)llSll

*

(b) Now suppose that S is injective and put A := llS-lll-l . Then there is an el-net (2i)!=i in the unit sphere of M and in the unit sphere of E’ such that

6. Duality of the Projective and Injective Norm

hence U, is invertible and llU;lll the perturbation W := R

75

5 (1+s)llS-'ll if €1 is chosen sufficiently small. Now

- Uo = (So- V0J)Q E 2 ( M , F )

which was found in (a) has to be controlled. The operator

is invertible if ]]U;lW]l

5 (1+ E ) 6 l]S1l1llQll is small and

by the Neumann series. It follows that

if 6 was taken sufficiently small. 0 An immediate consequence is the

COROLLARY: For every finite dimensional subspace M there is a subspace N c E with d ( M , N ) 5 1+ E .

t: > 0

c E"

of a normed space E and

This property is called: E" is 1-represented in E - and this notion serves the purpose that if F is represented in E, then F shares those properties with E which are determined by the finite dimensional subspaces. Clearly, the bidual E" is much more than l-represented in E : The subspace N and the bijection R : M + N can be chosen such that M n E c N and R t = t for all t E M n E . 6.7. An application: Every 'p E % i [ ( E ,F ) = (E @, F)' has, by 1.9., a canonical extension 'p" E %i[(E,FII) = (E @, F")' with llcpll = ~ ~ c p "What ~ ~ . if cp is continuous with respect to the injective norm?

EXTENSION LEMMA: Let E and F Be normed and cp E (E@,,F)'. Then 'p E ( E a t F)' if and only if

'p"

E ( E at F'I)'; in this case,

1

PROOF: Since 'p is the restriction of 'p" and E @c F E BCF" holds isometrically, it follows that 11'p11... 5 llcp"ll ... . Take, conversely, z E E a t F" and finite dimensional subspaces N c E and M c F" with 2 E N @ M . Then, by the weak principle of local reflexivity, there is (for every E > 0) an R E 2 ( M , F) with IlRll 5 1 + B such that for all y" E M and t E N

(d',&t)F",F' = (&Z,

R#)F',F

76

(L, :E

6. Duality of the Projective and Injective Norm

4

F' the linear operator associated with cp); this means

and therefore (y^,z ) = (cp, ( i d E C3 R ) ( z ) )for all z E E 8 F". It follows that

This lemma will be used in section 10 for the investigation of integral operators.

Exercises: Ex 6.1. Show that ( C ( E ,F ) ) ' & % ( F ,E ) under trace duality if the Banach spaces E and F are reflexive, and one of them is finite dimensional. In particular,

N(T : F

-+

I

E ) = sup((tr(S0 T)I S E C ( E , F ) , llsll I 1) .

Ex 6.2. Let T E $(F', E') , A E C(E', M I ) and suppose that M is finite dimensional. Then, given E > 0, there is an operator R E C ( M ,E ) such that IlRll _< (1 ~)llAlland R' o T = A o T . Hint: Apply the weak principle of local reflexivity to S := A' and N :=im(T). Ex 6.3. Check that the Johnson spaces C, have the metric approximation property if 1sp<0O. Ex 6.4. Lindenstrauss' compactness argument: Take Banach spaces E and F , an operator T E C(E,F ) and a net ( F a ) of subspaces of F , directed by inclusion, such that F,, := U, F, is dense in F . Let S, E Z(F,, E ) be operators such that TS,y = y for all y E F, and lim sup IlS,ll = A < 00 . (a) Define pa : E' x F, + K U ( 0 0 ) (onepoint compactification) by

+

Use the fact that (pa)is a net in the compact space (KU ( 0 0 ) ) ~ ' " ~ -(hence has a convergent subnet) to show that there is a (o E %i[(E',F,) such that IIp11 5 A and

for all s/ E F' and y E F,

.

7. The Natural Norm on the p-Integrable Functions

77

(b) Let U E C(E‘,F’) be the operator associated with 9. Show that UT‘ = i d p

.

(c) T’ is invertible on T’(F’) C E‘ and ll(T’)-lll 5 A ; the subspace T’(F’) is complemented in E’ with a projection of norm 5 AllTll . This result is quite relevant as the next exercise will show. The technique was refined to the so-called ultraproduct technique; see section 18, in particular Ex 18.9..

Ex 6.5. If F is a separable Banach space, then F’ is isometric to a 1-complemented subspace of each Johnson space C, . Hint: Choose an increasing sequence ( F k ) of finite dimensional subspaces and T n : Gn(k) -, Fk with IlTnll = 1 and llT;lll 5 l + l / n ; take in the foregoing exercise T = C Tn E 2(Cl ,F) and S, := T;l. This result is due to Johnson [130] . Ex

6.6. Use the principle of local reflexivity to show that a normed space E is an 2:,A-space if E” or the completion ,!? is f& . The full duality will be proved in section 23; for the converse of the statement about the completion see Ex 4.7..

Ex 6.7. Every abstract M-space is an 2$,1-space . EX 6.8. (a) Show that there are elements rn E P . , 8 Pm with

n(zn;e“,,e”,)1 f i

,

E(zn;e”,,P,)

5 1 and

Hint: Ex 4.3..

(b) If K and L are infinite compact sets, then there are llfnlloo = 1 and r(fn; C ( K ) ,C ( L ) )2 f i

fa

E C ( K ) 8 C ( L ) with

7. The Natural Norm on the p-Integrable Functions

The third type of natural norms for tensor products is the norm induced on L, 63 E from L , ( E ) . It behaves fundamentally different from E and n in one very important aspect: It does not satisfy the mapping property! Therefore, this section studies some additional conditions under which tensor product operators are continuous with respect to A, and gives a first look at the highly interesting consequences of this “failure”.

7.1. Let (52,p) be an arbitrary measure space, 1 5 p 5 00, and E a normed space. Then for p # 00 the space of (classes of a.e. equal) Bochner pintegrable functions R -+ E is denoted by L,(Pl@

and the space of (classes of locally a.e. equal) bounded p-measurable functions R

-+

,!?

78

by L,(p,

7. The Natural Norm on the p-Integmbte Functions

8),see Appendix B12.. Since the natural mapping LP(P)@E

f@z

4

L,@) f(-)z

is injective, the definitions

for 1 5 p < 00 and

A,(& L,, E ) := ess-sup

11f(.)ll~

give norms on Lp@ E; notation: L p ( p )@ A ~E and L,(P)&A$

for the completion.

PROPOSITION:

(1) (2)

(3)

A1 = a on Ll(p)@E . A , = € on L m ( p ) @ E . E < A , < ~on L , ( p ) @ E .

PROOF: (1) was one of the main examples for the projective norm (see 3.3.) and (2) was Ex 4.9.. For (3) observe first that for g @ z E L,@E the relation Ap(g@z)= [ ~ Y ~ ~ L ~ ~ implies A, 5 ?r by the triangle inequality. To see E 5 A, take h E BL,, (conjugate exponent), z' E BE' and fix f E L, @ E ; then

7.2. For 1 5 p < 00 the step functions

are dense in Lp(plk ) . This implies that L, @ E is dense in

L,(P)&,E

Lp(8)and therefore

=~ ~ ( p ) & A= , k~ p ( p ,

holds isometrically. For the product measure theorem shows that

p1 @ p2

on R1 x

$22

the Fubini-Tonelli

IZ~~E

7. The Natural Norm on the p-Integrable Functions

79

holds isometrically. Since the step functions in S(p1)@ S(p2) are dense in Lp(pl@p2), it follows that this injection has dense range and therefore isometrically 1

A

L ~ ( p l@ p2) = Lp(pl)~ApLp(p2)Lp(P1,Lp(P2))

(1 _< p c m). For p = 00 this is false (Ex 4.8. for discrete measures and Ex 7.2. for the general case). Recall from Ex 4.9. that, in general, Lm @ Eis not dense in Lm(k). Finally, it is convenient to define the “transpose”Ak of A, : 1

:=

E @ A ; Lp(p) x@f - the

L~(/l) @Ap

f@x

hu*

isometry being the transposition map. Clearly, Lp(p) @A; Lp(v)

1

= Lp(p)

@A,

Lp(v)

Lp(P @ v)

holds. is The norm in L,(Qi,p1) @A; LP(R2,p2)

and hence the continuous triangle inequality (see Appendix B5.) gives the

PROPOSITION: For measures p1 and p2 (dinerent from zero) and 1 5 q

[lid&

-

@ idLp : Lq(p1)@Aq Lp(p2)

5 p 5 00

Lq(p1)@A; Lp(p2)ll = 1 .

The case Am = E is trivial. This proposition has an interesting consequence for the continuity of operators S @ T : L, @ L, -, L, @ L,, see theorem 7.9..

7.3. Take operators S E 2 ( L p ( p ) ,Lp(v))and T E .C(E,F). Is it true that S 8 T Lp(p)@

AE ~

-

Lp(v) 8~~F

is continuous, in other words: does the mapping property which holds for s and E (see 3.2. and 4.1.) also hold for A, ? There are natural examples showing that this is not true for 1 c p < 00 (see 7.5. and 7.6.). But at least in some cases S @ T is continuous: Take S = idLp and T E 2 ( E ,F). Then

7. The Natuml Norm on the p-Integmble Functions

80

THEOREM: Let p and v be arbitrary measures, p , q E [l,m] and T E 2 ( E , F ) .

-

(1) If S E 2(Lp(p), Lq(v))is positive, then

= llsll ( 2 ) I f # is compact and S E 2 ( C ( K ) A((.)) , positive, then

11s@ T : L p b ) @Ap E

11s8 T : C ( K )8 c E

Lq(v) @Aq

FII

Lg(v) @Aq

Fll

-

= llsll

llTll

llTll

*

.

Since positive operators are involved, it is worthwhile t o note that the real and the complex case are treated jointly. Before beginning the proof recall the obvious (see also Ex 3.4.)

DENSITY LEMMA: Let 11 Ili be two norms on a vector space E with 11 112 5 cII 111for some c > 0 and T : E + F a linear map into a normed space F . IfT is 11 111-continuous and for some X 1 0

llT4lF I41412 holds for all z in a

11 Ill-dense

-

subset o f E , then

IIT : ( E , II 112)

Fll 5

-

PROOF of the proposition: Since S 63T = ( idLe(”)@ T )o (S@ i d E ) , it is enough t o take T = i d E . For G = L p ( p ) or C ( K ) ,clearly S@idE

G@=E-

Lq(v) @r E

Lq(v) @Aq E

is continuous, hence, by the density lemma, it suffices to check the continuity-inequality on a n-dense subset of G @= E . For (1) it is therefore enough to take step functions n m=l

with pairwise disjoint A , : Since ISx, I = Sx, , it follows that

7. The Natuml Norm on the p-Integrable Functions

81

- with obvious modifications if p = 00

(the case q = 00 follows also from 6 5 Ap ). For (2) take a partition ( p m ) of unity in K such that there are tm E K with p m ( t m ) = 1 and put n

f := C p m 8 Z m E C ( K )E~ m=l

(these functions form a dense subset of C ( K )@= E); then

and therefore

It is obvious that S@T is also continuous if the operator S E 2(Lp(p), L4(v)) is regular, i.e. S = S1 - S2 in the real case, or S = S1 - S2 i(S3 - S4) in the complex case, where the operators Sj are positive. The following result (due to Virot [275]) shows that this is best possible:

+

PROPOSITION: Let p , q E [l,001 with q < 00 and S E 2 ( L p ( p ) ,L,(v)) such that

i s continuous.

Then S i s regular.

The analoguous result holds for operators S E 2 ( C ( K ) ,L,(v)).The proof will show that both results also hold for q = 00 if v is localizable, i.e. L m ( v ) is order complete. Since A1 = A and A has the metric mapping property, the proposition shows that every Ll(v) is regular. See also Ex 7.19.. operator L1(p)

-

PROOF:Considering

cL=l

fk @ e k ,

it is obvious that continuity means that

The type of order completeness of L,(v) (see Appendix B5.) shows that S throws order bounded sets into order bounded sets. But this implies that S is regular (see e.g. Schaefer [244], p.229 and p.233). 0

7. The Natural Norm on the p-Integrable Functions

82

7.4. With regard to subspaces and quotients the norm Ap is well-behaved:

PROPOSITION: Let p be a measure and 1 5 p 5: 00. Then

respects metric injections and metric surjections.

PROOF:For p = 1 or 00 this was shown in 3.8., 3.10., 4.3. and 4.4. (recall that L , ( p ) = C ( K ) for some compact K). That L p ( p ) @A,, respects metric injections follows (for 1 < p < 00) directly from the definition of the norm. Now take Q : E --y F a metric surjection between normed spaces; by 7.3.

is continuous with norm 5 1 and clearly surjective. For every g E Lp @A, F and E it is necessary to find f E Lp @bP E with id @ Q(f) = g and

AP(f) I ( 1 +&)Ap(g) If g = Xk

X A @ ~

yk is a step function (the

>0

*

Ak pairwise disjoint), then there are . Hence f := XA,. @ zk covers

E E with Q(zk) = yk and 11zkl1 5 (1 + c)llykll

ci=l

g and

In other words, id@Q

sp(p) @Ap E

S p h ) @Ap F

(induced norms) is a metric surjection. The result now follows from the following lemma which is of independent interest (for the condition on the kernels see Ex 7.7.):

QUOTIENT LEMMA:Let E and F be n o m e d spaces, Q E C ( E ,F ) surjective, E, c E dense and Q0 := QIE. : E, + Q ( E , ) the (surjective) restriction. Then Q0 i s a meiric surjection i f and only if ker Qo = ker Q and Q is a metric surjection. PROOF: Assume first that Qo is a metric surjection and take z E kerQ. Then there are z, E E, with z, -+ z and, by assumption, z, E E , with Q0(zn) = &(zn) and

It follows that z,

- z,

E ker Qo and zn - z, + z,hence

k e r Q ~ m.

83

7. The Natural Norm on the p-Integrable Functions

Both implications of the claim now follow from the simple observation that

7.5. As was already mentioned, the operator s @ idE : L, @A, E 4 Lp @A, E is not always continuous. To see a concrete example, take S to be the Fourier transform 5 : L 4 R ) 4L2(R) defined by

[5(f)](t) := (2r)-’I2

Irn

exp(-ist) f ( s ) d s

-W

i f f has compact “support” - and then continuous extension; normed spaces E, consider the E-valued Fourier transform

5 is an isometry.

For

If H is a Hilbert space, a simple calculation with scalar products shows that 5~ is an isometry (see Ex 7.8.); considering H as some La, this also follows from 7.4.. But if E = 1 1 , the operator $11 is not even continuous: For the unit vectors en in el take

then

However, from

[z(x[a,b[)] ( t ) =

a

(exp(-ibt) - exp(-id))

it follows that

-

[exp(-i(m

(1” (C 6

1 --

1 7 -wt

“

rn=l

lexp(-i(rn

+ 1)t) - exp(-irnt)]

ernllildt)

+ 1)t) - exp(-imt)l)

112

2

dt)

112

=

=

84

7. The Natural Norm on the p-Zntegmble Functions

This shows that the operator 3cl = $ 8 idcl cannot be continuous. For the Fourier see Ex 7.18.. transform on &(Z) A celebrated result of Kwapieli [161] says that a Banach space E is isomorphic to a Hilbert space if and only if SE is continuous; this will be shown in section 30. 7.6. Another example is given by the discrete Ailbed transform A : tp(%) for 1 < p < 00 which is defined by

-+

lp(Z)

Due to a result of M. Riesz A is continuous (see for example Edwards-Gaudrey [62], theorem 6.7.4.), but A 8 idcl is not: Take zn := e k 8 e k E lp(zZ) 8 11; then A , ( ~ ~ ; 1 ~ ( Z ) , . t=1 )nilp and

n

n

-

Therefore

and since (l/n) log(n!/2)

+ 00

8 idtl cannot be continuous.

the operator

7.7. Though at first glance it is disappointing that Ap does not satisfy the mapping property, Kwapieli’s result shows that this behaviour gives new structural insights. It even can be said that this “misbehaviour” is an incredibly rich source for interesting properties of spaces and operators in modern Banach space theory: One of these sources is to study, for a fixed operator So E i ! ( L P ( p ) ,Lq(v)), the class of all those operators T E 2 ( E ,F) such that so

C3

-

: Lp(P) C ~ E A ~

Lq(v) @A, F

is continuous - and those spaces E for which idE is in this class.

To see an example of this kind, consider the Rademacher functions rn E &[O, 11defined in 1.5.(2). They form an orthonormal sequence, hence

85

7. The Natural Norm on the p-Intepble Functions

is a complemented subspace of L2[0,1]. If E is normed, define A2 on Rad B E to be the norm induced from Rad 8 E -+ L2[0,1] 8~~E . This means that

is the norm of f = C,"==l rncn in Rad @.a2E and not the norm coming from via the identification I :t 2 Rad (an)

-

e2

C3 E

Canrn

of Rad with t 2 . Actually it is this difference which counts: An operator T E C ( E ,F) is called of type p (for 1 5 p 5 2, hence Ip : tp4 t 2 L R a d is defined ) if 1 1 1 , C 3 T : ep @A, E

and of cotype q (for 2 5 q

llJ,

5 00,

-

L2[0,1] @A2 FII =: Tp(T) < I-'

hence J, : Rad-

8 T : Rad @ A 2 E

-

t,

Q

t, is defined) if

1, C ~ FII A ~=: C , ( T ) < 00

.

A normed space E is called of type p (resp. cotype q ) if idE is of type p (resp. of ) for the cotype). These notions are due cotype q ) and T,(E) := T p ( i d ~ (analogously to Maurey [285]and Hoffmann-Jfirgensen [106]. An appeal to the density lemma 7.3. shows that the inequalities for the mappings to be continuous need only be verified on r-dense subspaces of the original tensor product - for example, ep can be replaced by the dense subspace of finite sequences and E or Rad by any dense subspace. In particular, T E C ( E , F) is of type p if and only if there is some c 2 0 such that

for all

21,...,cn E

El and of cotype q if and only if

Since A, = A and Am = E , every operator/space has type 1 and cotype 00. In the exercises it will be shown that Hilbert spaces are of type 2 and cotype 2, for the converse see section 30 and also Ex 30.l.(b). Moreover, infinite dimensional C ( K ) have neither cotype < 00 nor type > 1 (one says: do not have proper coiype/type), and infinite dimensional L 1 ( p ) do not have proper type. In the next section type and cotype of

86

7. The Natural Norm on the p-Integrable Functions

L,-spaces will be studied and some results will be presented which begin to show the importance of these notions. For p > 2 (resp. q < 2) there are no non-zero operators satisfying the type p (resp. cotype q ) estimates; see Ex 7.15.. 7.8. For the Hilbert transform 3 : Lz(R) + L2(R) operators (and spaces) such that A @ A ~T is continuous were studied by Burkholder [19], Bourgain [15] and M. Defant [50], [51]; for the Paley projection La[O, 2x1 4 La[O, 2 4 by Pisier [223], see also the surveys of Figiel [64] and Pelczyriski [204] during the IMC 1983. For the projection R of La[O, 11 onto the Fbdemacher functions see Ex 9.16. and section 31.; the operators T such that R @ T is continuous will be called K-convex. Neither the el-valued Fourier transform (7.5.) nor and the el-valued Hilbert transform (7.6.) was continuous - and el did not have proper type. That the counterexamples were always in is not by chance: PROPOSITION: Let T E &(L,(p), L,(v)) such that

T 8 idE : Lp(p) @A? E

-

L,(v) @ A ~E

is continuous for E = el. Then it is continuous f o r all Banach spaces E. PROOF:T 8 idll being continuous implies that there is a constant c 3 0 such that

for all f1, ...,fn E Lp(p). The density lemma 7.3. gives that

T 8 idt,(r) : Lp(p) @A? e i ( r )

-

Lq(v) € 3 e i~( r )~

has norm 5 c for all r. Since every Banach space is a quotient of some tl(I'), the fact that L, 8~~ respects quotients (theorem 7.4.) yields the result. 0 Together with proposition 7.3. this implies for each T E &(L,(p), L,(v)) (where v is supposed to be localizable if q = co)the following: If T @ idE is continuous for E = el o r E = too, then T is regular and T @ idE is continuous for all Banach spaces E.

-

-

7.9. In particular, T @ S : L, @A, E L, 8~~F is continuous for all S E &(E,F) if T @ idll is continuous. For operators T 8 S : L, 8 L, L, 8 L, there is another quite useful positive result in this direction - which even holds for subspaces of L,. For this denote for a subspace El C L,(p) and a normed space E2 by

El @Aq

E2

E2 (see Ex 7.17. for more the space El @ E2 with the norm inherited from Lq(p) information about this definition). This notation was already used in 7.7. for Rad.

87

7. The Natural Norm on the p-Integrable Functions

THEOREM: Let p and v be measures, El C L q ( p ) a subspace, E2 and F normed spaces, T E C(Ei, F ) and S E C(E2,L p ( v ) ) .If 1 5 q 5 p 5 00, then

For Ei = L,(pi) this result is due to Beckner [8]; the present form is due to FigielIwaniec-Pelczyhski [65]. For another proof of Beckner’s result see 15.12.. Note the special case [IT @ : Lq @Aq Lq L~ @A, L ~ l l = llTll llsll

s

-

for q 5 p. This is false for q > p : see 26.3. for an example; some positive results will be presented in 18.9.’ 26.3. and 32.3..

PROOF: Recall that idL, @Sand T @ i d ~are , continuous. Now the continuous triangle inequality in the form of proposition 7.2. and the diagram

-

The general question about the continuity of operators S @ T: L, @ A ~E L, @A, F is rather involved; the reader will find at various places of this book more information about this problem - in particular, in the sections 18, 26 and 28 - 32. For a survey on this topic see [46].

Exercises: Ex 7.1. Let L p ( p ) be infinite dimensional (1 5 p 5 w). For a sequence (An) of pairwise disjoint integrable sets with positive measure the averaging operator P : L p ( p ) + L p ( p ) is defined by

7. The Notuml Norm on the p-Integrable Functions

88

Show that for any normed space E

.

E is a norm-one projection such that im(P @ i d E ) is isometric to eP E x 7.2. Use the fact that L,&L, = A(L1, L,) c (L1G3,L1)'to show that in general L,(Y) is not dense in L,(p @ Y) for finite measures (see also Ex 4.8.). L,(p) Ex 7.3. Let p be a measure, l/p = 1/r l / q and g E Lr(p). Then for every normed space E the multiplication operator

+

L,(P,E)

f

--

J%(P,E)

s*f

is continuous and has norm llgllr (if E # (0)). If Mg : L&) multiplication operator in the scalar case, then

llMB@ T : Lq @A* E

L p @ A ~Fll

+ L p ( p ) denotes

this

= IIgIIr IITII

€or all T E C ( E ,F). Hint: Holder inequality. Ex 7.4. If S E C ( L , ( p ) , G) and T E C ( E ,F), then

for each 1 5 p 5 00 . Ex 7.5. If T E C ( E ,F) factors through a positive operator To E C ( L P ( p l )Lp(p2)), , , the operator then for all S E 2 ( L p ( p )L,(v))

s @ T :Lp(p)

E

-

Lq(v) @ A ~F

is continuous. Ex 7.6. If E and F are Banach lattices, then every positive operator E F is continuous. Hint: Assume not, then there exist tn 2 0 such that lltnll 5 2-" and IlTZnII 2 2" . Ex 7.7. Use the metric surjection E'& F + %(E,F) to show that the kernel condition in the quotient lemma 7.4. is indispensable ! Ex 7.8. (a) If E C(Lp(pj), Lp(vj)), then -+

llT1@T2 : Lp(p1) @Ap Lp(P2)

(b) If S : L2(p) space, then

-+

-

Lp(v1) @Ap Lp(Y2)11

= 1IT1'111

llT2ll

.

L ~ ( Yis) a (not necessarily surjective) isometry and H a Hilbert

s 63 idH : L2(p) @ A ~H

-

L2(v) @ A ~H

is an isometry. Hint: (SflSg) = (flg) or look at A;

.

7. The Natural Norm on the p-Integrable Functions

Ex 7.9. If H is a Hilbert space, then T z ( H )= C z ( H ) = 1 . Ex 7.10- T p , ( T )I T p , ( T )and c , , ( T ) I C , , ( T ) if 1 I P I IP Z I 2 I Ex 7.11. If S o To U is defined for three operators, then

89

q1

i qz S 00 .

Cq(SoToU)I Ils~lc,(T)llUll

- and the same holds for the type constants T, .

Ex 7.12. (a) If F is a subspace of E , then C , ( F ) 5 C , ( E ) and T , ( F ) I T,(E) . (b) If (E,) is a family of subspaces of a normed space E with dense union such that for all a,P there is a y with E, U Ep C E,, then

C , ( E ) = SUP C q ( l a: E , + E) = sup C , ( E , ) Q

.

Q

The same holds for the type constants T , . Ex 7.13. Use the unit vectors e k to show that T p ( G )2 n1-l/p . Deduce that no space E which contains the uniformly (i.e. there are subspaces M, c E with sup d ( M n , q )< oo) does have proper type (Pisier [217]proved the converse; see also Beauzamy [7]). In particular, all infinite dimensional L1(p) and abstract L-spaces do not have proper type; by 23.3. this is also true for &:-spaces. Ex 7.14. Show that: (a) sup, Tp(Pw)= 00 for 1 < p 5 2. Hint: Ex 7.13.and lemma 6.3.. (b) C , ( c ) 2 nl/q for 2 5 q < 00. (c) A space which contains the uniformly has neither proper type nor proper cotype. In particular, this is true for all infinite dimensional C ( K ) ,L,(p) and abstract M spaces (use local reflexivity); by 23.3. this is also the case for arbitrary Z&,-spaces. Ex 7.15. Show with x k = 1 that the space R has neither type p for 2 < p 5 oo nor cotype q for 1 I q < 2 in the sense of the inequalities a t the end of 7.7.. Ex 7.16. Let E be a normed space. (a) For each finite dimensional subspace M c E and E > 0 there is a finite codimensional, closed L c E such that (1 - E)llxI} _< } / Q f ( x ) }I } 1 1 ~ 1 1for all x E M . L can be chosen to be weak-*-closed if E is a dual space. Hint: Proof of 6.3.. (b) Show that for f E L, @ E

Ap(f; L,, E) = inf{A,(f; L,, M) I f E L, 8 M , M c E finite dimensional} = = sup{A,(id~,. @ @(f); L,, E / L ) I L c E closed, finite codimensional} .

Ex 7.17. (a) Let E i j c L,(pi,j) be subspaces such that Ei,l is isometric to Ei,a (for i = 1,2).Use the generalization 7.9. of Beckner’s result to show that El,1 @A, E2,1 is . shows that, for fixed p , the definition of El @A? E2 is isometric to E1,2 @A? E ~ JThis independent of the L p ( p i )in which the Ei are embedded. However, Rad@A,E is, in general, not even isomorphic to t 2 @A, E , hence El @A, E (where E is an arbitrary 1 Banach space) depends on the embedding El + L p ( p ) .

90

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

(b) Use the fact that L1[0,1] has a non-complemented reflexive subspace E (e.g. isomorphic to &f via Rademacher functions) to verify that A, is not equivalent to ?i on E €4 Ll[O, 11. Ex 7.18. Consider the Fourier transform T : Lz(Z) + L2[0,2?i] Tek

for all k E Z

:= (2?i)-’l2exp(ik-)

.

Show that

T €4 i b , : t 2 ( Z ) € 4 t i~ ~L 2 [ 0 , 2 ~€ ] 4 t i~ ~ is not continuous. Hint: C e k €4 e k . Ex 7.19. Recall from 7.3.that an operator between complex Banach lattices is regular if it is of the form S1 - S2 + i(S1 - 5’2) with positive operators Sj. (a) Show that the Fourier transform on L 2 ( R ) and the discrete Hilbert transform on .fp(Z)(for 1 < p < m) are not regular. (b) Let v be a localizable measure. Then all operators in 2 ( C ( K ) ) L,(Y)) , and all operators in C(L,(p),L,(v)) (where p is an arbitrary measure and 1 5 p 5 m) are regular. Hint: 7.3. and the mapping property of E . -+

8. Absolutely and Weakly p-Summable Series and

Averaging Techniques

This section treats various types of series in Banach spaces which are related to E and Ap. Rademacher and Gauss averaging techniques will be introduced, in particular the Khintchine inequality will be proved. As first applications a theorem of Orlicz’ on the convergence of series in L p ( p ) and the calculation of certain nuclear norms are given.

8.1. A sequence (zn) in a normed space E is called absolutely p-summable (1 5 p if 00 tp(zn; E )

:= t p ( 2 n ) :=

(C

<

~ l z n l l ~ ~ ) ”00 ~

n=l

(the notation will be used for finite sequences as well); in other words, (Zn)

E t p ( Q = tp&A,E

.

< m)

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

91

p = 00 is just the case of bounded sequences: & ( E ) , and it is known (Ex4.8., Ex 7.2.)

that

L&E

= e&&,

E

c e&) #

.

The sequence ( z n ) is called weakly p-surnmable if

n=l

-

for all x' E E'; this means that the operator

T : E' 2'

lP

-((z',

zn))

is defined. The closed graph theorem shows that this operator T is bounded and hence wp(zn; E ) := wp(zn) := llTll < 00 :

defines a norm on the space q ( E ) of all weakly p-summable sequences in E . This formula shows that for each norming set D C BE'

Note that T' = Cen&znand T'(tPt)C B if 1 < p < 00 ; for p = 1 one has T'(c,) C k . The notation wp(zn)will also be used for finite sequences. The case p = 00 is, as before, the case of bounded sequences and w,(zn; E) = l m ( z n ; E ) .

PROPOSITION: If D C BE' is norming and 1 5 p _<

00,

then for all

21,

n

wp(zk; E ) =

SUP l l ( ( z ' , z k ) ) l l l ; Z'ED

=€

( C e k C?J X k ; $ , k=1

PROOF: Since

and

the relation

E

= wp on $ @ E follows from the definition of E . 0

E)

.

...,z n E E

92

8. Absolutely and Weakly p-Summable Series and Avemging Techniques

The following inequalities are simple but important: If T E C ( E ,F) and (zk)in E , then l p ( T z k ;F ) 5 IITllep(zs;E ) and wp(%; F) 5 llT1lwp(zk; E ) . 8.2. In order to describe the weak sequence spaces in terms of operators define the

closed subspace

$+'(E) := {(tn) E c ( E ) 1 W~((Z,);'N;E)

-

O for N

-* 00)

and give it the induced norm from c ( E ) . Clearly, Q o ( E )is the space c,(E) of zero sequences in E. It is easy to see that

E'&t, = f f ( E , ! , ) c C ( E , t , ) = t,(E') E'6eco = R ( E , c,) C C(E,c 0 ) = { ( x k ) E !,(El) ea&E = ff(&,E)c C ( t , , E ) = t , ( E )

I zk

+

0 weak*}

hold isometrically.

PROPOSITION: Let E be a Banach space. ( 1 ) The following relations hold isometrically: for l < p < 0 0

~ ( E ) = C ( ~ ~ ~ , E ) ~ R ( ~ ~ ~ , E ) = $ ~ O ( E ) = ~ ~ ~ ~ E

q ( E ) = C(c0,E ) 3 ff(c,, E ) = q ' " ( E ) = 4 & E . The identification is given by (xn)- Cz'I en&xn; the sum converges pointwise in C and with respect to the operator norm if it represents a compact operator. (2) Dually: c ( E ' ) = C(E,.tp)3 f f ( E , $ ) = c ? " ( E ' )= E'&eP hold isometrically for 1 5 p < 00; the identification is given b y convergence as in (1). ( 3 ) For p = 00 the following equalities hold isometrically: {(x,) E t , ( E ) I {z,} relatively compact} = e,&E c,(E) = co& E .

PROOF: That g ( E ) L, C(epl,E) (or C(co,E)) via S :=

llSll=

(zk)-+C;=l

xLsen,

= ff(t1, E )

en@,

and

wp(zn; E )

was already shown when defining wp(z,; E) in 8.1.. The norm relation also shows that S is compact if w P ( ( x n ) ~ ! , -+, ) 0. Conversely, take S 6 C(tpl ,E) (or C(c,, E)), define

x , := Sen and observe

8. Absolutely and Weakly p-Summable Series and Avemging Techniques

93

Therefore, C ( E )= C(t,/, E) holds isometrically; moreover,

t, atE

Q

C p o ( E ) ff(lp E/) ,= $&E Q

holds isometrically (tphas the approximation property, 5.3.) and (1) is proven.

For (2) take (z;) E C ( E ' ) . Then

this shows that

C ( E ' ) Q X ( E , t p ) and G+'(E')

L,

ff(E,tp)

hold isometrically. If T E C ( E , t p )and z; := T'ek (where the e', are the coordinate functionals), then, clearly, Tz = ((z;,~))and C ( E ' ) = X(E,tp)follows; the density of E' @tpin the space of compact operators gives, as in (l),the result. The first relation in (3) follows from

while the second is a consequence of the facts that

c,

at E

4

co(E)

holds isometrically (since both are isometrically in t , ( E ) ) and that c, @ E is dense in the Banach space c,(E): Just cut off the tail of the sequence! 0 One might be tempted to think that a description like that in (3) also holds for the space t p & E = c i o ( E ) if 1 5 p c m; but this is not the case. Take E = co and (2,) the following zero sequence in c,:

e1,2-1/pe212-'jPe2,3- '/peg, 3-'/pe3,3-'/~e3,

... ;

C(C,)

then (2,) E but ( z n ) 4 $*'(c,) . In Ex 8.4. a related question is treated: When is every (zn) E C (E) a zero sequence in E ? 8.3. A sequence (xn) in a normed space E is called unconditionally summable if for each bijective q : N -+ N the series C(zvcn,) converges; this is easily seen to be equivalent to the convergence of the net of finite sums

94

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

where I runs through the directed set of finite subsets of N.

PROPOSITION: Let E be a Banach space. Then a sequence (xn) in E is unconditionally summable if and only if it is in G ' " ( E )= I ! ~ & ~. E PROOF: Observe first the simple fact (decomposition in positive, negative, real, imagAnl 5 1 for all finite inary parts) that if (An) is a sequence of scalars such that ICnEI I C N ,then C,"=lIAnl 4. Now assume (xn) to be unconditionally summable. Then there is an no E IV such that llxnEI Zn 5 E for all finite I C {no,no 1, ...} and therefore lCnEI(t', zn)l 5 E for all these I and x' E BE' : it follows that

+

11

m

Viceversa: If w l

((.n)r!,,)

5 E , then for all finite I c {no,no+ 1, ...}

Bessaga-Pekzyriski proved (see Diestel [54], p.45) that a Banach space does not contain co isomorphically if and only if each weakly summable sequence is unconditionally

summable. Hence, it was not by chance that the example at the end of 8.2. was constructed in the space co. See also Ex 8.4.. 8.4. The following powerful tool will be used frequently:

Take a probability measure p on a set 52 and an orthonormal sequence of functions fn E L ~ ( / such J ) that span{ fn} C L p ( p ) for all 1 5 p < 00 and the norms coming from the L p ( p ) are equivalent:

Clearly, when investigating sequence spaces or function spaces these inequalities may be very useful. Additionally, the orthogonality has another quite interesting aspect: If z = C:=, x, @ y, is an element in the tensor product of two normed spaces, then n

n

8. Abaolutely and Weakly p-Summable Series and Averaging Techniques

95

- and the element z is represented in a new way as a sum/integral of elementary tensors; note that the Bochner integral in this formula exists in the finite dimensional space span{zm}@ span{y,). It will be seen that in many cases this new representation, together with the norm equivalences turns out to be extremely fruitful! The method is called the averaging technique. As is already indicated by the definition of type and cotype, averages of the form

for elements 2 1 , ...,z, in a normed space E will pIay an important role. Note that for Hilbert spaces H biorthonormality gives

(see also Ex 7.8.(b)). For calculations it is very helpful if the functions f, are of a simple nature, e.g. if 52 is a Cartesian product, p a product measure and f, the coordinate functions: The probabilist would say in this case that the (f,) are “stochastically independent random variables” - and this is why the averaging technique is often labelled a probabilistic method. Anyhow, this observation allows one to use powerful theorems from probability theory. There will be two examples used for averaging: the Rademacher functions 8.5. and the Gauss functions 8.7.. The Rademacher functions attain only values +1 and are actually defined on the countable product of a discrete space (as will be seen in a moment); in the case of Gauss functions R is RN or fl with the Gauss measure - and hence invariant under orthogonal transformations. 8.5. Rademacher functions: For every n E N define on D, := { - 1 , l ) ” the probability measure p, by pn({w)) := 2-” for every w E D,;it follows that p, is the 61) on D1 = {-1, 1). On D := {-1, l)N, n-fold product measure of p1 = 2-l(6-1 the countable product of the p1 is a probability measure p , which has, by definition, the following property: If f : D -+ C depends only on the first n coordinates, i.e.

+

f :D

projection

fn D,-C,

then

{-I, 1) (this will cause no The coordinate projections Ek : D -+ {-I, 1) or E k : D, confusion) have norm 1 in Lz(p) and L z ( p , ) and are biorthonormal: Take 6 < !5 n; then -+

&k(w)EL(W)p(dW)

=

k, [L1 wf

=

L*

WkWfpn(d(W1,

1

Wkpl(dWk)

pl(dWL) = 0

*

---I

wn))

=

96

8. Absolutely and Weakly p-Summable Seriea and Averaging Techniques

are called discrete Rademacher functions.

The functions

REMARK: If r k are the Rademacher junctions on [0,1] with the Lebesgue measure (see 1.5.(2)), E a normed space and zl,...,z, E E , then f o r all 1 Ip < 00

This follows from the fact that all Rademacher functions r k are constant on the intervals [m2-”, (m 1)2-”[ . The same argument gives that (A the Lebesgue measure)

+

for all a1,...,an E R and this means, in the language of probability theory, that the measure p, on 0, C R” is the joint distribution of the “random variables’’ ( r l ,...,r,). The basic inequality is the

KHINTCHINE INEQUALITY: For 1 Ip < 00 there are constants up and bp 2 1 such that

for all n E LV and a1,...,a, E a’.

This is false for the L,-norm, see Ex 8.18. (which also treats the case 0 < p < 1). The best constants ap and bp were calculated by Haagerup [96] in 1982; they are the same if one considers only real a k (see Ex 26.9.):

The number p o

%

1,84742 is the solution of r ( ( p

+ 1)/2) = &/2

in ]1,2[.

PROOF: Since the following proof (taken from Stein [260], Appendix D) will not give the best constants anyway, it is enough to check the inequalities for real ( Y k , the left

8. Absolutely and Weakly p-Surnrnable Series and Averaging Techniques

inequality for p = 1 and the right one for natural numbers p and put

> 2.

Take

97

a: = 1

n k=l

Then, for each natural number p

> 2 and A > 0

Since

it follows that ~ An estimate for

Taking A := f i ,gives bp 5 is a consequence of Holder’s inequality:

5

~ f ~ ~ ~ (2p!)l/PA-’ , , ~ p n ~ exp(A2/2p).

a.

5

8.6. For the Rademacher functions on [0,1] the Khintchine inequality implies that on the subspace Rad:= span{rn}c &[O, 11 all norms coming from L,[O, 13 are equivalent (1 5 p < m) :

fact which was already used in section 1; for p = 00 see Ex 8.18.. For p > 1 Rad is even complemented in L,[O, 11, me Ex 8.17.. It might be interesting to know that the only continuous functions f in Rad are the multiples of the function 1 - 2t (see Chatterji [26]). What happens if the scalars a k are replaced by ck in a normed space -a

98

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

E and the modulus by the norm? For p = 2 the definitions in 7.7. say that E is of cotype 2 if and only if the left inequality holds (for some constant) - and of type 2 if and only if the right one holds. Since there are spaces which have neither type 2 nor cotype 2 (for example C[O,11, see Ex 7.14.) there is no vector valued Khintchine inequality. However, the Kahane inequality says that there are constants K p such that for all normed spaces E and tl,...,zaE E

(for a proof see e.g. Lindenstrauss-Tzafriri [177], 11, p.74). By remark 8.5. the same holds for the discrete Rademacher functions. So it is important to have more examples of spaces with type and cotype. The Khintchine inequality will help. Moreover, in the following proof repeated use will be made of the “continuous triangle inequality” (see Appendix B5. and 7.2.)

PROPOSITION: Let ( $ 2 , ~be ) a measure space. (1) For 1 5 p 5 2 the space L p ( p ) i s of type p and cotype 2. ( 2 ) For 2 5 q < 00 the space L q ( p ) i s of cotype q and type 2. If L p ( p ) is infinite dimensional, then it is of no type larger than min{p, 2) and of no cotype smaller than max{p, 2}, see Ex 8.13..

PROOF: Discrete Rademacher functions will be used. A * will indicate the application of the “continuous triangle inequality”. If 1 5 p 5 2 and f1, ...,fa E L p ( p ) , then

which shows that Lp is of cotype 2; for type p

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

99

For 2 5 q < 00 the same type of argument applies - but we prefer to write them down as a chain of mappings: Here Rad, denotes the closed span of the Rademacher functions with the norm induced from L,(D). The continuous triangle inequality * will be used in the form of 7.2.:

and it follows that C , ( i d ~ , = ) 1. That L, has type 2 follows from

and T z ( i d L q ) 5 b, from the Khintchine inequality. 0 Of course, it makes no difference whether one writes the proof as a chain of inequalities or as a chain of maps. This is only a question of mathematical taste - we sometimes prefer maps. The proof showed that for all n E N

C Z ( $ ) _< a, C , ( c )= 1

and TP($)= 1 and Tz(G) 5 b,

for 1 5 ~ 5 2 for 2 5 q

< 00.

Ex 7.12. implies that for every 2i,A-space E C z ( E ) 5 AQ, C,(E) 5 A

and T , ( E ) 5 A and T z ( E ) 5 Ab,

for 1 I p s 2 for 2 5 p < 0 0 .

For the remaining type and cotype constants of P;: see Ex 7.13., Ex 7.14. and Ex 8.13.. It is obvious, by the proof, that one could have used other averaging methods instead of the Rademacher one to obtain that L, (as above) is of some type/cotype with respect to this method; this may improve the final constants in calculations for L, which use type/cotype constants. 8.7. Ga uss functions: The Gauss measure y,, on K" is defined by

100

8. Absolutely and Weakly p-Summable Series and Avemging Techniques

where = (27r)-"12 in the real case and cn = (27r)-" in the complex case (i.e. the Gauss measures on and R2"coincide); 7n is a probability measure - clearly the n-fold product of y1 . On K N the Gauss measure 7 is the countable product of yl on K ; as before, 8.5., functions on K N depending only on the first n coordinates can be integrated with respect to yn The coordinate projections gk : K N -+ K or K" 4 K are called Gauss functions; for K = R they have norm 1 in La(7) or L2(yn) and they are biorthogonal: if 1 5 k < 1 5 n, then

.

gkgfd7 = L

N

kn

gkgfdyn =

wk

(L

wf7l(dwf))Tl(dwk) = 0

.

Therefore the ( g k ) form an orthonormal system in Lz(7). If K = C, the Qk are biorthogonal as well, but have norm 4 - however, for our purpose there is no need to normalize them. Note that for a = (01, ...,a n ) E (P

and, by the transformation formula for the Lebesgue measure,

1,.

f ( ~ w ) 7 n(dw) =

1,.

f(w)yn (dw)

for all isometries A : t?j -+ 5 : The Gauss measure is invariant under isometries. This is the great advantage of Gauss averaging - as can already be seen in the proof of the following result of Khintchine type.

PROPOSITION: For 1 5 p < 00 and 0 1 , . . . , a n E K

PROOF: For every isometry A : 5 -+

5 and a E K"

by the invariance of 7,, under isometries. Therefore it is enough to prove the result for a=el: , It is routine work to calculate the constants:

8. Absolutely and Weakly p-Summable Series and Avemging Techniques

101

moreover,

It turns out that the real and complex constants are different - in some sense contrary to the case of the Khintchine inequality for Rademacher functions (but recall also that the gk are different in the real and complex case). As in the case of Rademacher functions, there is no vector valued Khintchine result for Gauss functions, and there is a type of Kahane inequality saying that on the subspace [span {gk}]@E all norms coming from Lp(7, E) are equivalent (see Marcus-Pisier [183], p.44). It can be shown that for all tl,...,zn E E

but there is no converse estimate (see Ex 8.9.); it is a deep result of Maurey and Pisier [187] that exactly the Banach spaces E which do not contain the Pm uniformly permit the other estimate. 8.8. The averaging technique will now be used to calculate various norms of the element ek @ ek E K" @ IK" ; considered in L(K" ,K")this is the identity operator. The

Holder inequality gives

PROPOSITION:

PROOF:E 5

T,

hence r( ...) 2 1. For the converse inequality Rademacher averaging

will be used:

COROLLARY:

102

8. Absolutely and Weakly p-Summable Series and Avercrging Techniques

PROOF: The mapping property for T gives

k=l

k=l

Since

(C@ r q)'=

@c

GI holds isometrically,

These three estimates give the result. 0 8.9. The cotype estimates serve nicely when investigating the following question: In

which Banach spaces is every weakly r-summable sequence absolutely s-summable? In other words, when is 4V4cW) ? The question goes back to a result of Orlicz 1933 [195] stating that in Lp[O,l]for 1 5 p 5 2 each unconditionally convergent series (see 8.3.) is absolutely 2-summable. A closed graph argument shows that c ( E ) c t z ( E ) if and only if there is a constant c with !,(Zn;E)

I cwl(zn;E)

- and clearly it is enough to check this inequality for finite sequences. So the question is whether or not sup ]lid @ id : n

@c

E

-

Ell < 00 ?

If E is a Hilbert space, a nice direct argument shows (Ex 8.10.) that this supremum is 1. Orlicz used Rademacher functions in his proof for L p ; let us do the same: First observe that

8. Absolutely and Weaklg p-Summable Series and Averaging Technique8

Now if E has cotype 2 the Rademacher average (for rk or 12 (zf ; E), hence

&k)

103

dominates, by definition,

PROPOSITION(OrliC2): If E has cotype 2, then every unconditionally convergent series in E is absolutely 2-summable; more precisely, for all X I ,...,znE E

eZ(zk;E)5 C Z ( E ) w l ( z k ; E )

*

By 8.6. the L p ( p ) (for 1 5 p 5 2) are of cotype 2 - hence Orlicz’ original theorem was also obtained (the work for the result was done when proving that the L, have cotype 2!). Maurey and Pisier [187]showed that each Banach space with the “Orlicz property” (stated in the proposition) has cotype ( 2 E ) for all e > 0, but Talagrand [266] recently constructed a Banach lattice with the Orlicz property which does not have cotype 2. Note that, by Ex 8.l.(c) the Orlicz property of E means that there is a constant c satisfying

+

for all 2 1 , ...,z, E E . In Ex 8.11. the reader finds a nice application of Orlicz’ theorem to pointwise convergence of series.

COROLLARY: If E has cotype 2 and 1/r 5 l / q - 1/2, then

i.e. every weakly q-summable sequence in E i s absolutely r-summable.

In particular] this applies to all L, with 1 5 p

5 2.

PROOF:For A = (An) E Biq, the diagonal operator D,+: t, therefore] by the proposition,

+ t1

has norm

5

which implies

for all z i , ...,znE E

for all r

2 ro . 0

. For

l/ro := l / q - 1/2 the left supremum is t r 0 ( Z k ; E), hence

1;

104

8. Absolutely and Weakly p-Summable Series and Averaging Technique8

The idea of Orlicz in 1933 to use the Rademacher functions and Khintchine’s inequality for the investigation of Banach spaces may be considered to be the beginning of the “modern” averaging techniques.

Exercises: E x 8.1. (a) Show that for all 2 1 , ...,2, E E

E x 8.2. If (cn)is weakly psummable and ( a n ) E c,, then (CrnZ,) E tp&E = q f o ( E ) . E x 8.3. Give an example of a zero sequence in a Banach space such that

is bounded but not relatively compact. Hint: 8.2.. E x 8.4. (a) Use the fact that every continuous operator c, -, tl is compact to show that (co & c,)’ = ll&t1. (b) Pitt’s theorem states that 2(tp,tq) = ff(tp,tq) whenever 1 5 q < p < 00. Use this = if and only if p‘ > q. Hint: 8.2.(1). to show for p, q E]1,00[that $(!,) (c) Show that e ( E ) C c,(E) if and only if e ( E ) = c e o ( E ) Hint: . Use the BessagaPekzyliski result mentioned at the end of 8.3.. (d) Every block basis sequence (yn) with respect to the unit vector basis ( e n ) in t,, (this means ark E span { e n k + i ,. . . , G , ~ + ~for } some 0 = nl < n2 < ... and llykll = 1) is weakly p’-summable. (e) Bessaga-Pelczyfa’ski selection principle: Let ( e n ) be a basis of a Banach space E with coeficient functionals ( e ; ) . If (yn) is a sequence such that JJy,,Il 2 c for some c > 0 and lim (eh,yn) = 0

cfo(t,)

n-m

8. Absolutely and Weakly p-Summable Series and Averaging Techniques

105

for all m E N ,then (yn) has a subsequence (yW(,,)) which is equivalent to a block basis sequence (z,) with respect to (en)' i.e. C(Qntn) converges if and only if C ( a n y q ( n ) ) converges. For a proof see Diestel [54], p.46. Use this to show the following result, which is due to Castillo [25]: Take 1 < p < 00. Then $(E) C c,(E) if and only if q ( E ) = q + ' ( E ) . Hint: An operator T : .fpt E is compact if and only if ~~2'yn~~+ 0 whenever yn converges weakly to zero. E x 8.5. Give examples of unconditionally summable but not absolutely summable sequences. --+

E x 8.7. Use Khintchine's inequality to prove that for all 21,...,2, E el C 4 .fl(zk;ez)

Ih%(zk;el)

*

Ex 8.8. Let 0 : 0,-* Dn be bijective; then for all f : Dn

a'

E x 8.9. (a) Show that for all normed spaces E and elements

21,

..., 2, E E

Hint: Integrate 11 x E k ( W ) g k ( V ) Z k I ( P = 11 c & k ( w ) signgk(v)lgk(u)lzkllp with respect to dyndpn and dpndyn; Ex 8.8. and 11 J 11% i J 11 11% . (b) Show by using the unit vectors in P , that there is no converse inequality for any 1 I p < 00. Hint: Use the fact that max{lgk( k I n} converges to infinity y-a.e. in

I

RN. Maurey and Pisier 11871 showed that for a Banach space E the converse inequality , uniformly. holds (with an additional constant) if and only if E does not contain the P Note that, by Kahane's inequalities for the Gauss- and Rademacher functions, this need only be checked for one p. E x 8.10. Let H be a Hilbert space and (zn) a sequence in H . Use the parallelogram identity to construct by induction a sequence (tun) in {-l,l} such that 11 qzkl12 2 11Zk112 for all TI. Deduce that every unconditionally summable sequence in H is absolutely P-summable. E x 8.11. (Orlicz) Show that for every unconditionally convergent series in L l ( Q , p ) the series C(lfn(z)i2) converges for p-almost all 1: E St . Ex 8.12. (a) For 1 5 s 5 r _< 00 call T E C ( E ,F ) absolutely (rl s)-summing if there is a constant c 2 0 such that

c;=,

c(fn)

.fr(Tzk;F ) I CWa(Zk; E )

106

8. Absolutely and Weakly p-Sumrnoble Series and Avemging Techniques

for all finite sequences (xk) in E; the smallest such c is denoted by Pr,#(T).By using the idea of 8.9. show that Pr,g(T)5 C2(T) if l / r _< 1/s - 1/2 .

(b) Why does the definition of absolutely ( r ,s)-summing operators only make sense for s 5 r ? (c) Show that T E C ( E , F ) is absolutely (r,s)-summing if (Txk)E tr(F) for all sequences (zk)E Gt"(E). Ex 8.13. (a) Show with the unit vectors e k that

(b) Deduce that infinite dimensional L p ( p ) do not have type for any r have cotype for any s < p .

> p and do not

Ex 8.14. (a) If 1 5 p 5 2 and T E C ( E ,F), then

= J ( C E ~ & . , C Q or T dualize Z ( ) ~the ~ ~tensor product Hint: Use that C(T'&.,xs) characterization. (b) Show that the converse inequalities are false - even with an additional constant. = .', . See Pisier [223] and also section 31 for positive results if T = idE . Hint: (c) With the weak principle of local reflexivity show that Cp(T")= C , ( T ) and Tp(T")= Tp(T)

for all T E C ( E , F ) . Hint: Use 4 E B p with (T"zg,y',)M ~ ~ T "forz the ~ ~type ~ ;use discrete Rademacher functions and the same idea.

Ex 8.15. For each finite set A c N define the Walsh function

if A # 8,and E ) := 1. Show that { E A I A c W finite 1 is an orthonormal basis of Lz(D,p). Hint: Do this first for D,. Ex 8.16. For 1 5 p 5 00 and elements 21,...,x, in a normed space E use the convexity of the function

(and Ex 8.8.) to show that (with CR = 1 and cc = 2)

8. Absolutely and Weakly p-Summable Seriee and Averaging Techniques

107

for all a1, ...,a, E P( . This is a very special case of Kahane's contraction principle (see e.g. Marcus-Pisier [183], p.45).

Ex 8.17. For 1 < p < 00 the space Rad of Rademacher functions is complemented in Lp[O,I]. Hint: L, L* L2 -+ e 2 if p > 2; for 1 < p < 2 define a projection on L, n L2 by Pf = C(flrn)rn, estimate IIPf(lp5 $(C J(fJrn)12)1/2and apply Holder's and again Khintchine's inequality to

to aee that P is continuous on L, fl La C L,. The Rademacher functions are not complemented in L1[0,1], see Appendix C7.. l a k l for real 0). In the complex Ex 8.18. (a) Show that 11 Crkrkll&~o,l]= case these two norms are equivalent. Note that (in the real case) this result provides an isometric embedding Q C , l!g .

(b) Modify the second part of its proof to show that the Khintchine inequality holds also for 0 < p < 1 .

Ex 8.19. Take zi,...,zk E E'. Show that

where the latter minimum is taken over all factorizations

with diagonal operators DX . It is enough to take Dx with non-negative entries.

108

9. Operator Ideals

9. Operator Ideals The notion of a Banach operator ideal is simple but nevertheless important since it structures the way of thinking and hence also the proofs. It was Pietsch who in 1968 noticed the relevance of this notion, and in the following years he and his school investigated all aspects of the abstract theory of operator ideals - a research which culminated in the incredibly rich monograph “Operator Ideals” in 1978. In this section we collect some (not all) of the fundamental notions of the theory which we will need later -on.

9.1. An operator ideal 31 is a subclass of the class C of all continuous linear operators between Banach spaces such that for all Banach spaces E and F its components

%(E,F ) := C ( E ,F ) n 21 satisfy (1) %(ElF) is a linear subspace of C(E,F ) which contains the finite rank operators. (2) The ideal property: I f S E 21(Eo,F0),R E C(E,E,) and T E C(F,, F ) , then the composition TSR is in %(E,F ) .

The ideal property of certain classes of operators has already been used quite often. The ideal 5 of finite rank operators is obviously the smallest operator ideal and C the largest one. Note that for the theory of operator ideals we are only considering operators between Banach spaces. 9.2. For a good analytic theory of operator ideals one needs convergence in 2l. Recall that a function 11 : E -+ [0, m[ on a K-vector space E is said to be a quasinonn if there is a constant c 2 1 such that

If E is complete with respect to the uniform structure coming from the quasinorm, it is called a quasi-Banach space. For 0 < p 5 1 a p n o n n is a quasinorm satisfying

(3’)

IIZ + YllP 5 ll~llp+ IIYIIP

for all Z,Y E E ;

in this case c := 2’IP-l may serve as a constant in the quasi-triangle inequality (3). If (E, 11 11) is quasinormed with constant c and p ~ ] 0 , 1such ] that c = 2l/p-’, then

9. Operator Ideals

109

defines a p n o r m on E satisfying

llzllo I 1141I 2”’11~110

for all z E E,

hence for every quasinorm there is an equivalent p n o r m . We omit the tricky proof since it is not important for our purposes; see e.g. Pietsch [214], p.92.

9.3. A quasinormed operator ideal (%,A)is an operator ideal % together with a function A : ?2l+ [0, CQ[ such that

(1) AI~x(E,F) is a quasinorm f o r all Banach spaces E and F. (2) A ( i d ~ = ) 1. (3) If S E %(E,, F,), T E 2(F,, F ) and R E 2(E, E,), then the composition satisfies A(TSR) I IITIIA(S)llRII. If all the components %(E,F) are complete (with respect to A) ,then (a,A) is called a quasi-Banach operator ideal. A simple argument (with lp-sums) shows that the quasitriangle constant may be chosen independently from (E, F);therefore the construction in 9.2. shows that there is always a p E]O,1] and an equivalent A, such that (a,A,) is a quasinormed operator ideal and A, a p n o r m on each component. The advantage of a p n o r m is that it is continuous (for the topology it generates) while a quasinorm need not be! See Ex 9.2. for an example of this “quasinorm catastrophe”. If A is a norm (resp. p-norm) on each component %(E,F), then (2, A) is called a normed (resp. p-normed) operator ideal. In the case that the components are complete one speaks of Banach and p B a n a c h operator ideals.

The following statements hold f o r each quasinormed operator ideal PROPOSITION: (%,A):

llTll 5 A(T) for all T E Q . A(z’&y) = 11x/11 llyll f o r all x‘ E E’,y E F.

(1)

(2)

PROOF:(1) follows from the fact that for all z E BE and y’ E B F ~

and (2) with the same idea of factorization: 1121 ‘1

IlYll = llZt&Yl1

I A(Z‘&Y) = A((idKBY) 0 idK 0 4 I

I l l ~ ’ l l ~ ~ ~ ~ K=~1 l1 4 l1 Y IlYll ll . 9.4. It is rather simple to check the following very useful

CRITERION: Let be given a subclass Q of 2,a function A : 31 + [0,CQ[ and 0 < p 5 1. Then (a,A) is a p-Banach operator ideal if (and only i f )

110

9. Operator Ideals

n=l

Observe that (3) accumulates linearity of the components, the triangle inequality and completeness. 9.5. A norm on a vector space E is good for analytic investigations of this space if it is unique, up to equivalence, under all “reasonable” norms; if E is complete under these norms, the closed graph theorem usually helps. The ideal norm on an operator ideal is always “reasonable” in this sense since it dominates the operator norm:

PROPOSITION: If (a,A) and (%, B) are two quasi-Banach operator ideals such that TZL C 23, then there is a constant p > 0 such that for all T E Q

PROOF:Since the embeddings Q(E,F) and B(E,F) into 2 ( E ,F) are continuous, the closed graph theorem (it holds for quasi-Banach spaces) shows that there is a constant ~ E , F such that B ~ E , F A on %(ElF). An obvious indirect argument with tp-sums shows that the P E , F can be chosen to be uniformly bounded. 0 The reader who is not used to the “obvious” argument with ep-sums (metric injection of the En’s into tp(Em),metric projection, ideal property) should check the details. 9.6. Examples:

(1) The approximable operators a Banach operator ideal

(3,II 11);

3 (see 4.2.(1))

together with the operator norm form

recall

and that z ( E , F) = ff(E,F) if E’ or F has the approximation property (see 5.3.). The ideal ff of compact operators and the ideal ?IDof weakly compact operators form (again with the operator norm) a Banach operator ideal; ideals of this type are called closed operator ideals (=: their components are closed in X ( E , F)). A famous theorem of Calkin states that the only proper (i.e. different from 2 and (0)) closed ideal in the algebra .C(t2,e2) is the ideal of compact operators; this is also true for c, and ep (if 1 5 p < w), but not for Lp[O,I] (if 1< p < 00 and p # 2, see Pietsch [214], p.84). Weakly compact operators satisfy an important factorization theorem due to Davis, Figiel, Johnson and Pelczyriski 1371: If T E m(E,F), then there is a reflexive Banach

9. Operator Ideals

111

space G and operators R E C(E,G) and S E 2(G,F) such that T = SR and IlTll =

IlSll IlRll : T

E-F G For a proof, including the relation about the norm, see Pietsch [214], p.55. (2) The class % of nuclear operators (see 3.6.) with the norm N is a Banach operator ideal: Completeness is clear since N is the quotient norm of

the ideal property is easy to verify (and corresponds to the mapping property of the projective norm) and N ( i d ~ )= 1 (- it is just a joke to argue for this with K = K = % ( K ) ) By . the very definition of the nuclear norm and the fact that K

for all operator ideals, it follows that (%,N)is the smallest Banach operator ideal. Recall that the nuclear norm of a Glinear operator may increase if it is considered as an BZ-linear operator; this follows from N ( i d ~= ) dimK E (see 3.7.).

(3) Let (Oi,pi) be two measure spaces, p , q E [ l , w [ , G a subspace of &(PI) and S an operator in & ( G , L , ( p 2 ) ) of norm 1. Take Cs to be the class of all operators T E C ( E , F) such that @ T : G @A, E

+

Lq(pLz)@ A ~F

is continuous. Together with Ls(T) := IlS@T : @ A + ~ @A,II the class 2 , is a Banach operator ideal (this is an exercise). As a special case one gets: The ideal ('Xp,Tp) of type p operators and the ideal (C,, C,) of cotype q operators are Banach operator ideals. Other ideals of this type and results about 2s are treated in Ex 9.15., Ex 9.16. and 17.4..

-

9.7. An operator ideal Q is said to be injective if for each metric injection I : F G an operator T E C ( E ,F) is in Q if I o T E 2l. A quasinormed ideal (a,A ) is called injective if moreover, A(T) = A(I o T ) . Clearly, the Banach ideals of compact and of weakly compact operators are injective, but the ideals of approximable and of nuclear operators are not, see Ex 3.17., Ex 5.13. and Ex 9.13. for the nuclear operators and Ex 9.12. for the approximable ones.

9. Operator Ideals

112

PROPOSITION: Let (24, A) be a quasinormed operator ideal. Then there is a (unique) smallest injective operator ideal 24'"j which contains 24. If IF denotes the canonical injection F 4 e , ( B F ~ ) , then T E C ( E ,F ) belongs to mini if and only i f IF o T E 24. With Ainj(T) := A(IFT) (2l'"J,A'"]) is a quasinormed operator ideal which is quasi-Banach if (24, A) is quasiBanach.

PROOF:Define ai"j to be the class of all T E ' C ( E , F) such that IF o T E 24 and A'"j as in the statement. Then it is easily seen that (24'"j,Ai"j) is a quasinormed

operator ideal (for the ideal property use the metric extension property of l,(r)) with complete components if % has. If I : F 4 G is a metric injection, the diagram

shows

A'"~(T)= A ( I F T ) = A ( P I G I T ) 5 I ~ P ~ ~ A ' " ~5( IA'"~(T) T) , hence (24inJ,Ai"j)is injective. If 23 3 24 is any injective operator ideal, then clearly 93 3 24'"j and the proposition is proven. 0

(Dinj,A'"j) is called the injective hull of (24, A). In Ex 9.13. it will be shown that 92'"' is the class of quasinuclear operators.

COROLLARY: If the Banach space F has the X-ettension property and (%,A) is a quasinonned operator ideal, then B(E, F ) = %'"j(E,F ) for each Banach space E and A(T)5 AA'"j(T) for all T E 24'"J(E,F)

.

This is immediate. 9.8. The dual concept is the following: An operator ideal 24 is called surjective if for each metric surjection Q : G --y E an operator T E C ( E ,F) is in 24 if T o Q E 24. A

quasinormed operator ideal (%,A)is called surjective if moreover, A(T) = A(TQ). The Banach ideals R . and !2D are surjective, 5 is not (Ex 9.12.) - and also the nuclear operators do not form a surjective ideal; this may be seen as in Ex 3.17. or by the following argument: Take a metric surjection Q : G 4 E and F with the approximation property such that id^ is not an isomorphic injection, for example (see 1.5.)

113

9. Operator Ideals

I : tzL, L1[0,1] the embedding of the Rademacher functions, Q := I' : L , F = ez. NOW

+t 2

and

commutes; since the subspace %(G,F ) t l imO is closed in %(G,F), it cannot coincide with O ( n ( E ,F ) ) by the closed graph theorem; in other words, there is a non-nuclear T E C ( E ,F ) such that TQ E %(G,F ) : The ideal % of nuclear operators is not surjective.

PROPOSITION: Let (%, A) be a quasinormed operator ideal. Then there is a (unique) smallest surjective operator ideal which contains %. If QE is the canonical surjection el(BE) --y E , then T E L!(E,F ) belongs t o %"' if and only if T o QE E a. With A'"'(,) := A(TQE) (Be"',A'"') is a quasinomed operator ideal which is quasi-Banach if Banach.

(a,A) is quasi-

PROOF:The argument follows the same pattern as that in the case of the injective hull - the metric extension property is replaced by the lifting property of tl(I'), see 3.12.; the central point is to show that %'"', defined as indicated in the statement of the proposition, is surjective: For every E > 0 there is a lifting S, of QE

hence

+ E ) IAdU'(T)(l+

A"'(T) = A(TQE) = A(TQQGS,)IAsU'(TQ)(l

E)

and this gives AsU'(T)= AdU'(TQ).0 (%(""', A'"') is called the surjective hull of

(a,A) . The same diagram gives the

COROLLARY: If (%,A) is a quasinorrned operator ideal, then for all sets Banach spaces F %(el(r),F ) = n y e l ( r ) ,F )

r

and all

114

9. Operator Ideals

holds isometrically.

A description of Wurwill be given in Ex 9.14.. 9.9. If

(a,A) is a quasinormed operator ideal, then %dual

._ .- { T E 1: I T' E a}

Adual(T):= A(T') defines a quasinormed operator ideal, the dual ideal of (a,A). If (a,A) is quasiBanach, then (21dua1,Adua1) is as well. Well-known results state: R = Rdual and ID = IDdua1; ideals Q with a = adua' are called completely symmetric. !Yl c !TIdua1 (see 5.9.), but % # ndua1 as will be seen in 16.7.. The approximable operators 3 are completely symmetric, see Ex 9.12.. 9.10. Finally, the notion of the product (or the composition) of two ideals 2l and % will be needed: T E &(E,F ) belongs to the product 2l o (23 of Q and '13 if there is a Banach space G and R E %(ElG) and S E a ( G , F) such that

E-F

T

G If % and % are quasinormed, then

A o B(T) := inf A(S)B(R) , where the infimum is taken over all such factorizations. It is not difficult to see that (a o '13, A o B) is a quasinormed operator ideal. The completeness (if IZI and 23 are) follows from the

PROPOSITION: If (a,A) is a q-Banach and ((23, B) a p-Banach operator ideal, then (Q o % , Ao B) is an r-Banach operator ideal where 1/r = l/p + l / q .

PROOF:Only property (3) of criterion 9.4. needs to be checked: For a sequence (T,) in rzL o %(El F ) with 00

n=l

choose factorizations T, = S, o R,, through G, with

B(R,,)'I (l+&)[AoB(Tn)]' A(&)' I(1 E ) [A 0 B(Tn)Ir

+

115

9. Opemtor Ideals

and put G := lz(Gn) (denote by I,,, and Qm the canonical injections and projections). Again by the criterion

n=l

with

n=l

C,"=lTn = SR and

n=l

hence ( A o B)(CZ==,Tn) 5 [(1+

E)

C:==,[A

0

B(Tn)]"]"' - for all

E

>0 .

Unfortunately, the product of two normed operator ideals need not be normed. For example, if 3 is the ideal of integral operators (to be defined in the next section), the composition ideal 3 o 3 is not normed (see 29.8. for an argument showing this). Moreover, %o% is not normed; this followsfrom the facts that operators in %o%(E, E) have summable eigenvalues, but nuclear operators have, in general, only square-summable eigenvalues (see e.g. Konig [147], 2.b.14. and 4.a.5.) and 'YI is the smallest Banach operator ideal.

Exercises: Ex 9.1. If

11 11 is a quasinorm with triangle inequality constant

Ex 9.2. (a) If (E,11 11) is p-normed, then (b) For S E 1:define

11 11 : E

-*

c,

then

R is continuous.

Then (2, A ) is a quasi-Banach operator ideal, but A is not continuous on 2 ( E ,E) (with respect to the A-convergence) if E is infinite dimensional. Ex 9.3. Check that the ideals 'XI of completely continuous operators ((Tzn)is norm convergent if (zn) converges weakly) and X of operators with separable range are closed Banach operator ideals.

116

9. Operator Ideala

E x 9.4. If 'I has a sufficiently large cardinality, then the algebra C(!Z(I'),&(I')) has infinitely many closed ideals. Ex 9.5. If T E ff(E,F ) and n E IV,then there are reflexive Banach spaces E l , ...,En and operators To E ff(E,El), T k E f f ( & , E k + 1 ) and T n E R(En, F ) such that T = Tn o ... o To and = IlTnIl ...I1Toll. Hint: 3.5., corollary 2. E x 9.6. Let 21, ...,z,, y1, ...,yn E K and M the matrix ( z k $ J ( ) ; , f = 1 considered as an operator K" 4 K". If (a,A) is an operator ideal, calculate A(M : + Ex 9.7. If (%,A)and (B,B) are quasi-Banach operator ideals, R E C(Eo,E) and S E C(F, F,) such that % ( E ,F ) WEO,Fo) T SoToR

C).

--

is defined, then this map is continuous. E x 9.8. If % is a quasinormed operator ideal, the class space(%) of all Banach spaces E with i d E E % is called the space ideal of a. Show that it is stable under isomorphic bijections, complemented subspaces and finite Cartesian products. If % is injective (resp. surjective), then space(%) is stable under forming subspaces (resp. quotients). Ex 9.9. For T E C ( E ,F ) put where the infimum is taken over all factorizations T = SR through any Hilbert space. Show that the class 2 2 for all operators with Lz(T) < 00 is (with Lz) an injective and surjective Banach operator ideal. Ex 9.10. Let 24 be an operator ideal, T E C ( E ,F ) and G a Banach space. (a) Let S E % ( G , F )such that im T c im S . If S is injective or % surjective, then T E %. (b) Let S E %(E,G)such that (lTc11 5 cIlSeII for some c 2 0 and all z E E. If S has dense range or % is injective, then T E %. Ex 9.11. Let (%,A)be a quasi-Banach operator ideal such that for every metric injection I the operator T is in % if and only if I o T E 8 .Then 8 = ainjand A and A'"' are equivalent quasinorms. E x 9.12. (a) Use the fact that there exist Banach spaces without the approximation property to show that the ideal 3 of all approximable operators is neither injective nor -8uc surjective. Moreover, $nJ = 3 = ff. -dual

-

(b) Show that f is completely symmetric: 5 = 5. Hint: If T E ff(E,F) and S E $(E,F") such that llT - sll I E , choose an e-net T x k of TBE and apply the principle of local reflexivity to span{imS U { T e a } } . ( c ) Deduce from (b) that f is regular, i.e.: if ICF o T E z ( E , FN),then T E $ ( E , F). Ex 9.13. An operator T E C ( E ,F ) is called quasinuclear if there exist ( z i ) E t!,(P) such that for all z E E n=l

117

9. Operator Ideals

(a) With Q N ( T ) := inf{C,"=l IIziII 1 (2;) as above } the class Q9l of quasinuclear operators is an injective Banach operator ideal. (b) If T or T' is nuclear, then T is quasinuclear and Q N ( T ) 5 N ( T ' ) . (c) Give examples of quasinuclear, non-nuclear operators. Hint: Ex 3.17. and Ex 5.13.. (d) If F has the X-extension property, then Q%(E, F) = %(E, F ) and N ( T ) X Q N ( T ) . Hint: Factor T through a subspace of L1 and extend. QN). (e) ( ( n i n j , N i n j ) = (f) Quasinuclear operators are compact. (g) Use the fact that there is a metric surjection Q : G --* F such that Q' @, idt, is not an isomorphic injection (since L, is no Ci-space by 8.6. and Ex 7.14., this will follow from 23.5.) to show that Q9l is not a surjective operator ideal.

(a%,

(h) Q%(co, F) = 91(co, F) holds isometrically. Hint: Ex 3.13.. Ex 9.14. Show that T E C ( E ,F ) is in the surjective hull ndur of the nuclear operators if and only if there is a sequence (yn) E t l ( F ) such that

Hint: Ex 9.10..

Ex 9.15. (a) Show that the operator ideal (Cs,Ls) defined in 9.6.(3) is always injective. If G = L p ( p l ) , then (Cs,Ls) is even surjective. Hint: 7.4.. (b) What does (a) imply for the ideal 2, of type p and the ideal Cq of cotype q operators? Show that Cq is not surjective; in particular, if G C L p ( p ) , then G @ A ~ does not in general respect metric surjections. (c) Let 1 < p < 00. If T is an operator which factors through a subspace of a quotient of some Lp(v)(or: through a quotient of a subspace of some Lp(v)-this is the same, as will be seen later on), then for all S E C ( L P ( p 1 ) L , p ( p 2 ) ) the operator

s @ T : Lp(P1) @ A p E

-

Lp(P2) @Ap F

is continuous. Hint: Fix S, consider first T = i d L p and use (a). It is a celebrated result of Kwapieli that these T are the only ones which make S @ T continuous for all S as above; we shall prove this in Chapter 111, 28.4.. Ex 9.16. Denote by R the orthogonal projection of L z ( D , p ) onto the closed linear span Rad of the discrete Rademacher functions (see 8.5.). (a) If En := L1(Dn, pn), then

llR @ j d E , : L2(D,p ) @A2 Hint: Use the function f := inequality.

En

-

nF=,((l+ Q )

L2(D, p ) @Aa En11 2

8

*

E L z ( D , p ) @ En and Khintchine's

@J ~ 6 )

118

10. Integral Opemtors

(b) Deduce that

R €3 idt, : La(D,P) @

A ~ti

-

L2(D1P) @ A ~t i

is not continuous. (c) Following Maurey and Pisier [187], an operator T E S ( E , F) is called K-conwez if

K(T) := llR @ T : Lz(D,P) € 3 E~ -* ~ L2(D,P) @ A ~Fll< 00

-

Show that the class of K-convex operators together with the norm K is an injective and surjective Banach operator ideal. It follows that a space which contains the uniformly is not K-convex; it is a deep result of Pisier [223]that the converse is true. See section 31 for more information about K-convex operators.

Ex 9.17. Show that Q o (!I3 o C) A (Q o 23)o C for quasinormed operator ideals Q, '13 and

e.

Ex 9.18. Let (a,A) be a quasinormed operator ideal. Then ('zr(E,F),A) is complete if E or F is finite dimensional. Hint: llTll 5 A(T) 5 IITIIA(~~F).

10. Integral Operators Grothendieck's integral operators are related to the injective tensor norm in the same way as the continuous operators are related to the projective tensor norm: the associated (bi-)linear form on E €3 F' is continuous with respect to E (instead of r). This section gives the basic factorization theorem for integral operators and some applications. Many of the arguments will be the same as those given in Chapter I1 where the study of maximal operator ideals using tensor norms will be taken up. So the present section is partly a training program for later on.

10.1. Recall that the continuous bilinear forms on E x G (i.e. the continuous linear forms on E & G) are exactly the continuous linear operators E + G':

119

10. Integral Operators

Every operator T E L!(E, F) defines a continuous linear functional

'p

on E & F' by

and hence 'p = P K F oE~(E @= F')' is the functional associated with K F o T : E .-,F". Recall that (E @t G)' C (E @% G)'. An operator T E C ( E ,F) between Banach spaces is called integral if its associated linear functional P n F o on ~ E 8 F' is continuous with respect to the injective norm E (i.e. is integral in the sense of 4.6.). Define the integral norm by

Then the class 3 of integral operators with I as a norm is a Banach operator ideal; the ideal property of J comes from the mapping property of E , obviously I ( i d ~ = ) 1 and it remains to show the completeness. Since (E @c F')' + ( E a,, 3'')' = C ( E ,F") is continuous and C ( E ,F) c C ( E ,F") is closed, the space

3 ( E ,F ) = ( E ~3~F')'

f l

C ( E ,F )

(the intersection taken in C ( E ,F")) is closed in the Banach space (E acF')', hence complete (with respect to the induced norm).

PROPOSITION: J(E,F') = ( E Qc F)' (via T u* pT) holds isometrically for all Banach spaces E and F . PROOF:T E C ( E ,F') defines a functional 'p = pT E (E @= F)'. The canonical right-extension 'p" of 'p to E @ F" (see 1.9.) satisfies

: extension lemma 6.7. for the injective norm e says that hence 9" = P K F r o ~The cp E (E F)' if and only if 'p" E (E @c F")' (with equal norms). This is exactly the statement of the proposition. 0 10.2. One should always keep in mind the tensor product representations of 3 ( E ,F) and J(E, F'). These representations imply the

COROLLARY 1: T E 3 ( E , F ) if and only if coincide: I(T) = I(KFo T).

KF

o T E J ( E ,F"); moreover, the norms

Ideals with this property are called regular.

COROLLARY 2: T E J ( E ,F ) if and only T' E 3(F', El); moreover, I ( T ) = I(T').

120

10. Integral Operatore

In other words, (3dua',Idu0') = (J,I) by the definitions in 9.9..

PROOF: If T E C ( E ,F) and

hence

'p'

'p

is its associated functional on E @ F', then

= TI . Since, clearly,

I I ~ I ( F J ~ , E ) # =IIPII(E~~F~Y the result follows from the proposition. 0

10.3. The definition 3 ( E , F) = (E@.,F / ) / n C ( E F) , of integral operators, the isometric equality F'@., E = 5(FlE) and the trace duality 2.6.imply that T E C ( E ,F) is integral if and only if there is a constant c 2 0 such that

clearly, I(T) is the minimum of these constants c . 10.4. Some examples: If E or F is finite dimensional, the duality of E and 6.4.) gives that %(E, F) = El @r F c* ( E a t F')'

T

(theorem

holds isometrically. Since C ( E ,F) = Z(E, F ) = E' @ F in this case, one obtains the

PROPOSITION: If E and F are Banach spaces one of which is finite dimensional, then for all T E C ( E ,F ) %(ElF ) = J ( E , F ) and N(T) = I ( T ) .

The results in 8.8. imply

EXAMPLE: The canonical injection I : -!I PROOF: It has to be shown that

'p

L,

: 11 @8 -!I

co is integral with integral norm 1

-

K defined by

.

121

10. Integral Opemtors

is continuous with norm 1 . Since p is clearly continuous on l 1 18C1 % with norm 1, it is enough (by the density lemma 7.3.) to check the &-continuity of p on the ?r-dense Q I8 : Since it was just seen that I(id : &) = 1, the result subspace follows. 0

U;=,

-

e

The ideal property implies that

which later in this section will be seen to have interesting consequences. Which operators are not integral? Integral operators are weakly compact (see Ex 10.1.) - this provides a good number of examples; the product of two integral operators is nuclear (this will follow from the corresponding property for absolutely 2-summing operators) which gives further examples. To see a more concrete example of an operator, recall the Fourier matrices from Ex 4.3.: They provide operators

A, : P , with llA,II

-

5 n and I(A,) = N(A,) = n3I2. Take 1 < d < f i and define T, := (2d)-,Ap

:

Then llTrnll 5 d - , and I(Tm) = (4. d-l)m c, el

lz

-+

-*

00;

2m

.

since

= co(t,2m ;m = 1,2, ...) = tl(e:"; m = 1,2, ...I ,

the operator T := @Tm: co -+ can be defined: it has norm 5 (d - l)-', but is not integral since the integral norms of T, = &,TI, (canonical injection and projection onto the m-th component) tend to infinity.

10.5. The integrating form /?I : ( j ,i )-+

f g d p on L , (Q, p ) C%

JL (Q, p ) has norm

p(Q) by 4.7., hence, by definition, the canonical embedding

I :L o ( P )

&(PI

is integral and I(I) = p(Q) for all finite measures p. If p is a Borel-Radon measure on a compact space Q, this also implies that

I : C(Q)

-

Ll(Q,P)

is integral and I(I) 5 p ( Q ) = 11111 5 I(I), hence this canonical map also has i n t e gral norm p(Q). These are the typical integral mappings - this will follow from the description of the elements of (E BCF)' as integrals.

122

10. Integral Operators

THEOREM: For a n operator T E C ( E ,F) between Banach spaces the following statements are equivalent: (a) T is integral. (b) There are a Borel-Radon measure p on a compact set R, operators R E C ( E ,C(R)) a n d S E Z ( L l ( p ) , F") such that

is commutative. (c) There are a finite measure space ( a , p ) , a n operator R E C ( E , L , ( p ) ) and a n operator S E C ( L l ( p ) ,F") such that

is commutative. Moreover, in both cases (b) a n d (c) the measure p and the operators can be chosen i n such a way that llRll= llSll = 1 (if E and F # (0)) and p(R) = I(T). In particular, it follows from the ideal property and llIll = p(R) = I(I)that

I(T) = minllRll

IlSll llIll

with factorizations as in (b) or as in (c).

PROOF:Only (a) c-+ (b) is not obvious: If 'p E (E BCF')' is the linear functional associated with T E 3 ( E ,F), then corollary 4.7. of the representation theorem for integral bilinear forms gives a Borel-Radon measure p on a compact Q with p(R) = I(T), and operators R1 E C(E,C(R)) and R2 E C(F',C(R)) with llRlll = llR2ll = 1 such that (with I. : C(Q) L,(R,p) canonical)

-

10.

Integral Operators

123

hence KFT= S I R which is the desired factorization. 0 This factorization theorem is very important for the understanding of integral operators.

10.6. The canonical maps L , ( Q , p ) Even more can be said:

-

L l ( Q , p ) and C(Q)

--+

Li (Q , p) are integral.

PROPOSITION: Let K be compact, p and v arbitrary measures. If E = C ( K ) o r L M ( p ) , then every positive operator T E 2 ( E , Ll(v)) is integral and I ( T ) = IlTll . This is also true for sublattices of El see Ex 10.5.. For the proof note first that theorem 7.3. implies that for all Banach spaces G

Therefore the conclusion follows from the following extremely useful tensor product characterization of integral operators:

10.7. THEOREM: A n operator T E 2 ( E ,F ) between Banach spaces is integral if and only if for all Banach spaces G (or only G = F')

is continuous. In this case,

PROOF:If T E J ( E ,F ) , then for each

'p

E ( F @r G)' = 2 ( F , GI) and

z

E E

@

G the

formula (PIT @ i d G ( z ) ) = (PL,oT, z )

holds (check for elementary tensors) and hence, since L, o T is integral,

which shows: x(T @ i d G ( z ) ;F I G ) I I(T)&(z; E , G ) - this is one direction of the statement of the theorem. Assume, conversely, that T @ i d p is continuous as indicated and let ' p E~ ( E @ r F')' be the functional associated with T : For z E E @ F'

(VT,2) = (h, T @i d ~ ~ ( . t ) ) (again a check for elementary tensors), hence

124

10. Integml Operators

with c := llT@i d p : E BEF' -, F was missing from the proof. 0

F'll. It follows that I(T) 5 c and this is all what

10.8. Another nice application of this characterization is the following

PROPOSITION: Let E be a Banach space, (zn) E q ( E ) and summable. Then

(2:)

E e ( E ' ) weakly

00

C I(zk,zn)I I

~ ( 2 ; E')wl(zn; ;

E)

*

n=l

PROOF: First recall that by 8.1. n

€(C.k@ p r ; G , G ) = wl((Yk);=l;G) k=l

for

91, ...,yn

in a Banach space G. Since I(q C?JC E

by the theorem, it follows that for

21,

E

and a shows that

It follows that for all zi,...,: z E E'

which implies the result. 0

COROLLARY: Ifp E %ii(co, co), then 00

n=l

PROOF: First note that

Pw) = 1 by 10.4. and hence

e", @ r Ell 5 1

...,2, E E

k=l

The duality of

4

C?Jr E

-

(G

El)' holds isometrically, hence

10. Integral Operators

If L , E E(co,ll) is the operator associated with

'p,

125

then

and hence the proposition implies that 00

00

n=l

n=l

These results (also due to Grothendieck [93], p.95; see also Ex 10.7.) are connected with an inequality of Littlewood's [178]from 1930:

for all 'p E !B3i[(co,co)where c is a universal constant - and the exponent 4/3 best possible. This will be shown in 34.11. as well as an estimate for the constant c.

10.9. The last corollary was a consequence of the fact that N(C c, Pm)= 1 holds; this was obtained by Rademacher averaging. Using this method directly, one obtains the more general

PROPOSITION: For l / p

+ l/q + 1/r

= 1 and 'p E %i[(lP,k',) the following inequality

holds: Il(v(eklek))k€NlIfr Ill'pll

*

One may replace &, by co. The result and the following proof are due to Aron (unpublished), see also a recent paper of Zalduendo [278]. PROOF: For

Ck

:= p(ek, ek) choose (Yk and

,fjk

E

(if p, q, r E [l,m[). Rademacher averaging gives n

n

k=l

k=l

with

126

10. Integral Opetutors

If some of the p, q, r are 00, some obvious modifications have to be made. 0

10.10. The ideal of integral operators is neither injective nor surjective: This can be deduced from the fact that E does not respect quotients (4.3.) - but the details are omitted since in Chapter I1 the relationship between E and 3 will be studied systematically in a more general context. On the other hand the fact that E respects subspaces implies extension and lifting properties of the ideal 3, see Ex 10.4..

Exercises: Ex 10.1. Every integral operator is weakly compact, but not necessarily compact (and hence also not necessarily nuclear). Ex 10.2. For which 1 5 p Hint: 10.4..

I q 5 00

is the canonical embedding

e,

c,

l,, integral?

Ex 10.3. If F is complemented in F”, then an operator T E S ( E , F ) is integral if and only if it admits a factorization E -+ L, 4 L1 -, F ; what about the integral norm I ( T ) ? Examples: F reflexive or a dual space, F = L1(p).

Ex 10.4. (a) Eztension property of integral operators: If G is a closed subspace of a Banach space E and T E 3(G,F ) , then there is a ? E 3 ( E , F”) extending KFTwith I @ ) = I(T).Hint: E respects subspaces or: the factorization theorem. (b) Lifling property of integral operators: If Q : G --y F is a metric surjection and T E 3 ( E , F ) , then there is a F E 3 ( E , G”)satisfying Q ’ o p = K F O T and I @ ) = I(T).

Ex 10.5. Let M be a norm-closed sublattice of an L,(p) or C ( K ) (with K compact) and T E S ( M , L l ( v ) )positive, then T is integral and I(T) = llTll. Hint: Kakutani’s theorem shows that T“ satisfies the assumption of 10.6.. Ex 10.6. Let p be a finite measure. Then T E SC(E,Ll(p))is integral if and only if TBE is lattice bounded. Hint: For one direction use an h 2 ITtI for all t E BE and the measure dv := hdp to obtain a good factorization; for the other direction recall L1(p)) is regular. from 7.4. that every operator S E S(LI(V), Ex 10.7. Use 10.8. to prove the following result: If ( E n ) and (F,) are two sequences of Banach spaces and A E %ir(e,(E,),C,(F,,)), then

Ex 10.8. Let Ti E C(Ei, Fi) be two operators one of which is integral. Then the operator TI 8 Tz : El QDrE2 4 F1 8-FZ is continuous.

11. Absolutely p-Summing Operators

127

11. Absolutely p-Summing Operators The Banach ideal of absolutely p u m m i n g operators is in some sense at the heart of Banach space theory. For p = 1 these operators are due to Grothendieck (“applications prkintkgrale droite” in the Rksumd or “semi-intkgrale droite” in his thesis); they were generalized in the mid sixties (Mityagin, Pekzyriski, Pietsch) to p > 1; the basic results on factorization and composition are due to Pietsch. The great advantage of absolutely p-summing operators is that - with the aid of various handy characterizations - it is often relatively simple to check whether or not an operator is of this type. Outside Banach space theory these operators had an important impact on the theory of nuclear spaces and the theory of distributions.

11.1. An operator T E S ( E , F) between Banach spaces is called absolutely p-summing (for 1 5 p < co)if there is a constant c 2 0 such that for all finite sequences ( 2 1 , ...,z n ) in E

Define P,(T) to be the infimum (=minimum) of these constants and write ppfor the class of all absolutely psumming operators; for obvious reasons it is sometimes P m ) := (C, 11 11). Clearly, the defining inequality for absolutely useful to define p-summing operators also holds for infinite sequences. Recall from section 8 that for the absolutely p-surnmable sequences

(vw,

and for the weakly p-summable sequences

Now using the definition and an obvious application of the closed graph theorem, gives the

PROPOSITION: Let E and F be Banach spaces. For each T E C ( E ,F ) and 1 5 p the following staternenis are equivalent:

( 4 7-E

V A E ,F ) .

< 00

128

11. Absolutely p-Summing Operators

(b) T maps weakly p-summable sequence sequences into absolutely p-summable sequences: T N ( G W )c tP(F) . (b') T N :$ ( E )

-,tp(F)is

continuous.

(c) T N ( q o ( E )c) t P ( F ) . (c') TN : c i o ( E )-,t p ( F )is continuous. (d) i d @ T : tpaCE --* tp@A? F is continuous. (e) There is a c 2 0 such that f o r all n E IV

In this case,

Recall that A1 = r;Ex 4.3.implies that P1 ( i d l ; ) 2 consequence of the definition:

fi.The following is an immediate

REMARK: (vp, Pp) is an injective Banach operator ideai. However, it is not surjective, see 11.12.. For p = 1 part (c) of the proposition means, by 8.3., that T E t ( E , F ) is absolutely summing (:= absolutely l-summing) if and only if it maps unconditionally convergent series into absolutely convergent series. The characterization 10.7. of integral operators by operators of the form idG @T shows (property (a)) that each integral operator is absolutely summing and

11.2. The following example is basic: If K is compact and /J a (positive) Borel-Radon measure on K , then the canonical map

is absolutely p-summing and P p ( I p )= llIpll= p(K)'/P. This follows from

Much more is true, namely the

11. Absolutely p-Summing Operators

129

COROLLARY: Let E = C ( K ) for a compact K or E = L,(p) for some measure p. If u is any other measure, then every positive operator T E C ( E ,Lp(u)) i s absolutely p-summing and Pp(T)= IlTll.

PROOF: It was shown in theorem 7.3. that

hence part (d) of the proposition gives the result.

11.3. To a large extend the theory of absolutely p-summing operators is governed by the following result:

GROTHENDIECK-PIETSCH DOMINATION THEOREM: Let 1 p < 00 and K C BE‘ a a(E’,E)-compact norming subset. Then T E C ( E ,F) is absolutely p-summing if and only if there is a (positive) Borel-Radon measure p on K with

for all t E E. In this case, Pp(T) = minp(K)l/p, where the minimum is taken over all such p .

PROOF: If T satisfies such an estimate, then n

. n

hence PP(T) 5 p(K)’/P. Conversely, assume Pp(T)= 1, hence n

n

It is easily seen that on the Banach space C R ( K )of reatvalued continuous functions on K the functional w defined by

is sublinear and satisfies

130

11. Absolutely p-Summing Operetors

The Hahn-Banach theorem gives a p E CB(K)’with p 5 w . Since p ( f ) 5 w ( f ) 5 0 for f 5 0, it follows that p is positive, hence a positive Borel-Radon memure by the Riesz representation theorem. For z E E consider -I(-, z ) l P E C*(K):

.>IP)

5 w(-l(.1 41P) 5 -llT4IP which is exactly the desired estimate since p ( K ) = p ( 1 ) 5 1. 0 P(-I(-1

?

The theorem can be rewritten in terms of a factorization - this is why it is sometimes called the Grothendieck-Pietsch factorization theorem.

COROLLARY 1: Let 1 5 p < 00 and T E C(E,F ) . Then the following are equivalent: (a) T E F) . (b) For each norming weak--compact subset K C BE, there are a Borel-Radon measure p on K , a closed subspace G C L p ( p ) , operators R E C ( E ,C ( K ) )and S E C(G,F ) such that S I p R x = Tx for all x E E : E - F

T

(c) There are a finite measure space ( Q , p ) , a closed subspace G c Lp(p)l operators

R E C ( E , L,(p)) and S E C(G,F ) such that E - F

T

I n this case, Pp(T) = minllRll llSll l[plll/P, where the minimum is taken over the factorizations in (b) or over those i n (c).

PROOF: Take T E Yp(E,F) and choose p from the theorem such that for all x E E and Pp(T)= p ( K ) l / p . Define RX := (*,z) E C(K)

G := Ipo R ( E ) S ( I p o R(x)) := T z ;

131

11. AbsoIuteIy p-Summing Opemtors

then S is well-defined by the inequality and both operators R and S have norm hence (b) is proved. The implication (c) A (a) follows from the fact that

5

1,

vP.

The equalities for the norm come by the corollary in 11.2. and the injectivity of Pp)is an injective normed operator ideal. 0 from the fact that

(vp,

In special situations the factorization can be improved.

COROLLARY 2: Let 1 p < 00 and T E Z ( E ,F ) . (1) If p = 2 or F has the metric extension property, then T E TP,(E,F ) if and only if at admits a factorization

E-F

T

(where, as before, K is any weablc-compact nonning subset of BE,). In this case,

= C ( K ) , then T E V p ( E F, ) if and only if there are a Borel-Radon measune K and S E C ( L p ( p ) , F )with

(2) If E p on

C(K)

T

F

In this case, P,(T) = min llsllP ( K ) ~ / P .

It followsthat in general a T E Z ( E , F) is absolutely psumming if and only if it admits a

factorization

132

and PP(T)= minllRll

11. Absolutely p-Summing Operators

llSll IIIpll.The space C ( K ) can be replaced by L m ( p ) .

For p = 1 the factorizations show that the “typical” absolutely l-summing and the “typical” integral map is the same (see 10.5.), only the way of factoring through this mapping is slightly different: The definition of the injective hull of an operator ideal (9.7.) and Ex 10.3. give the first statement of the following result.

COROLLARY 3: (1) ( W ,PJ) = (?I

,P I )

- the

absolutely summing operators form

the injective hull of the integral operators.

(2) If K is compact, then J ( C ( K ) , F )= ? P l ( C ( K ) , F )holds isometrically for all Banach spaces F. Clearly, (2) follows from 10.5. and corollary 2(2). Another immediate consequence of the Grothendieck-Pietsch domination theorem is that:

COROLLARY 4: If 1 5 p 5 q < 00, then

VP C Pqand P q ( T )5 Pp(T) .

The inclusion ppC !Jlq isstrict, see Ex 11.23.(a). One can also show that the canonical mapping C[O,11 4 L,[O, 11 is in no !JJp for p < q (see Pietsch [214]).

11.4. A first application is the THEOREM: A Banach space E is finite dimensional if (and only DVORETZKY-ROGERS if) every Prnconditionally Convergent series is absolutely convergent.

PROOF:Since, by 8.3., q l ” ( E )is the space of unconditionally convergent series, it follows from ll.l.(c) that the condition implies i d E E !& C p2, hence i d E factors through a Hilbert space by corollary 2:

H Since S must be surjective, E is isomorphic to a Hilbert space. But this implies that E is finite dimensional because ( t e n ) E q l o ( t 2 ) \ t , ( t 2 ) . 0 See

Ex 11.9. for the Dvoretzky-Rogers theorem for p-summable sequences.

11.5. For many applications Pietsch’s multiplication theorem is important:

11. Absolutely p-Summing Operators

THEOREM: If 1 S r, p , q C

00

with 1/r = l / p

+ l/q,

then gpo gqC

PROOF:One may aasume Pq(T : E + F ) = Pp(S: F that

Wr(t1

133

, ...,2,; E) 5 1 implies that

+ G)

Vr and

= 1. It remains to show

If p is a probability measure on BE' with

define C k := (lB,, I ( z ' , ~ k ) ] ~ p ( d z ' )and ) ~ / note ~ that assume T x k # 0, all C k are different from zero. Now

c;)=,5 1. Since one may c:

The Holder inequality for l/r'+ l / q + l / p = 1 implies for these ( L Y E ) and z' E BEIthat

It follows from this that

(*) and this inequality conclude the proof. 0 This shows in particular that V z o '& c and, consequently, that the composition of 2n absolutely psumming mappings is absolutely lsumming if 2n 1 p . Clearly, the

134

11. Absolutely p-Summing Opemtors

vp

inclusion o p qc Ex 11.23.(b).

is true if 1/s

5 l/p + l/q but it is false if 1/s > l/p + l / q , see

11.6. In order to study the relation between absolutely psumming operators and other operator ideals more presicely recall that an operator T E S ( H 1 , H 2 ) between Hilbert spaces is called a Hilbert-Schmidt operator if for some orthonormal basis (e,) of H i a

notation: T E A 6 ( H 1 , H 2 ) . It is easy to see that the definition of the Hilbert-Schmidt norm HS is independent of the specific orthonormal basis:

HS(T)~=

c

W e uI

=

C I(ea I ~

* f pt2 )= ~HS(T*)~ ~

a,B

a*B

where T* : H2 -+ H I is the Hilbert space adjoint; in particular, T E 4 6 if and only if ir E 46. The basic facts about Hilbert-Schmidt operators are treated in Ex 11.12., Ex 11.13. and Ex 11.18..

PROPOSITION: T E C(H1, H 2 ) is Hilbert-Schmidt if and only if it i s absolutely 2summing. Moreover, HS(T) = P2(T) .

PROOF: If T E v z ( H 1 , H 2 ) , then for all orthonormal ( e l , ...,e n )

(2

IITekll2) 1'2

5 ~ 2 ( T ) w ( e 1...,e,;H1) = P2(T)

k=1

and hence HS(T) 5 P2(T). If, conversely, T E A 6 ( H 1 , H 2 ) and an orthonormal basis ( e , ) of H2 -

21,

...,z,

E H I , then

- with

n

k

k=l

a

a

k

Khintchine's inequality actually implies that

Vp(H1,Hz) = A G ( H 1 , H z ) for all 1 5 p < 00 (with equivalent norms) - this will be done later in a slightly more general setting (see 26.6.).

11. Absolutely p-Summing Operotors

135

11.7. For testing whether or not an operator is Hilbert-Schmidt the following lemma is of great practical importance.

PIETSCH LEMMA: If H is a Hilbert spacel p a Borel-Radon measure on a compact set K and S E C ( H , C ( K ) ) ,then T := 12 o S : H 5 C ( K ) A L z ( K ,p ) ISHilbertSchmidt and H S ( T ) 5 ~ ~ S ~ ~ p (. K ) 1 ~ 2 This follows from the fact that Pz(12) = p(K)'I2, the ideal property and the last proposition. But, interestingly enough, this can be proved directly: If ( e l l ...,en) is orthonormal, then

5

J llS% llLP(d4 5 IlS'll"(K)

by the Bessel inequality (and H = H').

COROLLARY: The product of three absolutely 2-summing operators is nuclear. PROOF: Consider the factorization

and use the fact that the product of two Hilbert-Schmidt operators is a nuclear operator (see Ex 11.13.). Later on this result will be improved to the composition of two Operators, thus generalizing the case of Hilbert-Schmidt operators (see also Ex 11.14.). It follows that the composition of 3n absolutely psumming operators is nuclear if n 2 p / 2 . In many concrete cases this is an appropriate tool to show that a given operator is nuclear! In particular, absolutely p-summing operators in a Banach space are power compact and hence have a discrete spectrum; the sequence of eigenvalues of a T E v P ( E E , ) is in tP if p 2 2 . This famous result of Johnson-Konig-Maurey-ktherford I1311 is the basis for the recent theory on eigenvalues of operators (see the monographs of Konig [147] and Pietsch [216]); in 34.10. one step of the proof will be presented. 11.8. A nice consequence of !&(H, H) = 4B(H, H ) is

Pz(idp)= HS(idq) = f i

.

136

11. Absolutely p-Summing Operators

In order to have this for all n-dimensional spaces the following test, seemingly due to Kwapied, will be useful.

PROPOSITION: For T E E ( E ,F ) and 1 I p < 00 the following are equivalent: (4 T E 9 p m F ) * (b) There is a c 2 0 such that Pp(TS)I cllSll for all S E A!($#, E) and all n E N . (c) TS E Pp(tp#, F ) for any S E Z(lp#,E ) . Moreover, Pp(T):= sup(Pp(TS) n E N , 1 1s: -+ Ell 5 1) .

I

5,

PROOF:(a) implies (c) and (c) implies (b) by a closed graph argument. For the remaining implication recall from 8.2. that for 21,...,2, E E

and therefore k

k

- C“llS(lP . 1 = ePwp(z,; <

E)P . 0

See also Ex 25.14.. Comparing this result with ll.l.(e), one might be tempted to think that $ 8~~F = vp(.$,,F) holds isometrically. This is not true for p # 1 - a fact which will follow from the results in Chapter 11: Ap fails to be a “tensor norm”. 11.9. The proposition can be used to prove the aforementioned

THEOREM: If E is a Banach space of dimension n, then Pz(idE) = f i PROOF: To show Then

.

5 fi,use the foregoing test and take S E Z ( e , E) with llSll 5 1. S:

Q

G/kers% E ,

where 11Q11 = 1 and llSoll= llSll I 1. Since c / k e r S is a Hilbert space lt with k it follows that P2(S) 5 IlQllP2(4:>IlSoll IdZ I J;; . Conversely, the Grothendieck-Pietsch factorization theorem gives a factorization idE

E - E

H

5 n,

11. Absolutely p-Summing Opemtors

through a Hilbert space H with S is bijective, hence

137

llSll = 1 and Pz(R) I P z ( i d ~ ) .One may assume that

f i = PZ(idH) = P2(RS) 5 P2(R)IISII 5 PZ(idE) . 0 This result (due to Garling-Gordon [79]) has interesting consequences for the BanachMazur distance between Banach spaces of the same finite dimension and the existence of certain projections on finite dimensional subspaces of Banach spaces, see Ex 11.16..

11.10. Using Gauss averaging it is possible to calculate the p-summing norm of the identity map on 4; the following result is due to Gordon [82],[83] (see also Pietsch [214], 22.1.3.), who also calculated various other P p ( i d ~for ) finite dimensional E.

PROOF: The Khintchine-type equality for the Gauss measure says (8.7.) that for all

with vn(dw) := Ilglll&llwll;?i,(dw) (which is obviously a finite measure). This equality is the key to the proof (see also Ex 11.19.). The easy part of the GrothendieckPietsch domination theorem gives Pp(id~;)5 v,,(q)l/P . To obtain the converse, choose - again by the domination theorem - a measure p on the unit ball B of such that

It follows that

by the KhintchineGauss equality. This implies that

which is the claim. 0 The explicit calculation for p = 1 (for other p see Ex 11.24.) gives the following: If un-l is the size of the surface of the euclidean unit ball of En,then

138

11. Absolutely p-Summing Operntors

from 8.7.. Then (with an obvious notation for 6 )

and

What is the behaviour of these constants like as n + oo? Using again

and Wallis' formula a 2

- = lim n-m

one easily gets

since clearly Pl(idR;)

5 Pl(idRn+I)>this is also true for the odd n. It follows that:

COROLLARY:

11.11. Gauss averages will also give the

139

11. Absolutely p-Summing Opemtors

THEOREM: If H is a Hilbed space and T E 2(c, H ) , then

where KfG = @ an the real case and KEG = 2/+

in the complex case.

It is not by chance that the constants are the same as in the preceding corollary: In 20.19. it will be shown with the aid of this corollary that these constants are actually the best possible ones. The theorem (and the following corollary) is sometimes called the little Grothendiect theorem since it already has the flavour of Grothendieck’s fundamental theorem of the metric theory of tensor products which will be treated in Chapter 11.

PROOF:One may, by the ideal property of !l32, assume that H = (proposition 11.1.) that for all T E 2(c, 5)and all n E EV

c.It is to be shown

Since

by the duality of E and

A

(see section 6), this is the same as showing

for all S E 2(q,c). Fix S =

zcld e i and take

z

=

xk@ee, E

and hence

By the Khintchine-type equality 8.7. for the Gauss measure it follows that

@q.Then

140

11. Absolutely p-Summing Operutors

by the biorthogonality of the

gk

(see 8.4.). An appeal to 8.7. gives that

KLG

=~

b

1

~

~

~

,

~

~

b

1

~

~

2

,

~

is m indicated. 0

COROLLARY (“little Grothendieck theorem”): If E

is an C&,A-

or an C:,A-space and

H a Hilbert space, then

C ( E ,H ) = % ( E , H )

and

Pz(T) 5 KLG~ITII.

In particular, all operators from C ( K ) ,L , or L1 into a Hilbert space are absolutely a-summing. The “full” Grothendieck theorem will show that all operators L1 -+ H are even l-summing!

PROOF:(a) Assume first that E is CL,, and T E 2 ( E , H ) . To show the relevant !&-inequality take 2 1 , ...,x,, E E; then there is a factorization

em, and hence by the theorem

P2(TSR) 5 IIRIIP2(TS) 5 IIRIIKLGllTsll 5 KLGllTIIA(1 -k E )

.

(b) If E is an C : A-space, then the same reasoning shows that it is enough to demonstrate that for ali T E C ( c , H )

P2(T) 5 KLGllTll To do this take, by Kwapierl’s test 11.8., some S E C ( L 5 , q ) .Then P2(TS) = H S ( T S ) = HS(S’T’) = P,(S‘T‘) 5

5 p2(s’ :

+

e!>llT’ll 5 KLGlls’ll

11T‘11

by the theorem, hence Pz(T) 5 K L G ~ ~ T . IJ Note the simple dualization argument for ??32(., H) with the aid of Hilbert-Schmidt operators in part (b) of the proof. More systematic dualization arguments for operator ideals will be developped in Chapter 11.

141

11. Absolutely p-Summing Operators

One can also prove the corollary for Ci-spaces F instead of the Hilbert space H (with additional constant) just by adding a factorization of TS : -+ F through an 15. But Ex 11.17. implies that then F is isomorphic to a Hilbert space: nothing is gained but the 8n

REMARK:The 2i-spaces are exactly the Banach spaces isomorphic t o Hilbert spaces. This result will also be a trivial consequence of the general characterization of 2;spacee given in section 23.

vp

11.12. The operator ideal is injective - but not surjective: The little Grothendieck theorem implies that the canonical quotient map l,+& is in 9 2 but clearly idt2 is not since absolutely 2-summing maps are power compact.

-

The canonical embedding 11 4 1 2 is in ez

VZ (even in !?3,;

see Ex 11.5.) but its dual

co 4 1,

is not in pz : This follows from the fact that the sequence ( e n ) of unit vectors is weakly2-summable, aIthough C Ilenllzo = 00. This shows that the statements “T E !&” and “T‘E &” do not imply each other. Between Hilbert spaces, however, these statements are equivalent since v z ( H 1 , H z ) = AG(H1, H z ) . More generally, if T E f&(H, F),then T’E ?&(F’, H‘) (where H is a Hilbert and F a Banach space) since, by the factorization theorem, T factors through a Hilbert-Schmidt operator. Cohen [30] and Kwapieli [160] proved the following:

PROPOSITION: A Banach space E is isomorphic t o a Hilbert space if and only if the dual T’E f& whenever T E ‘&?(E,ez).

PROOF:One direction was just shown. Conversely, embed

I :~ - t , ( r ) . The idea of the proof is to show that the metric surjection I’ is in Qz, hence factors through a Hilbert space and consequently E’ is isomorphic to a Hilbert space. To see that Z’ E ‘J.32 take

vz(~w(r),~2)

(A) E q(t,(r)l)= .c(tw(r),~z) =

by 8.2. and the little Grothendieck theorem, and consider its associated operator S := CS/n&en. Since the composition SI E !J?z(E,~z), the assumption on E implies that Z‘S‘ E !$?z as well, therefore

C llI’Y‘n112 = C III‘S’enl125 P z ( l ’ S ’ ) 2 w z ( e n ;

12)’

n

n

< 00 ,

142

11. Absolutely p-Summing Operators

which shows that I' E V2

.

0

We shall come back to the relation between T and TI being in

VPin 25.9..

Exercises: Ex 11.1. Show that 3 c Vp and P,(T) 5 I(T) for all 1 5 p < 00. Ex 11.2. Prove that for all Banach spaces F

holds isometrically. Hint: T = C engTen.This result is true in many other cases: this can be shown with the Radon-Nikodfm property, see Appendix D7. and D8.. Ex 11.3. If a Glinear operator between complex Banach spaces is considered as &linear, does the p-summing norm change? Ex 11.4. For g E Lp(p)the multiplication operator

-

4 7 : ~w(c1)

LP(cL)

defined by M,(f):= fg is absolutely p-summing and P p ( M B )= llgllp. Hint: 11.2.. Ex 11.5. Use Khintchine's inequality to show that

p l ( q 443 I fi Pz(q+g)=l.

-

Deduce that the canonical embedding I : .tl .t2 is in C V Zwith Pz(I) = 1 5 PI(]),< 4.Recall from 10.4. that I is not integral! Ex 11.6. An operator T E C(E,F) is absolutely p-summing if there is a finite measure space (Q,p ) and a a(E', E)-measurable ip : Q -+ B p such that

In this case, Pp(T)I p(Q)l/P. Ex 11.7. For E = L w ( v ) show that T E v p ( E F) , if and only if it admits a factorization E 4 C ( K ) + L p ( p ) -+ F. Ex 11.8. (a) Every absolutely p-summing operator T is weakly compact and completely continuous (this means: if 2, 40 weakly, then llTznII + 0), but in general not compact. Hint: Lebesgue's dominated convergence theorem. (b) VpoVpC Ppo?2DCff, but I D o V p ff; moreover, V P= ?2DoVPif 1 < p < 0 0 .

11. Absolutely p-Summing Operators

143

(c) Every quasinuclear operator is absolutely l-summing, but not conversely. Ex 11.9. A Banach space E is finite dimensional if and only if $i"(E) = P,(E) (as sets) for some (and then for all) 1 5 p < 00. This is the Dvoretzky-Rogers theorem for p-summable sequences. Ex 11.10. Extension property for absolutely 2-summing operators: If G C E is a closed subspace of a Banach space E and T E T2(G,F), then there is an extension r?l E !432(E1F) of T with P z ( F ) = P2(T). Hint: Factorization through L , and 3.10.. Ex 11.11. If T E '&(El E ) , then there are a Hilbert space H, operators R E C ( E ,H) and S E C ( H , E) such that SR = T and T := RS is a Hilbert-Schmidt operator and such that HS@) 5 P2(T).One says that T and are related operators, a concept due to Pietsch which is extremely useful since related operators T and T have the same non-zero eigenvalues (with the same multiplicity). E x 11.12. (a) An operator T E C ( & , & ) is Hilbert-Schmidt if and only if H S ( T ) 2 = ck,( I(Teklef)12 < O0 * (b) An operator T E C ( L 2 ( Q 1 , p l ) ,L z ( Q 2 , p 2 ) ) is Hilbert-Schmidt if and only if there is a k E L 2 ( p 1 @ p2) with

In this case, H S ( T ) = ( l k ( (.~ ~ E x 11.13. (a) (46,HS) is a Banach operator ideal in the class of Hilbert spaces: (1) AB(H1, H2) C C(H1, Hz) is a linear subspace containing the finite rank operators. ( 2 ) HS(idK) = 1 . (3) H S ( T R S ) 5 ((TI(HS(R)l(S(( if the composition TRS is defined. (4) (AB(HI,H~),HS) is a Banach space. (b) If (e,) is an orthonormal basis of H , then

defines a scalar product on f i B ( H ,H) with associated norm H S : It follows that the space (AB(H, H), HS) is a Hilbert space. (c) The finite rank operators are HS-dense in AB(H1, Hz). (d) Hilbert-Schmidt operators are compact. (e) If T = C,"=lA n e n & f nwith two orthonormal sequences (en) and (f,) and a bounded sequence (A,) of scalars, then T is Hilbert-Schmidt if and only if (A,) E &. In this case, H S ( T ) 2 = C,"=l(A,('. (f) Use the spectral decomposition theorem for compact operators (see Ex 3.28.) to show that the composition of two Hilbert-Schmidt operators is nuclear and

N(ST) 5 HS(S)HS(T) .

144

11. Absolutely p-Summing Operators

(g) Show that (SIT) = trH(T*S)if S, T E A6;(H, H ) . Hint: Ex 5.15.. Ex 11.14. p z o ! & ( H ~ , H z ) C % ( H l , H z ) for Hilbert spaces H I and H z . Hint: FactorHilbert-Schmidt operators. This result also holds for arbitrary ization theorem for Banach spaces, see 19.4.. Ex 11.15. Pz(T) 5 I I T I I J a for all T E &(ElF ) . Ex 11.16. Prove the following corollaries of theorem 11.9.: (a) The Banach-Mazur distance from an n-dimensional Banach space E to is I f i . Hint: Use the fact that absolutely 2aumming maps factor through Hilbert spaces. (b)d(E,F)
vz,

for all finite Borel measures p on E; the infimum of these c is P P ( T ) .Hint: Choose a partition Dn of E with IIZ'zIIP - llTznllPl and 112 - zn[Ismall on Dn. (b) Extend (a) to u-finite Borel measures p. (c) If there is a u-finite Borel measure p on E with

for all 2' E El, then

11. Absolutely p-Summing Operetors

145

for each T E E(E, F). (d) Show that for ( x 1 , x 2 )E R2

Use this, (c) and the domination theorem to show that Pl(idR;) = r/2. This method was used by Gordon (see also Jameson [119]) for the calculation of various P , ( i d ~ ) . EX 11.20. Show that the class (qr,, ,P r , a ) of absolutely (r, s)-summing operators (see Ex 8.12.) is a Banach operator ideal. Moreover, T E E(E, F) is absolutely (r, s)summing if and only if i d @ T : e, @t E er @Ar F is continuous. In this case the norm of this mapping is Pr,a(T) . Ex 11.21. If r1 5 r2 and s1 5 s2 such that l/sl - l/rl 5 1/sz - l/rz, then Vr,,al C !J3r2,aa and Pr,,,, 5 P,,,,, . Hint: With l/r = l/rl - l/r2 and 1/s = 1/81 - l/sz it follows that

-

e+-1(akllT~kll) IPrl,sl(T)ea(ak)wa,(tk)

.

Ex 11.22. Use Rademacher functions and Ex 8.1. to show that for 2 5 p < 00 the identity map of any L p ( v ) is absolutely (p,l)-summing. Hint: CIfn(t)lP5 I CCn(W)fn(t)IPp(dW). For a more general result see 24.7.. Ex 11.23. (a) Take A E too and consider the diagonal operator Dx : co 4 co. Show that Dx is absolutely psumming if and only if A E ep. Hint: Ex 11.4. and unit vectors. (b) Take p , q, r, s E [l,00[ with l/p l/q = 1/r < l/s. Show with (a) that

+

Vqo'Pp

$ 'Pa

- 80 the index r in Pietsch's multiplication theorem Pqo qPc Vr is best possible. Hint: Dx o D,,= Dx,,. Ex 11.24. For the Gauss measure 7" on K" and 1 5 p < 00 define

and observe that Pp(idq) = c;,,/ci,, by theorem 11.10.. (a) Using polar coordinates as in 11.10., verify that

, coor(b) Use the Gauss-Khintchiee equality c:,,11z112 = ($ I(x,w ) l P y , ( d ~ ) ) ~ / Ppolar dinates and the calculation for (a) to show that

for every x E K", where qn-l is the normalized Lebesgue measure on the euclidean unit sphere S,-l of K " . The formula is taken from Pietsch [214], 22.1.2..

146

12. Definition and Ezamples

Chapter I1

Tensor Norms In this chapter the main topic of this book will be presented: The GrothendieckSchatten theory of tensor norms, with special emphasis on its interplay with the theory of operator ideals. Sections 12 - 15 and 20 give the basic definitions and properties of tensor norms, including Grothendieck’s fundamental theorem of the metric theory of tensor products -nowadays called Grothendieck’s inequality. Section 17 deals with the main theorem concerning the relationship between tensor norms and maximal Banach operator ideals - the representation theorem for maximal Banach operator ideals; the representation theorem for minimal Banach operator ideals will be given in section 22. Sections 18 and 19 study the (p,q)-factorable and (p, q)-dominated operators, the technique of ultraproducts, Maurey’s and Kwapieh’s factorization theorems. The central tool in all that follows will be the duality theory of tensor norms and later of operator ideals, in other words: a good understanding of trace duality is basic. Trace duality is “smooth” only for “accessible” tensor norms or under an additional hypothesis such as the metric approximation property. Due to the success of the notion of accessibility for tensor norms, accessible operator ideals will be defined and studied in section 21. It seems to us that the composition formulas for accessible operator ideals given in section 25 (which again are a sort of trace duality) fully justify the introduction of this notion. At the end of this chapter we hope that the reader is impressed with the power of also thinking in terms of tensor norms. In particular, she or he may have learned that it is natural to describe certain phenomena in terms of tensor norms, others in terms of operator ideals.

12. Definition and Examples In this section the definition and some basic examples of tensor norms are given. Two natural ways will be presented for extending a tensor norm from the class of finite dimensional spaces to all normed spaces - an inductive and a projective procedure. The inductive one (“the finite hull”) gives Grothendieck’s original definition of a tensor norm, the projective one (“the cofinite hull”) - essentially due to Lot2 - will turn out to be a useful tool for the understanding of various delicate phenomena related to the approximation property.

147

12. Definition and Ezamplea

12.1. A tensor norm a on the class NORM of all normed spaces assigns to each pair ( E lF) of normed spaces E and F a norm a( - ; E l F ) on the algebraic tensor product E @ F (shorthand: E Nu F and E& F for the completion) such that the following two conditions are satisfied: (1) Q is reasonable, i e . : E IQ 5 A . (2) a satisfies the meiric mapping property: If Ti E 2(Ei, Fi), then

llT1 @ T2 : El

@a

E2

-

Pl

@a F211

L 11TlIl llT2ll

‘

This definition goes back to Schatten [245] in 1943 (who called such an Q a uniform crossnorm) and Grothendieck [93] in 1954. It is clear that it can also be made for subclasses of all normed spaces: for the class FIN of all finite dimensional normed spaces, for the class BAN of all Banach spaces, for the class HILB of all Hilbert spaces or even for classes of pairs of normed spaces such as NORM x BAN or FIN x NORM. A tensor norm a being reasonable means

It can happen that all tensor norms are equivalent on some E @ F : the celebrated spaces P of Pisier, mentioned in 4.8., satisfy P P = P @I P isomorphically, and hence have this property. E and A are tensor norms since they have the metric mapping property (3.2. and 4.1.), but Ap is not a tensor norm:

PROPOSITION: For 1 < p and normed spaces F

< 00

there is no tensor norm Q such that for all spaces L p ( p )

L p b ) @A,

F =Lpb)

F

holds isomorphically. There exist, however, tensor norms “close” to Ap (see 15.10. and 15.11.) - and these tensor norms serve well for studying various phenomena related to Ap.

PROOF:The reason that no such tensor norm exists is that Ap does not satisfy the mapping property: For p = 2 it was seen in 7.5. that for the continuous Fourier transform 5 : L2(R) -,L 2 ( R ) the operator 5 @ idt, : L 2 ( R ) @ A ~ti -,L 2 ( R ) @A2 ti is not continuous, but 5 @ idt, : L 4 R ) @a t 1 -, L z ( R ) QU ti is continuous for all tensor norms a. For 1 < p < 00 the same argument applies using the continuous Hilbert transform tp(Z) tp(zZ)treated in 7.6.. 0 On the other hand, as was already shown in 7.1. and 7.3., Ap is at least a “one-sided” tensor norm: it is reasonable (i.e. E 5 Ap 5 T ) and llidLp @ T : Lp @ A ~E

for all T E C(E,F ) .

-

Lp @ A ~Fll=

IlTll

148

12. Definition and Examplea

12.2. The following test (formulated for NORM only) will be quite useful:

CRITERION: (Y is a tensor n o r m on NORM if and only if: ( 1 ) a( - ;E , F) is a seminorm on E €3 F f o r all pairs ( E ,F ) of nonned spaces. (2) a(1 c3 1; K ,K )= 1. ( 3 ) a satisfies the metric mapping property. Note the similarity with the criterion 9.4. for Banach operator ideals and also that the cases of real and complex scalars will be treated simultaneously.

PROOF: To show that the three properties suffice for a to be a tensor norm, one only has to check that a is reasonable, i.e. E Ia IA. For z' E E' and d E F'

-

x'@y':E@F-K@K=K, (x'@y',z)

hence l(z' €3 Y',

€3 d , 4; K ,K ) I11~'IlIldlla(z;E,F )

z)I =

by (2) and (3), and therefore e I a. To see that A dominates a,observe first that for x E E the operator T, : K -+ E defined by T,(A) := Az has norm 11211, hence

llG €3 q

:

€30

K

-

E

@a

Fll

I 1141llvll

9

and therefore (again by (2) and (3)),

ll4l llYll = 4 . €3 Y; E , F) I4

2

€3 Y;El F ) Illzll

which means that a ( z €3 y; E , F ) = 11x11

Ilvll.

IlYllQ(1 €3 1;K ,

w = 1141llvll

The triangle inequality gives a 5 A. 0

12.3. In general, tensor norms respect neither subspaces nor quotients isomorphically: this follows from the "bad" properties of E and A. The mapping property only implies continuity of the respective mappings. Clearly, if the subspace G c E is complemented and P : E + G a projection, then, again by the mapping property, 4 z ; E , F ) i 4 z ; G,F )IIIPlla(Z; E , F )

for all z E G €3 F. Recall the natural transposition map

12. Definition and Examples

149

If a is a tensor norm, then the transposed tensor norm at is defined by at(%; El F ) := “(2‘; F, E) . It is obvious that at is a tensor norm and art = a. The tensor norms symmetric, i.e. ct = E and rt = u.

E

and u are

12.4. Given a tensor norm a on the class FIN of all finite dimensional normed spaces, there are two natural ways of extending it to the claaa of all normed spaces. For this, denote for a normed space E

FIN(E) := {M C E I M E FIN} and COFIN(E) := {L c E I E/L E FIN}

,

the sets of finite dimensional subspaces and of finite codimensional, closed subspaces (with induced norms). Now take normed spaces E and F and define for z E E @I F + a ( z ; El F ) := inf{a(z; MIN) I M E FIN(E), N E FIN(F), z E M C3 N} t

a ( z ; El F) := sup{a(QE

@I Q r ( z ) ; E/K,

F/L) 1 K E COFIN(E), L E COFIN(F)}

(where QE : E * E / K is the canonical mapping); the arrows stem from the fact that the first procedure is an inductive, the second a projective one. By the metric mapping property it is enough to take cofinally many M, N resp. K, L in the definition. With the criterion it is very easy to verify that: 1

PROPOSITION: Let a be a tensor norm on FIN. Then the “Jinitehull” d o f a and the “cofinite hull” ‘Z of a are tensor norms on NORM wiih

Ifa as defined on NORM (and not only on FIN), then

holds on NORM. c

Clearly, ’Z 5 c /3 and d 5 c 7 whenever a 5 c p . Since E respects subspaces, E = T‘ and hence E = ?F; the description of the projective norm by an infimum yields that r = T‘ but it will be shown in 15.6. that u # f .A tensor norm a on NORM is called finitely generated if a = d and cofinitely generated if Q = %. The usual tensor norms are finitely generated - but the cofinite hull is natural as well and its consequent use (also for finitely generated tensor norms) helps to structure the theory, leads to a better understanding of various ideas and simplifies many proofs. In

150

12. Definition and Ezamples

his W u m 6 Grothendieck only considered finitely generated tensor norms - and this is quite understandable: It will be shown in 13.1. that

whenever E and F have the metric approximation property - and it was not until nearly 20 years after Grothendieck’s work when spaces without approximation properties were discovered (see section 5). It is obvious - but good to always bear in mind - that two finitely generated (or two cofinitely generated) tensor norms are equal if they are equal on finite dimensional spaces. This explains, for example, why these two types of tensor norms are often useful when investigating Banach spaces or operators in terms of finite dimensional subspaces.

12.5. Many of the interesting tensor norms can be obtained from the ones introduced by Laprest6 [166], who generalized those of Saphar [237], Chevet [27] and Cohen [31]. For p,q E [l,co]with l / p + l / q 2 1 take the unique r E [l,w] with l/r = l / p + l / q

-1

or equivalently,

1 = 1/r

+ l/p’ + I/q’

(where p’ is the conjugate of p) and define for normed spaces E and F and

(with the notation lp(*)and pup(-) from section 8). Obviously a1,1= a

t

E E@F

.

PROPOSITION: (1) ap,qis a finitely generated tensor norm on NORM. (2) apa,qa 5 Q P l , f l ifPl IPz and 41 I (3) a;,q = %,P *

4 2

*

PROOF:(1) By criterion 12.2.only the triangle inequality is not obvious: Take elements z1, z2 E E @I F and E > 0 and choose for j = 1,2 representations

such that

12. Definition and Examples

151

It follows that

(2) There is nothing to prove for l / p l + l / q l = 1, hence assume r1 < 00 and define

and (by Holder’s inequality)

12.6. To describe the completion of the normed space E @‘a,,q F recal1 from section 8 the notation $ ( E ) = ((211) E E~ I wp(zn) < 00)

$*‘(E) = ((zn) E C(f-9I wp ( ( Z n ) ? h )

-+

01

-

PROPOSITION: (1) For (An) E tr (in c, i f r = oo), ( t n ) E q ( E ) and (gn) E C , ( F ) the series

C ( X n t n 8 un) converges unconditionally in E&L,~,F .

152

12. Definition and Examples

( 2 ) For every z E EgUp,,F there are (An) E & (in co i f r = oo), (yn) E t$"(F) with

(2")

E elfl,"(E) and

where the infimum as taken over all such (finite or infinite) representations.

PROOF: (1) follows easily from the fact that (An) E tr (or co) forces the series to be an ap,q-Cauchy series - independent of the ordering. To prove (2) take for z E E~L+,F and e > 0 elements z, E E @ F with z = C,"=l zn and C,"==, ap,f(zn) 5 ( l + ~ ) a ~ , .~ ( z ) Choose (AT), (zr)and (yr) (finite) with zn = CiATz? @ gr and er

((Ar)i)

I (ap,q(Zn)(l+ &))'Ir

wql ((zr)i) I(ap,q(zn)(l+ wp'

((Yi")i) L

(ap,q(zn)(l+

&))"ql

E))"~'

.

Then

&I2

4, ( ( A T ) i , n ) wq' ((zT)i,n)Wpj (br)i,n) I a p , q ( z ) ( l + and (Ar)i,n E t, , (tr)i,n E Ct'"(E)and ($)i,,,E T J B o ( F. )Now (1) implies that m

m

n=l

n=l

n ,i

i

Moreover, if P denotes the seminorm defined by the infimum in the statement of (2), then for all z E E&,,, F . P(z) I aP,f(4 Conversely, if z = C,"='Anzn@ gn and z N :=

N

Anzn@ y,, then

4(An)Wq'(zn)wpl( ~ )n2 2 er

( < ~ n ) n N , iwp' ) ((2n)L)

wpl

(
2 a p , q ( z N )+ a p , q ( z ) ;

this implies P(z) 2 a,,,(%). 0

12.7. Special cases of ap,f-tensor norms are (1 5 p Im) gp := ap,1 dp := ( ~ 1 , ~ wp := ap,p' and therefore

g1=d1=n gp = d i wp

I gp

( g for "gauche") (d for "droite") (w for "weak")

, W1=dm ,w,=g,

,

wp =

1

WpI

I dp

153

1.2. Definition and Examples

For gp and dp the index r in the definition is p , for wp it is 00; therefore it is easy to see that n

gp(z; E , F ) = inf{lp(ti)wpt(yi)

I z=C

dp(z; El F ) = inf{ wp8(2i)lp(yi) I z =

x i QD yi}

i=l n

C zi QD

Vi}

i=l

Clearly, there is also a result for the completions which has the spirit of proposition 12.6.:

E l p ( E ) (in c,(E) i f p = m) and (yn) E T , ( F ) the series If, conversely, z E E g g ,F , then E l p ( E )(in c,(E) i f p = CO) and (yn) E c , * " ( F )with z = C,"=ltnQDPy,;

COROLLARY: (1) For C(Zn @

(tn)

yn) converges unconditionally in E g g PF .

there are ( t n ) moreover,

gp(z; E , F ) = inflp(xn)wp'(Yn)

,

where the infimum is taken over all such representations. (2) For 1 < p < 00 each series C(tn8 y n ) with (tn) E T * O ( E )and (yn) E T t ( F ) converges unconditionally in E g W p F . Conversely, each z E E g w p F has a representation z = Cr'l t n QD yn with (En) E $'po(E) and (yn) E T,"'(F); moreover, wp(z; E , F ) = infwp(xn)wp~(yn) ,

where the infimum is taken over all such representations. This follows immediately from the proposition - with the exception of the first part of (2): but this, as in the proof of 12.6., can be checked directly.

12.8. The following picture illustrates the situation:

1

2

P

Some of these tensor norms are equivalent (of course, two tensor norms called equivalent if there are c, d > 0 with CCY 5 p 5 da).

CY

and

p are

154

12. Definition and Examples

PROPOSITION: For p, q €11, co[with l/p

+ l / q 2 1,

where br is the constant from the Khintchine inequality. In particular, w2 and equiualeni if p, q €1 1,2].

Q

~ are , ~

PROOF: As the constants already indicate, Rademacher averaging will be used: For = Cy'l xi @ yi E E @ F take the new representation

z

Now the Khintchine inequality 8.5. gives

Consequently,

and therefore ap,q5 bq'bp'wz . 0 The tensor norms gp and dp cannot be estimated by w2; otherwise e.g. wm 5 gp 5 cw2 would follow which, by results to be presented in 23.2., would imply that Hilbert spaces are CLapaces, a contradiction. For the best constants satisfying ap,qI cw2 see Ex 24.7.. 12.9. It was shown in 8.2. that

Cso(E) =

E&tp m

(zn)

-Czn n=l

holds isometrically:

Ben

12. Definition and Examples

Since w p l ( e n ; t p )= 1 and e

155

5 w p , the representation 12.7. of E g W p t pimplies:

REMARK:The relations ~ ~ " (=EE&tp ) = E&,,tp c,(E) = E& C , = E g W mC ,

f o r 1 L p < 00

hold isometrically. In parlicular,

To obtain another useful description of wp consider the four-linear map

which can be linearized to a so-called tensor contraction

This mapping will turn out to be extremely useful; it mimics the composition of maps (see Ex 12.11.) and this has interesting consequences for theorems about the factorization of operators. A detailed treatment of tensor contractions will be presented in section 29; a first hint of its utility is given by the

COROLLARY: Let 1 5 p 5 00. Then the tensor contractions

are metric surjections. Moreover, the norms in the quotients can be calculated by only using elements in ( E m C t px) (tPlBCF ) or i t s completion, respectively. These statements also hold true if toois replaced by c,.

PROOF:Denote, for this proof, by sp c f?, the subspace of finite sequences with the induced norm. Then, by the definition of wp and the remark, the map

maps the Cartesian product of the open unit balls onto the open unit ball, hence

156

12. Definition and Ezamples

is a metric surjection; in particular, the completion of the bilinear mapping C is defined. Note that ( E QC sp)QT (sPt aCF ) is a dense subspace of (E&ep) a,, (eP#&F)if too is read as co. Since the second mapping in the statement is also surjective by the description 12.7.(2) of E & , ,F (and the remark), the quotient lemma 7.4. shows that both mappings are metric surjections. If 1, appears, note that both mappings have norm 5 1 by Ex 12,11.(c); the c,-case gives the result. 0 This result provides a description of the compact sets of E&,F, similar characterization holds for general ap,q, see Ex 35.10..

see Ex 12.12.. A

12.10. The last corollary, easy as it is, has quite interesting consequences. Take a tensor norm Q and consider the natural map C(K)&C(L)

-

C(K)&C(L)= C ( K x L )

for two compact sets K and L. For Q = ?r this map is injective since C ( K ) has the approximation property (see section 5 ) and it will be seen in 17.20. that this holds for all finitely generated a. What do the functions coming from C(K)&C(L) look like? For Q = ?r a function f E C(K x L) is in C ( K ) & C ( L ) if and only if there are fn E C ( K ) and gn E C ( L ) with

C,"=*Ilfnllmllgnllm < 00. For tr = wp the corollary above shows that c ( K , e p )x c(~,e,#) = (C(K)&~~,) x (ep,gcc(~))c ( K ) ~ L , c ( L ) c C ( K x L ) is onto - and this map is just (9, h) (g(.), h(.-))f,,l,, . To see this, calculate first for

-

and

u*

finite rank functions and then approximate. It follows that:

PROPOSITION: A function f E C ( K x L ) is in C(K)&,C(L) if and only if there are continuous functions g : K + 1, and h : L + e,, with for all ( s , t ) E K x L .

f ( s , t )= (!ds), h(t))f,,f,, Moreover, w P ( f ; C ( K ) , C ( L5 ) ) Ilgllmllhllm and for every can be chosen such that Ilgllmllhllm 5 wP(f)+ E .

E

>0

the functions g and h

Exercises:

Ex 12.1. (a) E @a K = E holds isometrically for all tensor norms a and normed spaces E.

12. Definition and Examples

157

(b) If Eo C E and Fo C F are dense subspaces, then E,, 8 F, is dense in E a0F for each tensor norm a. (c) If Ti E C(Ej,Fj) are isometries (onto), then

is an isometry for each tensor norm a. Is this true without surjectivity? Ex 12.2. If a is a tensor norm on FIN, then

Ex 12.3. Let a be a tensor norm on NORM; show that the “one-sided” finite and cofinite hull (also called right-finite and right-cofinite hull) a‘(%;

E, F) := inf {a(%; E, N) I N E FIN(F), t E E 8 N}

a‘(%;

El F ) := sup { a ( i d E 8 Q f ( z ) ;E , F / L ) I L E COFIN(F)}

are tensor norms such that ‘Z 5 a- 5 a 5 a- 5 d . Ex 12.4. Take S E f!(X,Y) with llSll = 1 and a and 9 , tensor norms (on NORM). Show that the class 2s of all T E 2(EIF) with

is a Banach operator ideal. Check exactly which of the tensor norm-properties are needed for the proof and note that a and /3 need only satisfy the metric mapping property on the right side of the tensor product. This means that X ma or Y @ p * could be replaced by L, @A, ; see also 9.6.(3), Ex 9.15., Ex 9.16. and 17.4.. Ex 12.5. Use Rademacher averaging to show that for all matrices (ajj) and all tensor norms a on FIN the relation n

n

holds. Hint: Iw((j):= ( ~ j ( w ) t j )for w E 0,defines an isometry. Ex 12.6. For each tensor norm a and p , q E [1,00]

i=l

where c := min{l/p

+ l / q , 1). Hint: 8.8..

158

12. Definition and Ezamples

Ex 12.7. For z E E @ F define the rank

of z as follows: rank(O):= 0 and for z

#0

(a) Show that rank(%)= rank(T,), where T, : E' -+ F is the associated linear operator. (b) For Banach spaces E, F and T E C ( E , F) the n-th approximation number a n ( T ) of T is defined by

I

an(T) := inf{llS - TI1 S E C ( E ,F),rank(S)

.

< n}

Show with the aid of these numbers that the set

I

Bn(E,F ) := { z E E @ F rank(%)5 n } is closed in E& F and in E @a F for each reasonable norm a on E @ F. If the map E&, F -+ E& F is injective, it is also closed in E& F. Hint: The u(E', E) - u(F, F')continuous operators in C(E', F) form a norm-closed subset, an is continuous and an(T) = 0 if and only if rank(T) < n. This result is due to Valdivia [274], see also [72]. Ex 12.8. Let l/p+ l/q 2 1. Then pp,4and 7p,qdefined by n

Pp,q(z; El F )

:= inf{ Il(ai,j) : GJ

+

I C

Cllwql(zj)wp'(Yi) z =

ai,jzj @ Yi}

i,j=l

Tp,q(z; E , P ) := inf{Il(ai,j) :

G,

-+

1

c n

~ l l e q ~ ( ~ j ) e p ' (z~= i)

ai,jzj 8 Yi}

i,j=l

are finitely generated tensor norms with PPA

Ia P , 9

and

PP,9 IIYPd

*

C,"".

@ Az : $l+nall _< Hint: Mimic the proof of 12.5. using the fact that (llA1ll' llA~l[r)l/r. These tensor norms will be studied in more detail in section 28. Ex 12.9. If (en) and (fn) are bases of Banach spaces E and F, respectively, then (el @ f 1 , e z ~ f ~ , e ~ @ f @fz,e3@ffi1e3@ffi, z,el ...) - the rectangular ordering - is a basis for E@,,F for each tensor norm a. Hint: Show that the expansion operators are uniformly bounded on E @a F and determine the coefficient functionals; use the fact that if the expansion operators of a biorthogonal system are uniformly bounded and converge on a dense subspace pointwise to the identity, then this biorthogonal system defines a basis. This result is due to Gelbaum-Gil de Lamadrid [80]. Ex 12.10. Let (%,A)be a normed operator ideal and define a ( . ; E F) , to be the induced norm from the injection E @ F L* Q(E', p). Then cr is a tensor norm on NORM; moreover, E @a F El' @a F holds isometrically. Which property does Q have if (Q, A) is injective? If (Q, A) is surjective?

+

-+

13. The Five Basic Lemmas

159

Ex 12.11. (a) Show that the tensor contraction

is the same as the composition map

(for p = 00 read co instead of lm). F ) + E 8 d p F is a metric surjection (1, can be (b) Show that (E tP)) BT ($ replaced by co). Hint: 12.7. and 12.9.. (c) For normed spaces E , F and G the tensor contraction

( E 8 G') 8 (G8 F )

-

E 8F

given by (z8 u') 8 (u 8 y) .r,(u', u)z 8 y is defined. Show that for an arbitrary tensor norm a ll(E 8 8 G') 8 r (G@a F ) E @a J'll I1 * +

[(En

-

Hint: En,m(uL, um)zn 8 Urn = u;@n) 8 id^] ( E m urn 8 ~ m ) Ex 12.12. Use corollary 12.9. and the description of compact subsets of the projective tensor product to show the following: For each compact subset K c E&,,,F there are a compact set D c L1 and zero sequences (zn)= ((zn(m))m)n in C * O ( E )and (yn) = ((yn(m))m)nin C ; " ( F ) such that for each z E K there is a (A,) E D with m

m

n=l

m=I

Note the case p = 1 : Each z, is an unconditionally convergent series and yn E c o ( F ) .

13. T h e Five Basic Lemmas This section contains five relatively simple lemmas which are basic for the understanding and use of tensor norms: the approximation, extension, embedding, density and local technique lemma. The power and importance of these devices will become clear while working with them.

19. The Five Basic Lemmas

160

13.1. Recall from section 5 that a Banach space E has the A-bounded approximation property if there is a net (T,) of finite rank operators E + E with llTqll5 X and T,z -+ z for all c E E .

APPROXIMATION LEMMA: Let Q and p be tensor norms (on NORM), E lF nomzed spaces, c 2 1 and on E@N ascp f o r cofinally many N E FJN(F). If F has ihe bounded approximation property with constant X 2 1, then a5Acp on E@F .

PROOF: It is easy to see that for each z E E €3 F

for the projective norm

A

and hence for all tensor norms. If 7 satisfies

and N is as in the hypothesis with T,,(F) C N , then, by the metric mapping property of tensor norms,

which implies the statement. 17 For the finite and cofinite hull of a tensor norm this lemma (and its transposed version) implies the following

PROPOSITION: Let a be a tensor norm (on FIN), E and F Banach spaces with the bounded approximation property with constants AE and X F , respectively. Then

I n particular, Z = d on E@ F i f both spaces have the metric approximation property. 13.2. Every 'p E (E@= F)' = 2 ( E ,F') has a canonical extension which satisfies the relation

'p"

E ( E @= F")'

19. The Five Basic Lemmas

161

where L, E C ( E ,F') is the linear operator associated with ifcpE(E@aF)'?

EXTENSION LEMMA:Let (on NORM). Then 'p

'p

E ( E Br F)' and let

E ( E @a F)'

i f and only i f

(Y

'p

(see 1.9.). What happens

be a finitely generated tensor norm

(p"

E ( E @ a F'I)'

.

PROOF:The metric mapping property implies IIE @a F Q E @a F"ll 5 1 and hence ll'pll... 5 ll'p"ll ... . Conversely, take z, E E @ F" and M E FIN(E) and N E FIN(F") such that zo E M @ N ; then the weak principle of local reflexivity (6.5.) gives for every E > 0 an R E Z ( N , F ) with IlRll _< 1 + such ~ that for ally" E N and z E M (Y", L , Z ) F ~ ~ ,= F '(RY", J?&Z)F,F! This means that (PA9 2 @ y") = ($7, ( i d E c3 R)(z@ Y'V

,

therefore (9" I 20) = (9, id^ 8 R(%,))

and hence I('p",zo)l

Ill'pll llRll4zo; E , N ) Ill'pll(1+

which implies the result since

(Y

&)"(to;

E ,N )

9

is finitely generated (see Ex 12.2.). CI

Sometimes the relation (*) is helpful. The reader may have noticed that the proof is nearly identical with the one given for cv = E in 6.7.. Strangely enough, we do not know whether the eztension lemma holds for cofinitely generated tensor norms. There is also a left-canonical extension "'p E (E" Br F)' of 'p E (E @ r F)' and it was shown in 1.9. that ("'p)" = " ( 9 " ) E (El'@,,F")' if and only if L, is weakly compact; clearly,

I I " ( ~ " ' I I ( E ~ ~ ~ ~ F ~ ~ ) =~ II("(PY"'I(Ew~F)J)J = IIPII(Es,F)~ by the lemma. Define two "natural" maps @j

: El' @ F"

C( E @ a F)"

162

13.

The Five Basic Lemmas

They obviously have norm 1 on E” @a FII . But we do not know which norms the Oj induce on E” @ F“. If the induced norm were a in reasonable situations, this would easily solve the problem of the bidual mappings which will be treated in Ex 17.16.. Clearly, (E €3 F”, (E @a F)’) is a separating duality system with the bracket (cp, 4 := (PA,4

*

The extension lemma and the bipolar theorem give the

COROLLARY: If a is a finitely generated tensor norm, then the unit ball BEB,F is u ( E €3 F N ,( E @a F)‘)-dense in the unit ball BE^,^ . 13.3. Tensor norms do not respect subspaces but the embedding into the bidual is usually respected:

EMBEDDING LEMMA:If a is a finitely or cofinztely generated tensor norm (on NORM), then idE69IcF : E B a F E m a F”

c ‘ .

is an isometry for all normed spaces E and F

.

See Ex 13.4. for a slight refinement of this result.

PROOF: The metric mapping property implies that for all z E E €3 F a(%;E , F”)

I a(%;E , F )

holds for all tensor norms a (the map idE €3 ICF will not always be written). (1) Let a be finitely generated. Then, by the extension lemma, a(%; E , F ) = SUP{I(V~,

%)II 9 E ( E ma F)‘, IIvII(EB,F)~ I 1) = ~ ~ P { J ( P I~P, E~ (I E ~ ~ F ) ~ , J l c p l lI ( ~1)~ , ~ y ISUP{ I($, %)II II, E ( E €3a F‘’)’, I I ~ I ( E ~ , F ~ ~ )IJ 1) = a(%;E , F”) ,

which is the reverse inequality. (2) If a is cofinitely generated and L E COFIN(F), then Loo (formed in F‘’) is in COFIN(F”) and the map KF/L :F/L

L(FIL)” = F”/Lo0

is isometric and surjective; moreover, Q f b o K F = K F / L o Q,“. Therefore, a(idE €3

QT(t.);E,F / L ) = a ( i d ~8 ( ~ F / L QT)(z); E , (F/L)“) = 0

id^ €3 K F ) ( J ) ; E , F”/Loo) 5 < Z ( i d E €3 K F ( Z ) ; E,F”) = Z(Z; E , F”) . = a ( ( i d E €3 Q$o)

o

IS. The Five Basic Lemmas

163

Taking the supremum over all L gives the missing inequality. 0

13.4. In addition, since F and its completion

have the same biduals:

COROLLARY: I f a is finitely or cofinitely generated, then

is an isometric and dense embedding f o r all normed spaces E and F . Density follows from Ex 12.1.. In particular, finitely or cofinitely generated tensor norms respect dense subspaces. This implies that the density lemma 7.3.can be applied to give the more specialized

DENSITYLEMMA: Let a be a finitely or cofinitely generated tensor norm, E and F normed spaces and E, and F, dense subspaces of E and F, respectively. I f G is a locally conuez space and T E Z(E @r F , G ) i s such that

then T E Z ( E 8, F , G)

.

Recall that T : L -+ G is continuous if each S o T : L + D is continuous, where D is normed and S E 2(G,D) . A particularly interesting special case is:

COROLLARY: Let a and ,d be tensor norms such that cr is finitely or cofinitely generated. Z E E(Ei, Fi) and Gi C Ei are dense subspaces such that

If

then TI @ T2 E Z ( E 1 @a E2, FI @g F2) . Since

Ti @ T2 : El Br E2

-

Fi Br F2

2 FI @ p F2 =: G

is continuous, the proof is obvious.

13.5. The local techniques for Zz-spaces (see 3.13. for the definition) are condensed in the following lemma; for E and ?r the argument has already been used several times:

Z P - L o c ~TECHNIQUE ~ LEMMA:Let a and ,d be tensor norms, c 2 0 , and E a normed space such that a
on

E@$

164

19. The Five Basic Lemmas

for all n E N. Then

a < ~ c 7 j + on E @ F for each i$x-space F

.

For a generalization of this from T = idE to arbitrary T see Ex 13.7.; see also 23.1. and Ex 23.13..

PROOF: For N E FIN(F) take a factorization

Then, for every z E E @ N,

this implies the statement. 0 Recalling from Ex 12.3. that a’ a-

is the right-finite hull of a,one notes that actually

5 Acp-

on E @ F

has been shown. Since wp = E on E @ $ (see 12.9.), it follows that E S W ~ ~ A on E E@F for each Cp,X-space F and arbitrary normed space E . Consequently, E6& F = E6&, F holds isomorphically whenever F is an 2;-space - and, e.g., Ex 12.12. gives a nice description of the compact subsets of E& F, which has an interesting application to compact operators (see Ex 13.6.). In section 23 it will be shown that E wp on @ F actually characterizes the Z;-spaces.

-

Exercises:

Ex 13.1. Let a be a tensor norm and E and F normed spaces with the metric approximation property. Then E

@a

F

c,

E @a F” holds isometrically. Hint: Use

z.

165

19. The Five Baaic Lemmas

Ex 13.2. Let El F, G be normed spacesl S E 2(E’, F’) and CY a finitely generated tensor norm. Then there is a linear S : (G @a E)‘ -, (G 80 F)‘ of norm llSll (if G # { 0)) such that

id,i@S

I

commutes; S is unique. Hint: Extension lemma. Ex 13.3. If a is finitely or cofinitely generated and E”&$ + C(E’, F ) is injective, then E&F -+ C(E’,F ) is injective. Is the converse true? Hint: Section 5. Ex 13.4. Check the proofs of 13.2.and 13.3. in order to show that: (a) If a = a* (i.e. a is finitely generated from the right, see Ex 12.3.), then the extension lemma holds for a. (b) If a = ‘ a or a = a+,then the embedding lemma holds for a. Ex 13.5. Use the special version Ex 13.1. of the embedding lemma and the tensor norm from Ex 12.10.to demonstrate the following statement: If @ , A ) is a normed operator ideal, E a reflexive and F an arbitrary Banach space such that E’ and F have the metric approximation property, then

A(T : E

-+

F)=A

( K ~o

T : E -,P I )

for all T E $(E, F ) . This is a sort of “regularity property” of finite rank operators with respect to all normed operator ideals. Ex 13.6. (a) Let E and F be Banach spaces, F an 2;-space and K C $ ( E , F ) a compact set of approximable operators. Then there is a compact D C PI , sequences (&) in $(E,Pp) and (Sn)in g(Pp,F ) with ll&ll + 0 and IISnll 0 such that for each T E K there is a (An) E D with

-

m

T=

AnSno R,,

,

-

for p = m replace t, by c, . Hint: e wp and a modification of Ex 12.12.. It will be shown later on (21.6.)that all 2g-spaces F have the approximation property, hence $(E, F) = R(E, F ) holds. (b) Show that every approximable (=compact) operator with values in an C;-space factors compactly through Pp (or c, if p = m). Hint: 12.9.. See Ex 23.9.for more information. (c) Use (b) to show the following result of Terzioglu [270]: Every compact operator t,(I’). factors compactly through a subspace of c,. Hint: E -,F

-

166

14. Grothendieck’s Inequality

Ex 13.7. Cp-local technique for operators: Let a and /3 be tensor norms and a be finitely generated. If T E C ( E ,F) satisfies Ilidt; @ T :

c No E

for all n E N ,then

-

5 @ p J’ll

llidc @T :G @ aE - G a p

5c

FIJ5 Xc

for each C;,A-space G.

14. Grothendieck’s Inequality Grothendieck’s “thbrkme fondamental de la thbrie mdtrique des produits tensoriels” ([93], 54’ thbrkme 1) states that the identity operator on a Hilbert space is in the operator ideal (3Jur)ini(see section 20); the operators in this ideal were called “pr6intdgral” by Grothendieck. A remarkable achievement of Lindenstrauss and Pelczydski in their celebrated 1968 paper [173] was to have noticed that this theorem can be reformulated in the form of an inequality concerning n x n matrices and Hilbert spaces - as well as in the form 2 ( t l , t , )= p~(t!l,tz). This is why the fundamental theorem is nowadays called Grothendieck’s inequality - in matrix form, in tensor form, or in operator form in order to highlight its various glamorous facets. Our path of entry to the many equivalent formulations of the inequality is establishing the fact that 7r and w2 are equivalent tensor norms on loo @loo; we follow Krivine’s proof [155], which was smoothed out by Pisier in his book [225] and polished into its final form by Pelczydski. Simple as it is now, the proof is in some sense dual to Grothendieck’s original one, but has still the same ingredients.

14.1. The idea of the proof is to take a function

f E C ( K )@w2 C ( K ) (where K = (1, ..., n}), factor it as in 12.10. through some Hilbert space

e,i.e.

n

with 11zk11 = llgllll = 1, and use an integral representation of (z,y) for elements in the euclidean unit sphere to obtain a new representation

167

14. Grothendieck's Inequality

with the functions g and h being uniformly bounded on Rn (with values in C ( K ) ) . However, the formula for (z,y) reads = sin [JR....dyn] , so it is better to represent sinf instead of f as in (*). In order to be able to define sinf (with control of the C ( K ) is a Banach algebra: norm) one needs the fact that C ( K )BW2

LEMMA: If K and L are compact, l / p + l / q 2 1 , then C ( K )8%

C(L)

i s a normed algebra (under the pointwise multiplication coming f r o m C ( K x L ) ) , i.e., :ffi,fa E C ( K )8 C ( L ) ,then aP,q(fl

I O b , q ( f 1 ; C ( W ,W m p , q ( f 2 ;

f 2 ; C(K)l C(JW

PROOF: Represent

fi

C ( K ) ,C ( L ) )

N

= En=l ALgi 8 hk. Then

c N

fl

f2

=

A:A;

(g:g;)

8 @Ah;) E C ( K )8 C ( L )

.

n,m=l

and likewise for 2 ~ ~ 1 ( ( h A h & )the ~ , ~result ) , is obvious. 0 In particular, the space PW&,a which is nothing more than (z1 * -72, ek

Pw is a Banach algebra under pointwise multiplication

8 e f ) = (z11ek 8 e f ) ( 1 2 , ek 8 e f )

in tensor notation. If C(amtm) is a power series with infinite radius of convergence, then E,"=,a,zm is defined in the Banach algebra & &i for each z E & 8 & and 00 W

for all k,A!= 1, ...,n by continuity of the point evaluation ek 8 el E (PwBW,PW)'. 14.2. The second ingredient in the proof will be factorization:

LEMMA: For z E Pw 8 PW with w2(z;P,,PW) < 1 there is an rn E EV and elements ...)2,, y1, ...,f/n E g of n o m 1 svcA that

21,

n

168

14. Grothendieck's Inequality

PROOF: Each representation z = allows one to define functions

xzl

Ui

8~ with W Z ( U ~ ; %< ) 1 and Z O ~ ( V ~ ; % ) < 1

The construction in this proof is the same as in 12.9. and 12.10..

14.3. Denote the sign function R + {-1, 0, +1} by sign. The main tool for the proof of Grothendieck's inequality is the following formula for the scalar product (=duality bracket) of vectors in the euclidean unit sphere of R".

where 7n is the Gaussian measure on

R"

.

PROOF: There is a 0 E [0, 7r] such that ( 2 ,y) = cost9 = sin (5 - e) . Let T : R" -, R" be a linear isometry (with respect to the euclidean norm) such that

Tx = e l and Ty = (cos 8, sin 8,0, ...0) . Since the Gaussian measure is invariant under isometries (8.7.),it follows that 7r

5

sign(2, w)sign(y, w ) m ( d w ) = J R n

7r

=Z J R m sign(e1,w)sign((cm 0, sin 8,0, ...0), w)7n(dw)= 1 -I . J J sign s .sign(s cos e + t sine) exp(--(s2 + t2))dsdt = 2 2 x R R 2

=a 1" Jdm

r2 sign(r cos 'p) sign(r cos 'p cos e + r sin psin e) exp(--)rdrd'p

=$l"

2

-

sign(cos 'p) sign(cos(9 - O))&

.

=

14. Grothendieck'a Inequality

169

The integration of this simple step function gives 1

=4 [ ( 2~ 28) - 281= 2 -8 , f

hence the result. 0 14.4. Now everything is prepared for proving

GROTHENDIECK'S INEQUALITY

(tensor form): There is a constant c such that

n(.;qL,CJI c

. w z ( * ; ~ , ~ )

for all n. If KG is the best such constant, 2hen (real theory) (complet iheory).

PROOF:The complex case will follow from the real one, which will be treated first: Take a := ln(1 + < 1 which satisfies sinh(a) = $(exp(a) - exp(-a)) = 1, and z E Pm @ P , with w ~ ( ~ ; P , ~ e " , < ) a . It has to be shown that n(z;Pa,Pm) 5 n/2 . The element

a)

22k+1

m

sin z := C ( - l ) k (2k + l)! E C @ W , e " , k=O

is defined in the Banach algebra from .&) and satisfies

BW2 P , (by 14.1., recall: pointwise multiplication m

= sinh [wz(.z;P , ,e",)]< sinh(a) = 1 . Lemma 14.2. provides an rn f I? and 2 1 , ...,zn, yl, ...,y, E (zk,

Since, by formula 14.3.,

yi)

= (sin 2, ek 8 e l ) = sin [ ( z , e k 8 e l ) ]

of norm 1 such that

.

14. Grothendieck's Inequality

170

it follows that

This implies that

which implies the inequality in the real case. To treat the complex case the notation @x,aK and w r will be used. It has to be shown that ac<2K$w? on GacG,

Kz

where denotes Kn with the l,-norm. The R-linear identity map I : 0 ,4Ez has norm 1 and its inverse has norm &. The space ( G ) R is nothing but iZ& considered as a real space; it was shown (Ex 2.11. and Ex 3.15.) that the natural map

has norm one (it is even a metric surjection). For the w2-norm observe first that for PmEx 8.1. implies that

21,...,zmE

NOW take z0 E G@'P,with w f ( z o ; G , G ) < 1. Then there are elements 21,...,zrn and yl, ...,ym E G with zo = C?=:=, zhmcyk and wf(zk) 5 1 as well as wf(yk) 1. Therefore

<

rn

21

:=

EzL aR yk E (G)R

@R

(G)R

k=l

satisfies O(z1) = zo and w f ( z 1 ;

( E ) R(G)R) , < 1. It follows that

171

14. Grothendieck'8 Inequality

14.5. By 1-complementation it is clear that r 5 Kcw2 on Pm 8 hence the &-local technique lemma 13.5. gives that:

COROLLARY 1: If E

is a CL,x-space and

for all n, rn E M ,

F a E&,,-space, then on

E 8 F .

In particular,

COROLLARY 2: For compact spaces K and L

PROOF: Since, by the preceding corollary, a 5 K G W on ~ C ( K )8 C ( L ) ,it follows that the natural map C ( K ) & , , C ( L )= C ( K ) & w , C ( L )4C ( K ) & C ( L ) = C ( K x L ) is injective. By 12.10, the assumption on the function f means that f E C ( K ) & , , , C ( L ) and wZ(f;C ( K ) ,C ( L ) )5 Ilglloollhlloo,which implies the statement. 0 Recall that f E C(K)&,,C(L)means that f has an absolutely convergent representation

which easily gives a representation f = (9,h) as in the corollary.

COROLLARY 3: If H i s

a Hilbed space and X I ,...,xn,y1, ...,yn

E H , then

172

14. Grothendieck'a Inequality

or For H = t z or LZ it does not matter whether one writes the scalar product (-1 the duality bracket ( ., ..), since, clearly, there are elements #j with 11#j:jll= llyjll and . a )

(zilyj)

= (zi,#j)

*

An immediate consequence is the Lindenstrauss-Pekzyriski formulation:

GROTHENDIECK'S INEQUALITY(matrix form): Let (ai,j) E C(Kn,K")be an n x n matrix. Then for each H i l b e r i space H

PROOF:Define cp E (& Br C)' by

then

Hence for zi, yj E BH

by the preceding corollary. This is the claim.

14.6. Note that each of the last four statements was deduced directly from the former one with the same constant; hence the best constants in the last four statements are non-increasing. Actually they are the same:

14. Grothendieck'a Inequality

173

PROPOSITION: The best constants in - Grothendieck's inequalitiy, tensor form - Grothendieck's inequalitiy, function form (corollary 2) - corollary 3 - Grothendieck's inequalitiy, matriz form all coincide. In particular, it will be shown that all of the last four statements are equivalent formulations of the same fact.

which is a metric surjection: For wz(z;Pm,Pm) ...,znand y1, ...y, E B12 with

<

1 there are (by 14.2. or 12.9.)

21,

=C

( z i , Yjtj)ei 8 e j

9

i ,j

hence ~ ( zPm, ; &) 5 KG , which implies the tensor form with the constant from the matrix form. 0

14.7. Considerable effort has been made to find the exact values of the Grothendieck constants K$ and K$ but they are still unknown. A lower estimate (due to Grothendieck) can be obtained as follows. Denote for the moment by KLG the best constant in the little Grothendieck theorem (11.11.); then: PROPOSITION:K ~ I GKG

PROOF: For E > 0 there is an ra E N and T E Z(P,,&) with JlTll= 1 and P Q ) > KLG - E ; this establishes the existence of 21,...,zm E P , with m

C l l T ~ ~ 12 1( K~ L Gk=l

E ) and ~

wa(z1,

...,zm;e",)= 1 .

174

14. Grothendieck's Inequality

(c c)'

The functional cp E ar defined by ( c p , ~8 y) := (Tz,Ty) has norm 1 and therefore (with some flk of modulus 1 in the complex case) m

m

m

It will be shown in 20.19 that the best (real and complex) KLG are actually the ones given in 11.11. : K f G = and K f G = 2/+, hence A

1.57079... = - < K E 24

1.27323... = - 5 K c

5 <

A

+ 4)= 1.78221...

2 1 41 A

- ln(l+fi)

= 3.56442...

Pisier [220] showed that K,$ _< exp(1 - 7) = 1.5262... (where y is Euler's constant) and Haagerup [98] improved this to KE 5 1.40491 using a representation of the scalar product

for z,y E e 2 (complex) with 1 1 ~ 1 = 1 ~ llyl12 = 1 (sign denotes the complex sign function) which he deduced from the representation of the real scalar product. The difficulty is that the function

is much more complicated to handle than the inverse of the sine function, but apart from these tough analytical calculations, the proof runs along the same lines as the proof of the real case since W 1can be expressed as a power series (in X and I). In particular, these estimates show that KZ < K Z ; Krivine [154] showed directly by an estimate of the norm of the complexification of a real operator Pm -, that K g 5 f i K $ (see Ex 14.4. for a slightly worse estimate). Concerning the lower bounds it seems that Krivine showed that Kg 2 1.676 and Davie that K g 2 1.3381, but these results are unpublished (see Pisier [225], p.68, Jameson [119], p.114 and Konig [I481 ).

14.8. We resist the temptation of giving the operator form of Grothendieck's inequality now (see 17.14.) and prefer instead to continue with the theory of tensor norms in order to discover the precise connection between operator ideals and tensor norms. Nevertheless, we prove a result, which will be seen to be nothing but c(tl,tz)= Vl(t1,ez).

175

14. Grothendieck’s Inequality

PROPOSITION: a I K c d , on e;t 84 for all n E N.

PROOF:For ti E e with

ti

n = Cj,l ai,jej

Hence, for every z = CEl Zi 8 Vi E

84

i=l

j=1

i=l

by Grothendieck’s inequality (matrix form). Taking the infimum over all representations of z gives a(z;G,G)5 K c d , ( z ; G , G ) . 0 It will turn out that the best constant in this relation is also the Grothendieck constant K c (see 17.14.and 20.17.). For other versions of Grothendieck’s inequality see 17.14., 19.5., 20.17.,22.4., 23.10.and Ex 23.18.. Grothendieck’s original formulation will be presented in 20.17..The so-called “finite dimensional” Grothendieck constants will be treated briefly in 20.17.~ well. Other proofs of Grothendieck’s inequality will be given in Ex 14.1.,31.6. and Ex 32.7..

Exercises:

Ex 14.1. Take elements t 1 , ...,tn,yl, ...,yn E R” of euclidean norm 1 and denote by 0(ti,yj) E [0, a] the angle between ti and yj. (a) Prove that for all k E IV

176

14. Grothendieck's Inequality

Hint: 14.1. and 14.3.. (b) Check that

(c) Prove that A(

2

A

(zi,yj)ei @ e j ; c , c )

5 sinh(-)2

.

iJ=l

(d) Show that these arguments prove Grothendieck's inequality in tensor form with 5 sinh(~/2).Hint: 14.6.. This is a slight modification of Grothendieck's original proof of the "th6orhme fondamental" . Ex 14.2. Let K and L be compact sets, (o E BiI(C(K),C ( L ) )and g k E C ( K ) ,hk E C ( L )be given. Then

Kg

What happens if (o comes from a signed measure on K x L? Ex 14.3. Let Sn-lbe the unit sphere of q and sn(z,y) := (sly) for (a) Show that sn E C(Sn-l) 8 C(Sn-l) and

2 ,y

f

5 1 , A(&; c(sn-I),c(sn-1))5 K G . yl, ...,yn E Sn-l (b) Prove that for all 21, ...,zn, W ( s n ; C(sn-l),

c(&-1))

n

A(

C (zilYj)ei 8 e j ; c ' G )

5 ?r(sn;~(~n-l),~(~n-l))

-

i,j=l

Hint: A ( C i , j g ( z i ) h ( y j ) e i 8 ej;&,&) 5 Ilgllmllhllw for all g , h E C(Sn-1). (c) Conclude that li%+m r(sn;C(Sn-l), C(Sn-l))= KG . Hint: Corollary 3. For a more precise statement of this result see Ex 20.16. (and 20.15.). Ex 14.4. Recall from Ex 11.19.(d) that for every A E CT 1A1=

1'" 0

.

Ih(eitA)ldt

(a) If A : Pw-+ Ry is an R-linear operator and Ac its complexification then

llA": Cw

-+

A q I I 5 5llA : Rk

-+

-+

q,

Ry(I .

See also 26.3. and Ex 28.14. for the complexification of operators.

(b) Show that K g (c) For

A :=

(i )1:

5

5 KE . Hint:

Matrix form.

: RL + R: the following holds: llA"ll

the best constant in hence obtains KF 5

is 2

f i .Krivine

= ./211All. This shows that

[154] calculated that it is actually

f i and

25. Dual Tensor Norms

177

15. Dual Tensor Norms

The purpose of this section is to study the norm which is induced on E’@ F’ by its embedding into (E @p F)‘. This investigation is quite fundamental for applying tensor product methods to operator ideals, as will be seen shortly. It also gives new insight into the importance of the cofinite hull of a tensor norm. Moreover, some notions (“accessibility”) concerning the relationship between and are defined and tensor norms close to Ap are investigated.

15.1. For two separating dual pairings ( E i ,Fi) of normed spaces the bilinear mapping

(El 8 E 2 ) x (Fl 8 F2) N

M

n=l

m=l

N

K

M

n=l m = l

gives a separating dual pairing (El 8 E z ,FI @ Fz) according to Ex 2.4.. This simple and natural pairing is also called trace duality since it is the restriction of the trace duality studied in 2.6.: Clearly, the duality bracket induces a mapping 0 CP : Fi @ Fz

-

(El @ E2)’ = L(E1, E ; )

which is injective (since the pairing is separating) and the elements of the range of 0 are certain finite rank operators El + E i . For u E El @ Ez and v E F1 @ FZ let Tu E $(,Ti, E2) be the operator associated with u and L, E $(Ez, E i ) the operator associated with w := C P ( V ) ~ E (E2 8 E l ) * . Then (21%

4 ~ ~ c x m=,(@(v), ~0~ ~ (~ E ~ @ E ~ ) * ,= E ~tm2 B E(T~’

-

0

L,) = t r E ; ( L ,

0

T,) .

Note again that transposing z E E @ F 4 2 ( E ‘ , F) means passing to the dual of the associated operator T, - more or less: Tl = ICE o T,t . 15.2. The interesting cases for the (restricted) trace duality are the injections

E @F E l @F E’@F’

(E‘ @t F’)’ ( E QDC F’”’ (E&F)

(E‘ F’)’ ( E @o F’)’ c (E@pF)’

(E’ & F‘)’ ( E ~F’)‘ x (E@rF)’

(p any tensor norm). Which norm is induced on E @F by (E’@pF’)’? TOanswer this question dual tensor norms will be introduced. Recall that M

@N

= (MI@* N’)’

178

15. Dual Tensor Norms

holds algebraically for all M, N E FIN.

PROPOSITION: Let a be

a iensor norm on

FIN. Then a' defined by

I

a ' ( z ; M ,N) := sup{((z~,~)Ia ( u ; M',N') 5 1) (for all z E M 8 N ) is a iensor norm on FIN.

PROOF: To apply the criterion 12.2. (for FIN), observe first that a'(-;

r) is a norm,

(2) is obvious and the mapping property (3) follows from

((Tl @ Tz)z, u) = (2,(T:63 Ti)u)' In other words,

M

@at

N := (M'

*

0

ma N')'

~-

A

holds isometrically by definition. The finite hull a' of a' will be called the d a d tensor norm a' (on NORM) of the tensor norm a (on FIN or NORM). 15.3. The following properties are obvious: (1) If a 5 cp, then 5 ca'.

( 2 ) a = a" on FIN and = a". (3) a =a ' on NORM if and only if a i s finiiely generaied. By dualization, the relation

and hence

E

5 a' 5 R implies for a = E that

d = E and E' = R .

This is part of the duality relation between the projective and injective norm which was treated in section 6, where, in fact, much more was achieved. Note that for two finitely generated tensor norms a and p the relation /3 = a' is just a statement about finite dimensional spaces. It is obvious that (a')' = (a')' and this norm is denoted by a*;it is called the adjoint or contragradient tensor norm of a.At this point it may not be apparent why one would want to consider a*.The reason will become clear in 17.9., where the relationship to adjoint operator ideals will be explained. 15.4. There is an interesting connection between Saphar's tensor norms gp and its duals:

REMARK: g; = db 5 gpj for 1 5 p 5

00

(where p' i s the conjugate exponent of p ) .

179

15. Dual Tenaor Norms

PROOF:From dl = g1 =

?r

it follows that d: =

1 < p < 00 it has to be shown that

llM @gp, N

(M’

E

5

g m and db,

5

?r

= 91. For

a d p N’)’ll 5 1

for all M , N E FIN since it is enough to check d; spaces. In other words, it has to be shown that

I gp’ on finite dimensional Banach

I(u, 41 I dp(u; M’, “)gp’(Z;

M ,N )

for allu E M ’ @ N ‘ and z E M @ N . Take = Enz;@d and z = Em zm@vy, (finite representations with all entries different from zero); then Holder’s inequality gives

n

m

I epj (zm )wp’(zk )wp (3/m )ep ( A1

*

Passing to the infimum over all representations of z and u yields, by the representation 12.7. of dp and gP’, the desired inequality. 0 At the end of this section this remark will be used for determining the tensor norms close to Ap. The inequality g;‘ 5 g p will be improved in a certain sense in 20.14..

15.5. It is clearly of fundamental importance to know in which sense the isometric embedding for finite dimensional spaces N‘

L( M

N)’

extends to infinite dimensional spaces. The answer is given by the

DUALITYTHEOREM: Let a be a tensor norm (on FIN). Then for all normed spaces E and F the following natural mappings are isometn’es:

15. Dual Tenaor Norms

180

PROOF: To prove (3), observe first that FIN(E') = { K o l K E COFIN(E)} and that, for ( K , L ) E COFIN(E)x COFIN(F)

c E' @ F'. Now,by the valid duality relation

whenever z E E @ F and u E KO @ Lo for finite dimensional spaces

and this is exactly (3). The extension lemma 13.2. and the commutative diagram

imply (2) - and (1) follows in the same way. 0 The proof showed that the duality theorem is nothing but a reformulation of the definition of the cofinite hull. The duality theorem indicates that the cofinite hull of a tensor norm a will be an extremely helpful device. Since & 5 a,the theorem implies in the theorem (8: replaced by BU)are conin particular that all mappings @a + tinuous and of norm 1 (if E and F are different from (0)). The approximation lemma 13.1. gives = a on E @ F if both spaces have the metric approximation property, 1 (El F')' in this case. But what is the situation in general? hence E BU F

-

a

c

z

15.6. To get information in this direction recall first from section 13 that 5a5 holds for all tensor norms a on NORM. A tensor norm a is called right-accessible if

z

+

a(.; M , F ) = Z(.; M , F )

for all ( M , F) E F I N x NORM, left-accessible if a ' is right-accessible, and accessible if it is right- and left-accessible. a is called totally accessible if = i.e. if a is finitely and cofinitely generated. The injective norm E is totally accessible (this was already

z z,

181

15. Dual Tensor Norm8

mentioned in 12.4.), the projective norm A is accessible by theorem 6.4. (duality of and E ) and the duality theorem - but it is not totally accessible: The mapping

E’63rF’-W g(E,F’)

4

?r

J ( E , F’) = ( E B CF)’

is in general not injective (see section 5 about the approximation property and also 6.2.), hence A # by the duality theorem. It will be shown in section 21 that all ap,q are even totally accessible. are accessible and that all Is every finitely generated tensor norm accessible? This problem seems to be quite hard since - by the approximation lemma - the non-accessibility of a tensor norm can only appear on spaces without the metric approximation property. In section 31 Pisier’s space P (which does not have the approximation property, see 5.2.(1)) will be used to construct a tensor norm which is neither left- nor right-accessible. It is obvious from the definitions that the presence of accessibility facilitates the investigation of questions involving trace duality. It will turn out that in some situations an accessibility assumption is even necessary, see 25.4., proposition 1. In section 21 the accessibility properties of tensor norms will be “translated” to conditions on operator ideals. Concerning the distinction between right- and left-accessible it is worthwhile to note that in specific situations it is sometimes enough (and easier) to only check that a tensor norm is accessible from one side - and that there are even tensor norms which are only accessible from one side (see 31.6., corollary).

PROPOSITION: Let a be a tensor norm on NORM. (1) a is right-accessible if and only if a’ is right-accessible. (2) If a is accessible, then the transposed tensor norm a t , the dual tensor norm a‘ and the adjoint tensor norm a’ are accessible.

If a is totally accessible, a‘ is accessible (by (2)) but in general not totally accessible: take a = E as an example.

PROOF:Clearly, only (1) needs

a proof. Take a right-accessible; then the duality

theorem gives

M I aallF’

A M I a;

F’

A M I a:

F’

A (M

F)’

for finite dimensional MI hence

+

+

-

which means that a’ = a’ = a‘ on M @J F (again by the duality theorem). This shows that a’ is right-accessible. Conversely, if a‘ is right-accessible, then a’’= Q is right-accessible and this is the same as a being right-accessible. 0

182

15. Dual Tensor Norms

15.7. Recall that it is an immediate consequence of the approximation lemma 13.1. that a = fl on E 8 F whenever a = p on M 8 F for all M E FIN(E) and E has the metric approximation property. This observation and the duality theorem show that for a tensor n o r m a the relations

E @a F

1

A E 8-a F and E&F

(El

F‘)‘

hold in each of the following three cases: (1) E and F have the metric approximation property. ( 2 ) a i s right-accessible and E has fhe metric approximation property. (2’) a is left-accessible and F has the metric approximation property. ( 3 ) a i s totally accessible. So, “two ingredients” are necessary in order for a “good” relationship to hold between

a and a’.For the bounded approximation property the relations hold isomorphically. It will be seen in the next section that for a = T the metric approximation property is in some sense even necessary for the “good” duality. Note also that each of the conditions (1) - (3) (with “metric” replaced by “bounded”) implies that the natural map E&F E&F is injective, but this observation will be improved in 17.20.. The definition of the dual tensor norm (as the finite hull of what one might have expected) and the duality theorem (explaining the role of the cofinite hull) appear innocent enough. However, in the forties Schatten defined the dual norm by using the embedding E 8 F C (G ma F‘)‘ and ran into severe difficulties; it was Grothendieck in his Rksumk who used the definition of the dual tensor norm as a finite hull - and it turned out that this is the appropriate notion for the investigations he had in mind, of a for example, for the study of operator ideals. By the way: the cofinite hull (see [93], p.11). tensor norm a is, by the duality theorem, Grothendieck’s norm 11 Nowadays, when it seems natural to study Banach spaces and operators in terms of ++finite dimensional spaces, the definitions of a,a and a‘ fit easily into the accepted philosophy - a successful philosophy whose beginning was marked by Grothendieck’s definitions (and results) in his %sum6 1954.

-

15.8. Two tensor norms being equivalent on some E 8 F is always a fruitful situation. Up to now, Khintchine’s and Grothendieck’s inequalities have produced the most powerful equivalences. In 13.5. the relation E 5 wp 5 XE on E 8 G was proven to hold whenever G is an 2;,Aapace. Dually: PROPOSITION: Let E and G be normed spaces, G an 2;,A-space. Then w;

5 f 5 Xw;

PROOF: For NE FIN it follows from E = wp#on N‘ 8

e;:

N = (q,8 t N‘)’=

($1

-

onG8E.

c,that

Bw;, N’)’ - (GI QW,

Nt)I = q Bw;N

.

15. Dual Tensor Norms

183

The &-local technique lemma 13.5. implies the result since wh is, by definition, finitely generated. 0 The trace on E‘ @ E is always continuous with respect to the projective norm z. Even more holds in the above situation: The proposition implies that the traces

and

(for a Hilbert space H) are continuous. It will be seen in section 23 that the statement of the proposition is actually characteristic for C;-spaces.

15.9. A nice analytic application is the following: In 8.8. the values of

were calculated explicitly by using some averaging and the knowledge of the &-norm of - e k 8 e k (which is the operator norm of id : $, -+ To obtain a result of this type for general tensor norms a define for the moment

q).

and the operators

S(Z1,- - - I Mw(21,

2,) -.-I

:= (

2,)

~ 2 1 ~ - .3. I 1t n l ~ 1 )

:= ( E I ( W ) Z I , - - - , E n ( W ) Z n )

for w E D, = {-l,+l}n, where the ~k are the Rademacher functions from 8.5.. It is easy to see that

for all z E K” 8 Kn (duality in (Kn8 K”,Kn@ Kn)).If z = t o:= ELzla k k e k 8 ek is the associated “diagonal” element, then

a i j e j 8 e j and

184

15. Dual Tensor Norms

(where pn is the "Rademacher" measure on Dn)and n

C sk8 sk(t0) = t r ( t ) Jn) = ( z , Jn))Jn) .

(3)

k=l

This is obvious for z = ei @ ej and the general case follows from linearity. Now everything is prepared for demonstrating the following result whose idea is due to Garling and Gordon [79] (see also Jameson [119], p.72):

PROPOSITION: Let E and F be Kn eqvipped with n o m s such lhat llMwII = llSll = 1 (in E(E, E ) and 2 ( F , F ) ) for all w E Dn.Then

k=l

k=l

for all tensor norms a.

The conditions on E and F are satisfied for the tp-norms and more generally for Orlicanorms on K". Note further that the result implies a relationship between certain norms of the identity operators E' -+ F and E --+ F' (see section 17). A result which is a bit more general will be treated in Ex 15.4..

PROOF: Since n = (u("), d"))5 a(u(");E , F)a'(u(");E', F'), one inequality is obviF')' = E a a F holds isometrically; ous. To see the other one, observe first that (E' therefore there is a t E E 8 F with a ( z ;E, F) = 1 and (t,d"))= a'(u(");El, F'). Then, by (2) and (3) a(u(");E , F)a'(u(");E', F') = ( 2 , d")) . a(u(");E , F ) = n

8 sk(Z,); E , F ) I n +,;

=

E,F ) I

k=l

a(Mw8 M W ( z ) ;E , F)pn(dw) L n

by the mapping property of a and pn(Dn) = 1 . 0 15.10. The duality relation for the "one-sided" tensor norm Ap is exactly what one would expect:

PROPOSITION: For 1 5 p 5 00, any normed space E and any measure p

15. Dual Tensor Norms

185

PROOF:Since A1 = A and Am = E (see 7.1.), the claim for p = l,00 follows from the duality theorem because e and A are accessible and L1 and L, have the metric approximation property. For 1 < p < 00 the assertion is an immediate consequence of the isometric embedding

(see Appendix B12.). 0

COROLLARY 1: For every measure p , every normed space E and every 1 5 p 5 00 the following inequalities hold: on L p ( p ) @ E . dp 5 A, 5 s;‘ PROOF: Since d, = w1 = E on L, @ E and A = w< = g& on L1 @ E (see 13.5. and 15.8.), the result holds for p = 1 and p = 03. Let 1 < p < 00. Since dp and Ap are smaller than the projective norm, the relation dp 5 Ap only needs to be verified for step functions n

the (Am) being pairwise disjoint and of positive measure (by the density lemma 7.4.). 0bviously, Ap(f) = l p ( p ( A m ) l / P t m ; E ) and

dp(f; L p , E ) 5 ~p’(p(Am)-l’PXA,; Lp)ep(p(Am)l’PZm; E ) -

For g E B L ~ ,

and it follows that w ~ ’ ( ~ ( A ~ ) - ~ / P x A 5 ,1.) This shows that dp 5 Ap. To see that the other inequality holds use duality: By the proposition and the duality theorem there are isometries

t86

15. Dual Tensor Norms

and the vertical mapping has norm 5 1 by what was already shown. Therefore, c

Ap 5

4

1

5 dil = giI

and the result is proven. 0 Using g i l

gp from 15.4., the Cheuet-Persson-Saphar inequalities $8

I dp I Ap I $ + I ‘9p

on

L, 8 E

are obtained (due to [27], [208], [242]), where E is an arbitrary normed space.

COROLLARY 2: d;, = dp = Ap = A; = g;l = gp on L p ( f 2 ~ , p 8 l ) Lp(R2,p2) for arbitra y measures pi.

PROOF:Since dil 5 dp 5 Ap 5 g;l 5 gp on L p ( p l ) 8 E and gp 5 A; 5 dil on F 8 Lp(p2)by the Chevet-Persson-Saphar inequalities, the result follows. 0 Later on (25.10.) the spaces E such that Ap = dp (and so on) on Lp 8 E will be characterized.

15.11. To finish this section it will be shown that dp and norms closest to Ap, a result of Gordon-Saphar [87].

are actually the tensor

PROPOSITION: Let a be a tensor norm, p a measure such that L p ( p ) i s infinite dimensional and c 2 1. (1) If a 5 cA, on L p ( p )8 E for all normed spaces E , then a I cdp. (2) If Q i s finitely generated and Ap 5 ca on Lp(p)8 E for all normed spaces E , then g;1

I CQ.

Actually, the condition that a is finitely generated in (2) is superfluous, see Ex 15.6.. See also 25.10. and 29.11. to understand how close dp and Ap are.

PROOF: Since lpis “nicely” 1-complemented in L p ( p ) (see Ex 7.1.), it is enough to consider tPinstead of L p ( p ) . (1) For the tensor contractions Ci (see 12.9.) the following diagram commutes

187

15. Dual Tensor Norma

Ex 12.11. showed that (71is a metric surjection and llC2ll I 1; since the left vertical map has, by assumption, norm I c, it follows that the right one also has norm I c. This means that a 5 cd, on G @ E. Note that E could be fixed.

(2) If Ap 5 c a on for A,)

+

and therefore a' c

ep @ E, then

I cApt on 4,)

a'

I c$,

on

M 8 E for all M

giI

Ica. 0

(by the duality theorem for a and the duality 15.10.

@

El. From the first part of the proof it follows that

on M' @ E' for all M E FIN. Dualizing again yields

E FIN. Since gi, and a are finitely generated, this implies that

For a fixed normed apace E the proof shows that a 5 cdp on all G @ E if (Y 5 CA, on L p ( p ) @ E for some infinite dimensional L,(p); using the fact that g i l is totally accessible (this will be proven in section 21), statement (2) reads: gi, G 8 E if Ap I c a on L p ( p )@ E (where a is finitely generated).

I c a on

all

15.12. A neat application of the fact that there are tensor norms close to A, is the following alternative proof of the extension of Beckner's result given in 7.9. (we omit the case of subspaces El @A; E2) :

PROPOSITION: Let 1 5 q I p 5 some measures p and v. Then

00

and T E 2 ( E , L p ( p ) )and S E 2 ( L , ( v ) , F ) for

PROOF: For z E E @ L, the Chevet-Persson-Saphar inequalities and 15.4. imply that

This result does not hold for p < q : This is the content of Maurey's factorization theorem which will be treated in 18.8.. In 26.3. the reader will find an explicit counterexample and other positive results in this direction. See also section 32.

15. Dual Tensor Norms

188

Exercises: Ex 15.1. Let a be a tensor norm on NORM and k E HV. Then a('), defined by a(')(z;E, F )

:= sup{l(u, z)I

I

II

E E' 8 F', rank(u) 5 k,a ( u ;E', F') 5 1) ,

is a tensor norm on NORM. If a is finitely generated, then c

lim a ( ' ) ( ~ E l; F) = a'(z;E , F )

k-wx

for all normed E l F and z E E 8 F. Hint: Duality theorem. Ex 15.2. Recall that there is a natural continuous mapping

J : E&&F

-+

E&F

C(E',F ) .

(a) Take z E EgaF.Then (p6&idp(z) = 0 E K 8 F = F for all cp E E' if and only if the associated operator J ( z ) : El -* F is zero. (b) Formulate conditions on the spaces El F and the tensor norm a in terms of accessibility and the bounded approximation property such that the following holds for each z E E g a F : If ( ~ & i d ~ ( z=) 0 for all cp E E', then J = 0. Hint: 15.7.. For another weaker condition see Ex 17.11.. Ex 15.3. (a) Show for E = that for every E > 0 there are R E C(E,$) and S E C ( G , E ) w i t h S o R = i d E and

C

(c n

Ils((llRll 5 w2

ek

8 e k ; E', E) (1 4- €1.

k=l

Hint: 12.9.. (b) Show that the trace on f!, @ l p t is not continuous with respect to w i if p # 2. Hint: z ) this would imply wZ(u("); $,, 5 c for some c and hence, by With (tr,z) = (d"), (a), d ( c ,6 )5 2c; but d($, 4) 00 (a well-known fact, see also Ex 26.4.). This is a proof by local technique. Later on (23.2.) a more natural result will be given: if the trace on E is wi-continuous, then E is isomorphic to a Hilbert space. Ex 15.4. Using the ideas of 15.9.show that: If E and F are lK" equipped with some norms, then

C)

-+

where

k=l

is the symmetry constant of the canonical basis

(ek)

of E .

15. Dual Tensor Norms

189

Ex 15.5. Show that L z ( p ; H) = L z ( p ) 6 d a H = L z ( ~ ) holds ~ ~ ~isometrically H for all Hilbert spaces. Hint: 7.2.. Ex 15.6. Use the fact that g; is totally accessible (to be proven in section 21) to show that for each tensor norm a the inequality Ap 5 c a on all .tp 8 E implies that gp+/5 ca.

E x 15.7. Is the natural duality ( E g 0 F ,E’ @ F’) separating? Ex 15.8. Show by using the duality theorem that

Ex 15.9. If E is an Ci,x-space, then dp 5 Asp+, on E 8 F for all normed spaces F. Hint: 13.5.. Ex 15.10. Let a be a tensor norm. (a) If A c E&F is bounded, then A is u(EI&F, E’ 8 F‘)-bounded. (b) The converse of (a) is true for a = E , but not for a = ?r. Hint: Uniform boundedness principle. (c) If a = e or E’ @ F’ is norm-dense in (EGQF)’,then every U ( E & I ~E‘F@ , F‘)convergent norm-bounded sequence in E g 0 F is weakly convergent. Hint: 4.6.. Note that if for two locally convex topologies on a vector space the convergent sequences coincide, then the Cauchy sequences also coincide (proof?). A subset A of a normed space is called weakly conditionally compact if every sequence in A has a weakly Cauchy subsequence. Assume now for (d), (e) and (f) that E and F are Banach spaces, F separable and a is such that every norm-bounded, a(E&F, E‘ 8 F’)-convergent sequence in E&F is weakly convergent. This implies, in particular, that the duality C(E‘, F ) is injective). system (E&F, E’B F’) is separating (and 80 E&F (d) A bounded subset A c E&F is weakly conditionally compact if it satisfies (denote J1 : EGaF C(E’, F ) and JZ : F & I ~ ~ C(F’, E E ) the canonical maps) (1) ([J~z](z’)~z E A} c F is weakly relatively compact for each 2’ E E‘. (2) ([J&](y‘)lz E A} c E is weakly conditionally compact for each y’ E F‘. Hint: Take a countable D c F‘ which is u(F’,F)-dense, hence dense for the Mackey topology p(F’, F ) . For a given sequence in A choose a subsequence (2”) for which ([JzzA](y‘))converges for all y’ E D. For 2: 8 yb E E‘ 8 F‘ use (1) and the p(F’, F)density t o show that ( z ” ,2: @ 6,)= ([J1zn](z~), y’,) = ([Jz.z!,](y’,),2:) converges. (e) If, additionally, E and F are reflexive, EGa F is reflexive if and only if it is weakly sequentially complete. (f) Rosenthal’s theorem (see Diestel [54]) states that .t, is not contained isomorphically in E if and only if every bounded subset of E is weakly conditionally compact. Use this to show that, if F # (0) is in addition reflexive, then .tl is not contained isomorphically in E g 0 F if and only if it is not in E .

-

-

-

16. The Bounded Approzimation Property

190

(g) Using only the fact that E &

-

respects subspaces, show that (d) and (e) hold for a = E if F is not necessarily separable. These results are due to Lewis [167] and Rivera Ortun [232].There are many interesting situations in which E'gaF' = (E&F)' holds isometrically, hence the condition on E&F is satisfied, see section 33.

16. The Bounded Approximation Property This section resumes the study of the bounded approximation property introduced in section 5. A characterization in terms of T = 'ir will be given. With the aid of the Radon-Nikodim property (see Appendix D) the full duality of E and T will be investigated - as well as the duality of the spaces of compact, nuclear, and all continuous operators.

16.1. According to definition 5.1. a normed space E has the &bounded approximation oS.property (for X 1 1) whenever idE E A B ~ ( E , E ), where T,, stands for the topology of uniform convergence on the absolutely convex, compact subsets of E; in other words, there is a net (Tq)of finite rank operators of norm 5 A such that Tqz + z uniformly on absolutely convex compact sets. There are two other locally convex topologies on C ( E ,F) which will be needed: r, := the topology of pointwise convergence rw := u ( C ( E ,F ) , E

8 F'), the topology of pointwise weak convergence.

LEMMA:Let E and F be normed spaces. (1) On bounded subsets of C ( E ,F ) the topologies rcoand r8 coincide. (2) ( C ( E ,F ) , T ~ ) = ' E 8 F' with the obvious duality bracket n

n

k=l

k=l

(3) For bounded, conuet subsets A

c 2 ( E ,F ) the

closures with respect t o rco,rr and

rw coincide: -rce

A

1 . =A =xY~C(E,F).

26. The Bounded Approzimation Property

191

PROOF: (1) is simple and was already used in 5.1.(2). Tosee (2) take (o E ( C ( E ,F ) , r,)‘; then there are 21, ...,znE E with

and hence, by the Hahn-Banach theorem, (o factors through

rv

Since (F“)’ = (F‘)”, the result follows (this was Ex 5.1.). But now = -7. A (for convex A) is an immediate consequence of the following well-known fact: if (G, H) is a separating dual system, then the closures of a convex set A c G coincide with respect to all locally convex topologies 7 on G satisfying (G, 7)’ = H. This, together with (1) gives (3). 0

REMARK: A normed space E has the A-bounded approcimation property if and only if idE

E JBZ(E,E)‘,

16.2. Since A is accessible, it follows that A = f holds on E 8 F whenever one of the spaces is finite dimensional. The approximation lemma 13.1. gives A 5 A f on E @ F if E or F has the &bounded approximation property. The key for proving a sort of a converse to this observation is the following lemma - which will even be stated for arbitrary tensor norms. It characterizes a 5 A= in terms of approximating operators in ( E & F‘)‘ c C ( E ,F”) by finite rank operators E -+ F :

LEMMA:Let a be a t e n s o r n o r m , E and F n o n n e d spaces. T h e n a 5 A ‘ Z o n E 8 F’ if and only if

c

B ( E ~ , F J ) ~ABEIB,,F~

c C ( E ,F”) ,

where u := a ( C ( E ,F”), E 8 F’).

PROOF: This sounds more complicated than it is. Clearly, both E‘8 F and ( E & F’)’ are regarded as subspaces of C ( E ,F”). For each z E E 8 F’ the duality theorem 15.5. gives c a (z;E , F‘) = S U P I(T, .)I T E BE’O,,F}

{

1

and it is straightforward that “(z;

{

E , F’) = S U P I(T %)I

IT

E qEOoF’)’}

*

Now the result is obvious if one applies the following simpIe fact to the separating dual system ( C ( E ,F“), E 8 F’) : If (G,H) is a separating dual system, A and B are

192

16. The Bounded Approximation Property

subsets of G , then A C XBoo if and only if Bo c AAO. Just take A := B ( E B , ~ )and , B := BE#B,,Fand apply the bipolar theorem. 0 The proof shows that the result also holds with the topology u replaced by 5 := U ( ( E @a F I Y ,E ~ x ) . In section 21 this lemma will be used to investigate the so-called bounded a-approximation property. For the moment only Q = ?r is interesting. From T' = E it follows that the relevant spaces in the lemma are

(operator norm in both cases), and hence

?r

5 A% on E 8 F' if and only if

c AB%(E,F)*

B~(E,FII)

(with u := u ( C ( E ,F"), E @ F')). In particular, i d E E B C ( E , E ) c A B 3 ( E , E ) * n & ( E ,E )

A

5 A% on E 8 E' implies

= A B 3 ( E , E ) " Y = AB3(E,E)

2.2.

(by 16.1.), hence E has the A-bounded approximation property. Collecting all this, the following result has been proved:

THEOREM: A normed space E has the A-bounded approximation property i f and only if ?rsAf

onE@F

for all normed spaces F (or only F = El). In particular, E has the metric approtimation property af and only af c A = T onE8F for all normed spaces F (or only F = El).

16.3. From the structural point of view this result is very important. Some immediate consequences are:

COROLLARY 1: If for u normed space E the dual E' has the bounded approximation property, then E has the bounded approximation property as well, with the same constant. PROOF:By the easy part of the theorem (which was just the approximation lemma) T 5 A% on E' @ E . Since A = A' the other part of the theorem gives the result. In the following statement the constants for the bounded approximation property are omitted - since they will be clear anyhow.

16. The Bounded Approximation Property

193

COROLLARY 2: Let E be a normed space. (1) E has the metric (resp. bounded) approximation property if and only if

holds isometrically (resp. isomorphically) for all normed spaces F (or only F = E l ) ; moreover, this is equivalent to saying that

holds isometrically (resp. isomorphically) for all normed spaces F (or only F = E). (2) E' has the metric (resp. bounded) approximation property if and only i f

holds isometrically (resp. isomorphically) for all F (or only F = E"); this is also equivalent to the statement:

holds isometrically (resp. isomorphically) for all F (or only F = El).

PROOF:This is an immediate consequence of the theorem and the duality theorem 15.5.. 0 Clearly, these injections extend to the completions isometrically, resp. isomorphically. For the embedding of the nuclear into the integral operators

this implies: If El o r F has the A-bounded approximation property, then

for all T E %(ElF ) . Under what conditions is % ( E ,F ) = J ( E , F ) ? To answer this question the Radon-Nikodfm property is the appropriate tool. 16.4. A Banach space E possesses the Radon-Nikodim property if for every finite measure p every operator T : Ll(R, p ) + E is representable, i.e. there exists a bounded p-measurable function g : R + E with

T f = J f g dP

for all f E L l ( R , p )

16. The Bounded Appron'mation Property

194

The results needed concerning representable operators and the Radon-Nikodfm property can be found in Appendix C and D, as well as the notion of Pietsch integral operators. The Radon-Nikodfm property has powerful consequences for the questions treated in this section. For tensor norms a variant of the Radon-Nikodfm property will be investigated in Chapter 111, section 33.

COROLLARY 3: Let E be a Banach space with the Radon-Nikody'm property which is A-complemented in its bidual (i.e. there is a projection P : E" --y E with llPll 5 A). If E has the approzimation property, then it has the bounded approzimation property with constant A. PROOF: It follows from Appendix D7.that %(E,E) coincides isometrically with the space 'J?3(E,E) of Pietsch integral operators. The factorization properties of integral and Pietsch integral operators and the existence of a projection E" -+ E imply that 3 ( E , E) = V 3 ( E ,E) and PI(T) 5 AI(T). Since E has the approximation property, it follows that 1 E'&E V 3 ( E ,E ) = 3 ( E , E ) ( E El)' and hence, by the duality theorem, T(Z;

E', E ) = PI(T,) 5 AI(T,) = A?i(z; E', E ) ;

therefore, the theorem implies corollary 3. 0 Reflexive spaces and separable dual spaces have the Radon-Nikodfm property and are clearly 1-complemented in their biduals, hence:

COROLLARY 4: If E is reflexive or a dual space which is separable, then E has the metric approximation property if it has the approzimation property. 16.5. The Radon-Nikodfm property is intimately connected with the duality of the tensor norms E and T .

PROPOSITION: Let F be a Banach space. Then F' has the Radon-Nikody'm property i f and only i f the canonical mapping J E : E'& F'

-

( E @= F)'

is surjective f o r all Banach spaces E (or only E = C ( K ) with K compact, or only L,(p) for arbitrary finite measures). In this case, J E is even a metric surjection for all Bonach spaces E .

PROOF: Surjectivity of JE means that %(E, F') = 3 ( E , F') by the very definitions of nuciear and integral operators, so the result is nothing but a reformulation of corollary 1 in Appendix D7..0

195

16. The Bounded Approximation Property

The result cited from the appendix even implies that in order for F' to have the RadonNikodgm property it is enough to take only E = C[O,11or only E = L, [0,11(Lebesgue measure) in the proposition. 16.6. Combining the results of 16.4. and 16.5. easily gives the following result on the duality of the projective and injective norm:

THEOREM: Let E and F be Banach spaces. (1) If E' or F' has the approximation property and: E' or F' has the Radon-Nikodim property, then

E'& F' = ( E @< F)'

isometrically.

( 2 ) Conversely: If E'&G' = ( E G)' (as vector spaces) for all Banach spaces G , then E' has the Radon-Nikoddm property and metric approximation property.

PROOF: Since E' or F' has the approximation property, the canonical map J : E'GXF'

-

(E

F)'

-

is injective (see 5.6.), hence proposition 16.5. gives (1). For the converse, proposition 16.5. again shows that E' has the Radon-Nikodim property. The injectivity of the mapping E'&G' (E G)' for all G implies that E' has the approximation property (again by theorem 5.6.) and hence the metric approximation property by corollary 3 from 16.4.. 0 16.7. These results have nice applications for spaces of operators. From section 5 on the approximation property it is known that

n ( E ,F )

E'&F

and f f ( E F , ) A E'&F

if E' or F have the approximation property. Proposition 16.5. gives

% ( E ,F') = (E&F)' if E' or F' has the Radon-Nikodgm property and, clearly,

C ( E ,F') = (EGXF)'. The duality is always the natural trace duality.

PROPOSITION: Let E and F be Banach spaces such that E" or F' has the approximation property and: El' or F' has the Radon-Nikody'm property. Then ( f f ( EF))' , = %(El,F') ( f f ( EF))" , = (%(El,F'))' = C(E", F")

isometrically isometrically .

196

16. The Bounded Approzirnation Property

The duality brackets are given by m

n=l m

if T = C,"==l 2," @I b, E %(E', F') = E"&F'. The natural embedding of f f ( E F , ) into its bidual ( f f ( EF))" , = C(E",F") ff(E,F ) is passing to the bidual: S-S"

.

A

PROOF: The statements ff' A % and %' 2 follow from what was said before. The duality brackets originate from the natural embedding

and the identification

(%(El,F'))' = (E"@I~F')' = C(E", F")

.

It also follows (check for elementary tensors) that for S E ff and T E %

which is nothing but the statement that ff

LI

ff" = 2.0

I f E" has the approximation property, then the duality brackets are

An immediate consequence is

COROLLARY: Let E and F be Banach spaces (f { 0 } ) , one of which has the approzimation property. Then E&F is reflexive if and only if E and F are reflexive and every T E C(E,F') as compact. PROOF: If E and F are reflexive and C(E,F') = R(E,F'), then

( f f ( EF'))'' , = C(E,F') = R(E,F')

16. The Bounded Approximation Property

197

by the proposition, therefore C(E,F’) = (E6=F)’is reflexive, hence also E&F. Conversely, if E&=F is reflexive, then E and F are reflexive (as complemented subspaces of E&F), moreover C ( E ,F’) = (E&F)’ and its subspace f f ( E ,F’) are also reflexive. The proposition gives

R(E, F’)= ( f f ( EF’))‘’ , = C(E,F‘) and the result is proven. 0 For an example, recall that Pitt’s theorem (see e.g. Kothe [152], 11, p.208) says that

if 1 5 g < p

< 00. It follows that for 1 < q < p < 00 the space

is reflexive. Recall that 1 2 6 r t 2 is not reflexive (Ex 3.9.). For some more details in this direction, see Ex 16.4. - 16.8.; in section 33 the reflexivity of other operator ideals will be treated. 16.8. Finally an answer to the following question concerning dual nuclear mappings is obtained: If T E E ( E , F ) is such that T’ is nuclear, T is nuclear as well? In 5.9. it was shown that the answer is positive if E‘ has the approximation property. But in general this is false; this can be seen as follows: Take a Banach space E with the approximation property such that E‘ has the Radon-Nikodim property (e.g.: E’ is separable). Clearly

commutes. The Radon-Nikodim property (16.5. or Appendix D7.) implies that 1 2 is an isometry. Since the subspace of dual operators is closed in L!(E’,E’), the space G is closed in %(El,E’). So if11 is surjective, the closed graph theorem yields that 11 is a topological isomorphism and hence J is as well: Corollary 2 in 16.3. now implies that E has the bounded approzimation property. However, the Figiel-Johnson example 5.2.(5) is a Banach space E with the approximation property and such that E’ has the Radon-Nikodim property (since E‘ is separable), but without the bounded approximation property. It follows that:

PROPOSITION: There is a non-nuclear operator the dual of which is nuclear.

198

16. The Bounded Approzimation Property

Note that the counterexample is an operator T E C ( E ,E ) , where E has the approximation property and E’ is separable. For another counterexample, see 22.9..

16.9. There is a simple characterization of the bounded approximation property which has interesting consequences for the factorization of finite rank operators:

PROPOSITION: A normed space E hos the A-bounded approximation property if and only if for every M E FIN(E) and E > 0 there is an R E S ( E ,E ) with IlRll 5 A(1+ e) ond Rz = 2 for all 2 E M . PROOF: Clearly, the condition is sufficient (see also Ex 5.7.). Conversely, take a basis ( e l , . . . , e n ) of M with coefficient functionals eh E E’ and choose T E S ( E , E ) with IlTll 5 A and IITek - ekll 5 6 for all k = 1, ...,n. Then

R := T +

n

eiB(ek - Tek) k=1

COROLLARY: Let E and F be normed spaces, T E $(El F ) and E > 0 . (1) If F has the A-bounded approximation property, then there is an R E S ( F , F ) with llRll
+

((R’oI E E ) z , T ‘ ~ / = ) (Sc,T’y’) for all z E E and y’ E F‘. This implies T o S = T . 0

Exercises: Ex 16.1. A Banach space F has the A-bounded approximation property if and only if for every Banach space E each operator T E & ( E , F ) can be approximated with respect to the topology of pointwise weak convergence by finite rank operators of norm 5 A.

16. The Bounded Approximation Property

199

Ex 16.2. (a) If Go and G are normed spaces and I E C(Go,G)is of norm 5 1 such that ~ ( B G ,is) norming for G’,then I’ : G’-+ G‘, is an isometry, hence f : &, 4 (? is a metric surjection. C ( E @c F)’ is norming for (E QCF)”. (b) Use (a) to decide whether or not BE’ @ B F ~ Ex 16.3. (a) If K is compact and F a Banach space the dual of which has the Radon-Nikodjm property, then ( C ( K ,F))’ = M(K)@,*F’, where M ( K ) is the space of signed regular Bore1 measures on K.In particular, for every (o E ( C ( K ,F))’there are measures pn E B q q , elements y’, E B p and (An) E .f, such that

for all f E C ( K ,F). Hint: Recall that M ( K ) has the metric approximation property by Ex 5.6.. (b) Is statement (a) true for all Banach spaces F? More explicitly, show that

C([0,11)‘63,*C([O,11)’

5 C([O,112)’

Hint: C([O,11)‘ does not have the Radon-Nikodjrm property and C[O,11 is a “testspace” for this property, see Appendix D7.,proposition (2). The following results (Ex 16.4. - Ex 16.8.) are due to Holub [114]; in section 33 the reflexivity of other operator ideals will be treated. Ex 16.4. Let E and F be Banach spaces one of which has the approximation property. Then E& F is reflexive if and only if El& F’ is reflexive. Ex 16.5. Show that for p , q E [l,oo]: (a) ep6,*.fq is reflexive if and only if 1 < q‘ < p < 00 . (b) e p & ’ , is reflexive if and only if 1 < q < p’ < 00 . Ex 16.6. If E is an infinite dimensional Banach space with the approximation property, then neither E& E‘ nor E& E‘ is reflexive. Ex 16.7. Lp[O,l]&Lq[O, 11 is not reflexive if p, q E [l,001 and a = ?r or E . Hint: The Rademacher functions are complemented in Lp[O, 11 if 1 < p < 00, see Ex 8.17.. Ex 16.8. Use the fact (to be proved in 26.1.) that every operator from a n C:-space into an Zi-space factors through a Hilbert space (if 1 5 p 5 2 5 q 5 00) and Pitt’s theorem to show that W , I LP(P)) is reflexive if 1 < p 5 2

< q < 00,

and Z(Lq(P),.fP>

200

17. The Representation Theorem for Maximal Operator Ideals

is reflexive if 1 < p < 2 5 q < 00 . What does this imply for the reflexivity of Lq&d, and La&& ? E x 16.9. If T E 2 ( E ,F ) has a nuclear dual, then it is integral. E x 16.10. It may happen that N(T’) < N(T) even for finite rank operators. Hint: Look at the diagram in 16.8.. E x 16.11. An operator ideal a is called regular if T E % ( E ,F ) whenever the operator K F o T is in a ( E ,F”). Using 16.8., show that the ideal ?TI of nuclear operators is not regular. For positive results in this direction see 22.8.. E x 16.12. If .C(tm, F) = !l31(tm, F), then F is finite dimensional. Hint: The assump1 tion implies that t16bCF 3(&, F ) c* (t, mr F’)‘ ; 11.3., corollary 3.

-

17. The Representation Theorem for Maximal

Operator Ideals

This section deals with the one-to-one correspondence between finitely generated tensor norms and maximal Banach operator ideals - a simple, but extremely powerful correspondence, which will be central to the investigations which follow. It links together thinking in terms of operators with “tensorial” thinking - and as a result both theories, the theory of tensor products and the theory of Banach operator ideals, are incredibly enriched (and very often simplified). There will be two types of characterizations for operators T which are in a maximal operator ideal: firstly, in terms of the associated bilinear form and, secondly, in terms of the continuity behaviour of T @ id : @a + @ p for appropriate tensor norms.

17.1. If

(a,A) is a normed operator ideal, then M@* N

: A rn(M‘,N)

defines a tensor norm a on FIN; in other words, if z E M @ N and T, E 2 ( M ’ , N) is the associated operator, then a(%;M ,N) := A(T, : M’+ N )

.

That a is a tensor norm is an immediate consequence of the criterion 12.2.: the m e t r i c mapping property of a i s the s a m e as the ideal property for % (for finite dimensional spaces) since for z E M 8 N

R o T, o S‘ = Ts@qt)

($1

17. The Repmeentation Theorem for Mazimal Operator Ideals

201

holds for all S E E(M, M I ) and R E E(N, N1). Note that the criteria 12.2. and 9.4. for tensor norms and operator ideals are quite similar.

17.2. Conversely, let a be a tensor norm on FIN. Define, for finite dimensional Banach spaces M and N, a norm A on C ( M ,N ) by A(T : M

+

N ) := ~ ( z TM', ; N),

where ZT E M' @ N = C ( M ,N) is the associated element in the tensor product. It follows from formula (*) that in this way a normed operator ideal (%,A)for finite dimensional spaces is defined. How to extend it to all Banach spaces? (Recall that operator ideals were only defined for Banach spaces, not for general normed spaces.) Of course there are several different ways to do this - for our purposes the smallest and the largest extensions will be the most interesting ones. The smallest will be treated in section 22; for the largest take the following

DEFINITION: Let (a,A ) be a (quasi-) normed operator ideal. (1) For T E E(E,F ) define Amaz(T):= sup {A(@ o T o I g ) %"""

E FIN(E) I LM COFIN(F) E

:= {T E E I A"""(T) < a}

and call (a,A)""" := (%mazIA""") the maximal hull of (%, A). (2) (%, A) is called maximal if (a,A ) = (%maz, Amas).

It is clear that (%"'az, Amas)is always a maximal (quasi-) normed operator ideal. It is also immediate from criterion 9.4. that maximal (quasi-) normed ideals are necessarily (quasi-) Banach ideals. The following observation is simple, but useful. REMARK: Let (n,A) be a maximal quasinormed operator ideal and (23, B) a quasinonned operator ideal which coincides with (%, A) for finite dimensional spaces. Then 23 c 2l and A _< B. In particular, two maximal ideals coincide if they coincide on finite dimensional spaces. Returning to the tensor norm a on FIN, for T E C ( E ,F) define

A(T : E

-+

1

F) := sup a ( ~ ~ ; ~M', ~ F ~ / ,L ;) ;

M E FIN(E) E COFIN(F)

and a := { T E E I A(T) < a}. It is clear that ( % , A )is a normed operator ideal, that % ( M ,N) = M' & N holds isometrically for MIN E FIN, and hence that (a,A) is even a maximal normed operator ideal.

202

17. The Repreaentation Theorem for Mazimal Operator Ideals

17.3. Both finitely generated tensor norms and maximal Banach operator ideals are, by their definition, uniquely determined by their behaviour on FIN.

DEFINITION: A finitely generated tensornom a (on NORM) and a maximal Banach operator ideal (a,A ) are said t o be associated, notation: (%,A) a

if for all M ,N E FIN % ( M ,N ) = M‘ @a N holds isometrically: A(T, : M + N ) = a(%;M’, N). The constructions in 17.1. and 17.2. show that in this way a one-to-one correspondence between macimal Banach operator ideals (24, A ) and finitely generated tensor norms a is established. This correspondence is basic to the understanding of all that follows. 17.4. Before examining this correspondence more precisely, the maximality of various operator ideals will be checked. Obviously, (C,II 11) is maximal and (ff, 11 11) is not. A quick look at the definition shows that the absolutely psumming operators ppare maximal - but there is a nice, rather general result which gives this and a great variety of other maximal Banach operator ideals. For this, take a normed space G and a tensor norm a. Then (1) E 5 a 5 ?r on G @E f o r all normed E ; (2) a(i& @ T ( z ) ;G, F ) 5 llTlla(z;G, E ) for all T E C(E,F). If G is a subspace of some L p ( p ) and a = Ap is the norm induced on G 8 E by L p ( p ) @A, El then (1) and (2) also hold by the results in 7.3.. This is why we adopt a more general notion and call a a right-tensor norm (with respect t o a fixed normed space G ) if it is defined on all G 8 E with E E NORM, and satisfies (1) and (2). We repeat: we do not want to make a fuss over this definition - it is just more practical to treat a and Ap jointly. A right-tensor norm is finitely generated (from the right) if

a(z;G, E) = inf{a(z; G, N) I N E FIN(E), z E G @ N } for all E E NORM and cofinitely generated (from the right) if

a(z;G, E) = sup(a(idc @ Qf(z); G, E / L ) I L E COFIN(E)}

,

Clearly, finitely/cofinitely generated tensor norms are finitely/cofinitely generated from the right (see Ex 12.3.) and it was shown in Ex 7.16. that Ap is finitely and cofinitely generated from the right with respect to every subspace G c Lp((cl).

PROPOSITION: Let G, H be normed spaces, S E C ( G , H ) with llSll = 1, a a finitely generated right-tensor norm with respect t o G and fl a cofinitely generated right-tensor norm with respect t o H . Then the ideal (.Cz*p,Lzlp)defined by

204

17. The Representation Theorem for Mazimal Operator Ideals

and

Q(E,F ) = ( E @at F')' n S ( E , F )

hold isometrically.

By definition 10.1. of the integral operators, 3(E, F) = ( E @c F')' n S ( E ,F) holds isometrically. In particular, 3 ( M , N) A ( M&C N')' A M' @= N and hence A 3""";

-

the representation theorem gives

F*"(E, F ) A ( E @c FI)' n C ( E ,F )

-

J(E,F )

-

and therefore 3 = 3""" and A 3 (see also Ex 17.2.). This example explains why operators in 2i are sometimes called a-integral operators whenever Q a. This theorem is due to Lotz [180]. His approach to tensor norms was different from ours and was very influential in the development of the theory of operator ideals: Roughly speaking, he took the representation theorem as a definition for tensor norms and consequently emphasized the one-to-one correspondence between maximal normed operator ideals and tensor norms; see also Ex 17.2..

PROOF: The second formula will be proven first: it needs to be shown for T E C ( E ,F) that T E %(E,F) if and only if the associated bilinear form P l c F 0is~ a'-continuous: P s F o ~E

( E Bat F')'

(with equal norms). But this is easy: T E Q(E,F) and A(T) 5 c if and only if

for all (M, L) E FIN(E)xCOFIN(F); by B ( M , F / L ) = (M Bat Lo)' this is equivalent to ((Pfi~oT .)I , = I(pQ:oT.Izt z)l 5 ca'(z; M , L o ) for all z E hf @ Lo Therefore, the result follows since a' is finitely generated. To see that the first formula holds just look at the diagram

and use the extension lemma. 0

17.6. An obvious combination of the representation theorem and the duality theorem 15.5. gives the

17. The Representation Theorem for Maximal Operator Ideals

EMBEDDING THEOREM: If (a,A)

-

a are associated, then the relations

E' 8F' Q

a(E , F')

E @I-Q F

%(El,F )

E'

% ( E ,F )

@+Q

205

F

hold isometrically. In particular, the extension

E'&F

-

%(ElF )

of E' @a F + %(E,F) is well-defined and has norm 1 (if E and F are not trivial). In 16.3. this map was studied for the case a = T and was useful in understanding the relationship between nuclear and integral operators. More generally, in section 22 the embedding theorem will be crucial for the investigation of minimal operator ideals.

-

17.7. An essential goal of the theory is to compare different tensor norms/maximal operator ideals. The very definition of (%,A) a using finite dimensional spaces shows the

-

-

REMARK: Let (%,A) a and (%,B) /3 6e associated, only i f A 5 c B . In this case, 23 C %.

c

2 1. Then a 5 c p i f and

Very interesting phenomena can be deduced from estimates on special Banach spaces. The representation and embedding theorems

E' 8F' Q

CZL(E,F')

( E @IQ, F)'

imply the following:

-

-

COROLLARY 1: Let (%,A) a and (%,B) p be associated, Banach spaces. Consider the following conditions: (a) p' 5 ca' on E 8 F. (b) B ( E ,F') C % ( E ,F') and A 5 c B . (c) z Then

c

2 1 and E and F

5 c F on E' 8 F'.

(1) ~a)c+(b)-+)* (2) If E' and F' have the metric approximation property, or: a and /3 are accessible and E' or F' has the metric approximation property or: a and p' are totally accessi6le, then (a), ( b ) and (c) are equivalent.

206

17. The Representation Theorem for Mazimal Operator Ideals

PROOF: (1) follows from the observation above. So it remains to show that (c) implies (a) under the conditions of (2). But this follows from the diagram

c

since 'Z = a on E'@ F' and p' = p' on E @F under the assumptions of (2) (use 15.7., 15.6. and 16.3.). 17 This result, although simple, is often quite useful and will be fundamental in the understanding of the interplay between maximal ideals and tensor norms. Since it transfers statements about operator ideals to inequalities involving tensor norms and vice-versa, it will be referred to as the transfer argument. Note that (2) gives conditions under which: a 5 cp on E' @ F' if and only if p' ca' on E @ F.

17.8. The representation theorem has simple, but important consequences for the behaviour of maximal operator ideals. Looking at

Q(E,F )

(E

F')'

A %(ElF") ,

gives the

COROLLARY 2: Maximal operator ideals (a,A) are regular, which means: T E C ( E ,F ) is in ?2I if (and only if) K F o T E % ( E ,F"); moreover, A(T : E

--+

F ) = A( ICFo T : E

-+

F")

.

Recall from 9.9. that Qdual := {T E C I T' E a} and AdUar(T) := A(T').

COROLLARY 3: If (Q, A) is a maximal normed operator ideal associated with a finitely generated tensor norm a, then (Qduar,Adual) is maximal and i s associated with the transposed tensor norm at :

Adual)w a* .

PROOF:Denote by (23,B)the maximal normed operator ideal associated with a'. Then, by the representation theorem, F')' n 2 ( E ,F ) = %(I?, F ) = ( E = { T E 2 ( E ,F ) E (F'

= {T

1 E C ( E ,F ) I T' E %(F',

E)'} = E ' )}

17. The Representation Theorem for Maximal Operator Ideals

207

with B(T)= A(T'). In other words, (23, B) = (adua', AdUa'); this was the claim. 0 Since (!21dua')dua'

-

ati = a

-

2l, one obtains

COROLLARY 4: Let (24, A ) be a maximal normed operator ideal. Then (24dua')dua' = 2f holds isometrically. In other words, T E 24 if and only if T" E a; in this case, A(T) = A(T") . Another way to formulate this is: T is &-integral if and only if T' is &integral.

17.9. Let

-

(a,A ) a be associated.

Then for M , N E FIN

N)' = ( N Bat M')' = N'

( % ( M ,N))' = (M'

M

holds isometrically. Hence, if (a*,A*)is the maximal normed operator ideal associated with (ai)' = a*,one obtains

(2l(M, N))' = 24*(N,M )

- the duality bracket given by (T,S)a. ,a = t r N ( S o T ) = trM(T o S) whenever T E 2l*(N,M) and S 6 O ( M ,N). This is why (a*,A*) is called the adjoint maximal normed operator ideal of (a,A).In section 22 it will be seen how the duality relation is extended to infinite dimensional spaces. When is an operator T E C(E, F) in a*?Observe first that, by duality,

sup

{

M E FIN(E) L E COFIN(F) A(S : F / L M ) 5 1

-

( t r M ( S o Q: o T o

E-F

M

T

FIL

+---------

.

208

A*(T : E

17. The Representation Theorem for Maximal Operator Ideals

-+

M E FIN(E) L E COFIN(F) A(S : F/L + M )5 1

{

F) := sup ltrM(S o Q[ o T o I:)I

17.10. The maximal normed operator ideal associated with the tensor norm ap,*of Laprest6 (see 12.5.) is the ideal ('cp,q, LP,*)of ( p , q)-factorable operators; recall that l/p l / q 2 1 and 1/r := l/p l / q - 1. If for finite dimensional M , N

+

+

T E zp,*(M,N ) = M' @Qp,p N , then, by definition of the norm, Lp,*(T)= a , , , ( z ~ ;M',N), there is for every e representation

>0 a

n

T

M-N I

with llRll llDxll llSll that

t

5 LP,*(T)(l + E ) . Reading this argument backwards, it follows LPJT) = infllRll

llW list1

17. The Representation Theorem for Mazimai Operator Ideals

209

where the infimum is taken over all such factorizations. An example: Since llid : TI + $11 = rillr, it follows that

id : cl -+ 5 )= nilr . These factorizations help to explain the name of the ideal; it will be proved in section 18 that an operator T is in Cp,q(E, F) if and only if it factors

. T

1

note that q’ 2 p : if q’ > p, then the measure p is understood to be finite and I is the canonical inclusion, whereas in the case q’ = p the measure may be arbitrary and I is just the identity map. There are several interesting special cases: the ideal (Cp,Lp)associated with wp = ( Y ~ , is ~ I the ideal of all p-factorable operators. By what was just said these are the operators T such that n~ o T factors through some L p ( p ) . Since A = q 1 , the (1,1)-factorable operators are exactly the integral operators and the factorization theorem mentioned above is already known from 10.5.; more generally, the ideal (Jp,Ip) := (Cp,l, LP,1)associated with gp = aP,lis called the ideal of p-integral operators. Note that 3, = .I& since , wm = gm . It is important to reformulate the factorization of T E C,,,(M, N‘) = (M @a;,QN)’ in terms of bilinear forms: The diagonal operator Dx E f!(ql,$) = (qlBX q l ) I corresponds to the “diagonal” bilinear form n

6x :=

C

Xkek

QD ek

E


$8)‘

.

k=l

Therefore, for every

’p

E (M@* N ) ‘ , Il’pIl(M@.;,*N)’

= inf IPll IlSll 1164

9

where the infimum is taken over all factorizations

where R E C ( M , q , ) , S E f!(N,$,) and X E K ” . Clearly, it is enough to take only X E lK“ with non-negative components.

17.11. Take p, q E [l,m] such that l / p + l / q 5 1. Then l/p’ + l/q’ 2 1 and ( Y , I , ~ I is defined. The ideal (9p,q, DP,,) associated with (Y:~,~, = ( Y ~ ~ , is, ~ the maximal normed

17. The Representation Theorem for Mazimal Opemtor Ideals

210

operator ideal of so-called (p, q)-dominated operators. To describe the operators in 9p,q, define r E [1,00] by 1/r := l / p + l/q ,

+

+

so that l/p l / q l/r' = 1. The representation theorem for maximal operator ideals if and only if the associated bilinear form is Q ~ I , ~ ~ shows that T E &(E,F ) is in continuous: PICPOT E ( E 8 a p t , , ,F')' - with equal norms. This means that DP,q(T)5 c if and only if for every element z = Cr=lA k x k 8 4 E E 8 F' ~ ( P I C F O T I= Z )( ~e A k ( d , T z k ) l5 k=l

C

e,l(Ak)wp(Xk;E)wq(3/I;F') ,

which is equivalent to c

er((!h,T2k))

for all c

21,

...,x,,

E E and

wp(xk;E)wp(d;F')

&, ...,b, E F'; clearly, DP,q(T)is the

minimum of all such

2 0.

-

Special cases: (SP,Dp) wit = ail, is the maximal operator ideal of p-dominated operators; the operators T E DP(E,F? = !I)p,p~(E, F ) are characterized by

4((dVTXk)) I

c

Wp(xk;E)wp'(d;F')

For q = 00 one obtains r = p and the operators in !I)p,m(E,F) are those satisfying

44d,TXk))5 c wp(zk;E)%o(3/I;F') or equivalently (choose appropriate

4EBp)

ep(llTxkll)

I

c

wp(w

E)

for all 21, ...,x,, E E. These are exactly the absolutely p-summing operators from section 11: ('Pplpp)

= (Bp,m,Dp,m)

N

(~;t,i

= $*'

*

In section 19 the Grothendieck-Pietsch domination theorem for ppwill be generalized to JDPlqr thus explaining the name (p, q)-dominated. 17.12. Before going into some examples showing the interplay between maximal operator ideals and tensor norms, it might be helpful to have a list of the operator ideals connected with Q ~ , It~ is . clear from the definition of the adjoint operator ideal and a**= Q (if (Y is finitely generated) that

=

!Dptrqt

and

hold isometrically for p , q E [l,001 with l/p

= &,

+ l / q 5 1 . Hence

- -

17. The Representation Theorem for Maximal Operator Ideals

211

all operators integral operators (p, q)-factorable operators (p‘, q’)-dominated operators p-factorable operators p’-dominated operators p-integral operators absolutely p’-summing operators , where qJm := l:, which is a reasonable definition from several points of view.

17.13. Transfering results from tensor norms to operator ideals and viceversa, is a sort of “game” whose value stems from the fact that it is quite often easier (or much more natural) to prove a result in one form than in the other. We are going to illustrate this now, although not always with the intention of getting the best result; this will be done later. The Khintchine equality gives that aP,(Ibq’bp~wzwhenever p , q E]l,oo[ (see 12.8.). From the very definition of rzL being associated with a it follows that Q(M,N)

M’rXlaN,

hence l:2(E, F

) c Cp,q(E, F ) and Lp,q(T) I bqJ,lLZ(T) for P, Ell,I..

for all Banach spaces E and F. Since w2 that

I ap,q,whenever p, q E]1,2], it follows even

.C2(E,F ) = l : p , q ( E , F ) and LZ(T) 5 Lp,q(T) IbqJJp’L2(T) for p , 4J €11921 * The relevance of these results stems from the fact that the 2-factorable operators are exactly those which factor through a Hilbert space (to be shown in the next section), thus emphasizing the very special role of Hilbert spaces in the theory. What does g,; 5 gp (see 15.4.) imply? Since v p

-

$1

5 Sp

-

this clearly means (again by the definition of 2t sional spaces) that

Jp

-

9

a and extension to infinite dimen-

Jp(E,F) c Vp(E,F) and Pp(T) I Ip(T) (this inclusion will also follow easily from the factorization behaviour of p-integral and absolutely psumming operators). These results were obtained by using the simple transfer argument : a 5 cp if and only if A 5 cB.

212

17. The Representation Theorem for Maximal Operator Ideals

For special pairs of spaces the method of arguing with the representation and embedding theorem is the transfer argument from corollary 1 in 17.7.. An example: From the Chevet-Persson-Saphar inequalities (see 15.10.) it is known that dp 5 gi, on L, @ * which implies (Lp @g;, E’)’ll 5 1 ll(Lp @ d , El)’

-

-

-

Since d i = g; Y,I and (g;,)’ = g;l 3Pa‘(by the description of the dual operator ideal in 17.8.), it follows from the representation theorem for maximal operator ideals that Yp1(Lp, E ) c 3$’”’(L,, E) and I,I(T’) = q a ’ ( T )5 PPd(T). For a “transposed” result see Ex 17.7., for p = 2 see Ex 17.8.. The spaces F which satisfy this property of F = Lp will be characterized in 25.9..

17.14. As has already been mentioned, Grothendieck’s fundamental theorem of the metric theory of tensor products (section 14) has powerful consequences for operators; at this point we could examine the general versions in terms of C:-spaces, but we will postpone this task to section 23 and restrict ourselves for the moment to considering the sequence spaces e, for p = 1,2, and 00. Since T 5 K G W on ~ loo €31co and w~ 5: dz (see 12.5.), it follows that

by the representation theorem, and therefore:

THEOREM: Every operator T E C(e,,tl) is 2-dominated and (hence) absolutely 2summing; moreover, pZ(T) 5 DZ(T) 5 KGllTll * In 20.18. and Ex 20.14. it will be shown that this result is also equivalent to Grothendieck’s inequality: For 2 = ID2 with the same constant (this is obvious), for C = ’&, with another one; see also Ex 25.5.. It is customary nowadays to call another operator form of Grothendieck’s inequality

THEOREM: Every T E C(41,ez) is absolutely l-summing and GROTHENDIECK’S

PROOF: It was shown in 14.8. that T 5 KGd, on 8 and hence, by l-complementation, on @I q for all m,n. The density lemma (or the C!-local technique lemma) implies that this holds on 1 1 @&, hence C(e1 ,t z ) = (el €3IX f2)’ = (!I ad,,,ez)’ = P1(.f1,&) since PI g& = d k . 0

-

17. The Representation Theorem for Maximal Operator Ideal8

213

20.17.concerning the fact that the best constant in Grothendieck’s theorem is the Grothendieck constant KG. The little Grothendieck theorem (11.11.) says that

See

whenever E = 4, or

el.

Since

it follows:

PROPOSITION: If E = tl or .too, then

17.15. Apart from the representation theorem there are other extremely useful characterizations of a-integral operators T € C ( E ,F) in terms of tensor product mappings idc @ T : G@pE

-

G a yF

with appropriate tensor norms p and 7 - as has already been seen in the characterization 10.7.of integral operators and 11.1.of absolutely p-summing operators. Actually the following theorem will be proved along the lines of the proof of 10.7.; the key to that proof waa to use two simple formulas (check for elementary tensors) which connect TEC(E,F)andT@idG: (1) For ‘pE ( F & G)’ and z E E @ G

THEOREM: Let (%,A)N a be associated and T E C ( E , F ) . Then the foflowing two statements are equivaleni: (a) T E %(E,F ) . (b) For all Banach spaces G (or only G = F’ or G = L if L‘ = F isometrically)

is continuous. In this case,

214

17. The Repmaentotion Theorem for Maximal Opemtor Ideals

-

We prefer to formulate the theorem for idG 8 T instead of T 8 i d c , even though the proof will be for T 8 id0 : @.(In 8=.

L, o T E a ( E ,G’) = ( E

G)’

and formula (1) implies that

which shows that

r ( T @ i d G ( 2 ) ; F, G)5 A(T)a‘(z;E , G ) .

Conversely, assume (b) to be satisfied for G = F’. Since 2l is regular (1?.8.), it is enough to show that PspoT E %(El F“) = ( E @an F’)‘ . For z E E 8 F’ formula (2) implies that

-

J(PKpO z)I~=, J(trF,T8 i d ~ f ( z ) 5 ) ) r ( T 8 i d d z ) ;F, F‘) 5 5 llT @ i d p :Bun QD,*lla’(z; E , F’) which gives T E %(ElF ) with the norm estimate. For F = L’ the proof is nearly the same for checking that pT E (E L)‘ . 0 It is important to note that (b) is actually a statement about the composition of operators

S ( G , E )= G‘ 8 E S

id@T

G ’ 8 F = S(G,F ) ToS

(see also 17.19. and Ex 17.6.).

17.16. In order to obtain characterizations involving E - which will turn out to be a sort of a dualization of the last theorem - the following lemma will be needed; the reader will find it to be a simple consequence of the metric mapping property of tensor norms and the fact that for every M E FIN and E > 0 there is a 1-complemented subspace N of the Johnson space C, (see 6.3.) with d ( M , N) 5 1 E .

+

LEMMA: Let p and 7 be tensor norms, P finitely generated, c 2 0 and T E 2 ( E , F). (1) If for a normed space G

17. The Repwsentotion Theorem for Mazimal Operator Ideals

215

for cofinally many M E FIN(G), then

( 2 ) If for some Johnson space Cp Ilidc, @ T : Cp @p

E -* Cp 8-yF l l 5

-

then IlidG @ T : G @ p

E

c

9

G @7 F i l s c

for all normed spaces G . Everything is now prepared for the

COROLLARY: Let (24,A )

-

Q be associated and T E C ( E ,F ) . I f a i s a right-accessible and finitely generated tensor norm or if F has the metric approximation property, then the following statements are equivalent:

(a) T E %(E,F ) . (b) For all Banach spaces G (or only G a Johnson space Cp for some 1

5 p 5 cm)

is continuous. In this case,

-

at and in this case A(T) = Adua'(T') PROOF:T E 24 if and only if T' E holds. So the theorem and the lemma imply that A(T) 5 c if and only if

for all M E FIN. Since the map (see 6.4. for the duality of E and

is the restriction to M'

T)

F' of the dual of

it is clear from the lemma that (b) implies (a), and the converse is true if the relation 1 F holds for all M E FIN; but this is true if a is right-accessible or M @a F = M approximation property. 0

216

17. The Representation Theorem for Maximal Operator Ideals

Note that the assumption about the accessibility of Q or the assumption on F was only used for the proof that (a) implies (b).

-

17.17. An example: For the ideal (cfp1Ip) gp of p-integral operators the following three statements are equivalent: (a) T E E(E, F ) is p-integral. (b) For all Banach spaces G (or only G = F')

is continuous. (c) For all Banach spaces G

is continuous.

For (a) or (b) implies (c) the fact that gp is accessible is used - a fact which will be proved in section 21. However, this will not be needed in the following nice application of this result.

PROPOSITION: Every positive operator T E 2 ( E ,L p ( p ) ) , where E is a C ( K ) or some L,(v), is p-integral and I,(T) = IlTll.

It has already been seen in 11.2. that these operators are absolutely p-summing and P,(T) = IlTll; see also Ex 18.3..

PROOF: It is known from theorem 7.3. that for all Banach spaces G IlidG €4 T : G @c E

-*

G @A;

Lpll

= llidGII llTll

I

Since gp 5 A: by the Chevet-Persson-Saphar inequality 15.10., it follows from the corollary (since L, has the metric approximation property) that T is p-integral and IP(T) = IlTIl * 0

17.18. Sometimes it is possible to find better test spaces than G = C, in corollary 17.16.. Using the fact that gp, is accessible (this will be proved in section 21), it is clear that for T E gp(ElF) and G = t, the map

is continuous by the corollary. Since A,, 5 g i , on t, @ F, it follows from proposition 11.l.that this actually characterizes psumming operators:

17. The Representation Theorem for Maximal Opemtor Ideals

217

REMARK:T E C ( E ,F ) is absolutely p-summing if and only if

Another description of Q p ( E , F) is given in Ex 17.9. with the help of Kwapieli’s characterization 11.8.. 17.19. The theorem (and its corollary) also has structural consequences.

PROPOSITION 1: Let (a,A ) be a maximal normed operator ideal such that the associated tensor norm a is accessible. Then Q* o

C3

and

I(T o S) 5 A * ( T ) A ( S );

the composition T o S of operators S E Q and T E a* is integral.

In 25.4. this result will be generalized to the so-called “cyclic composition theorem”; it will also be shown in 25.4. that Q* 0 % C 3 implies that a is necessarily right-accessible. In section 21 the accessibility of a will be explained in terms of the operator ideal a. If a is not accessible (recall that there is an example known), the proof will also show that % * ( F , G ) o Q ( E , F )C 3 ( E , G ) , with the norm inequality whenever F has the metric approximation property. In 19.4. the reader will find examples in which a* o Q consists of nuclear operators only. For another “converse” of the proposition, see Ex 17.6., Ex 17.15. and Ex 25.10..

-

-

PROOF:Since a a and a* a*,the theorem and its corollary show that for S E n ( E ,F) and T E Q’(F, G ) the composition

-

has norm 5 A*(T)A(S); since ?r 3 and integral and I(T o S) 5 A*(T)A(S).0

A*

= E , the theorem implies that T o S is

If dim E = n, then I ( i d E ) = N ( i d E ) = n (see 10.4. and 3.7.). It follows that for each isomorphism T E C ( E , F ) A(T : E

+ F)A*(T-’ : F -+

E) 2 n

for each normed operator ideal ( Q , A )(maximal or not: E and F are finite dimensional). For special cases there is equality: Recall the investigation 15.9. of certain

218

17. The Representation Theorem for Maximal OpeMtOr Ideals

norms on K" which are invariant under the isomorphism M, (multiplication of the components by f l ) and the cyclic shift S. In terms of operators, 15.9. reads as follows:

PROPOSITION 2: Let E and F be K" equipped with norms such that llM, an C(E', E') and C(F, F ) for all w E 0,.Then A ( i d ~ :-E for all normed operator ideals

-+

F ) A * ( i d K m: F

+E)

11 = llSll = 1

=n

(a,A).

xi=,

PROOF: The element u(") := ek 8 ek corresponds to the identity operator. If a is the tensor norm associated with am"*, then A(id : E

+F ) =

A*(id : F

+E)

a ( u ( " ) ;E', F )

= a*(u("); F', E ) = o'(u("); El F')

- and the result follows from a(u(");E', F)a'(u(");E l F') = n which was shown in 15.9.. 0 This implies, for example, that the results in 8.8.about the nuclear norm of the identity id : $ -+ can be directly calculated from the operator norm of id : -+ $ since C E and E* = A N 3, which is n on finite dimensional spaces. The exact calculation of P l ( i d t ; ) in 11.10. gives the exact values of the norm of idt; in 'Qt = 3, = ,C,. that is, the exact values of the oo-factorable norm:

-

It follows from corollary 11.10. that

These results are quite interesting since it turns out that L,(idE) constant of E whenever E is finite dimensional, see 23.6..

is the projection

17.20. The final topic of this section is the following question: Under what conditions is the natural map

injective? If a is totally accessible, then the duality theorem 15.5. for tensor norms implies that

17. The Representation Theorem for Mazimal Operotor Ideals

E h Q F= E g - F I*

1

(E’c&JF’)’

219

C(E’, F“)

hence I is injective.

PROPOSITION: If a is a finitely generated tensor norm and E and F are Banach spaces, one of which has the approximation property, then the natural map

is injective.

PROOF: Assume that F has the approximation property, z E E g a F and I(z) = 0. It has to be shown that (cp,z) = 0 for all

By theorem 17.15. (and obviously also by the correspondence between maximal operator ideals and tensor norms)

is continuous. The lower map in the diagram

is injective by the approximation property of F (see 5.6.), hence

Formula (2) in 17.15. implies that

which was to be shown. 0 The diagram

220

17. The Representation Theorem for Mozimal Operator Ideals

shows that:

-

COROLLARY: If (%,A)w a, then the natural map

E'haF

%(E,F )

is injective if (Y is totally accessible or i f E'

or

F has the approximation property.

See Ex 25.4. and Ex 17.11. - Ex 17.14. for more results connected with the injectivity Ofg3a+g)r.

17.21. The components Q ( E ,F) of a maximal operator ideal are not in general closed in C(E,F) (with the operator norm) (why?) - however the unit balls are. Even more is true:

PROPOSITION: Let (%,A) be a maximal normed operator ideal. Then the closed unit ball of g ( E ,F ) is closed in the space C ( E ,F ) with respect t o the weak operator topology u ( C ( E , F ) ,E 8 F'). In other words: If (T,)is a net in % ( E ,F ) which converges pointwise weakly to T E C ( E , F ) and sup, A(T,) < 00, then T E % and A(T) 5 liminfA(T,) .

PROOF:Since the unit ball of % ( M ,N) is closed for u(...,M @ N') (finite dimensional spaces), the second assertion follows from the definition of Amax . 0

Exercises: Ex 17.1. (a) Let {(%A, A ~ ) l ) rE A) be a family of maximal normed operator ideals. Then the class of all T E 2 with

A(T) := SUP Ax(T) < 00 A€A

is a maximal normed operator ideal. (b) Deduce from (a) and 10.7. that the ideal (3, I) of integral operators is maximal. Ex 17.2. A normed operator ideal (a,A) is maximal if and only if there is a finitely generated tensor norm p such that

B ( E ,F )

( E @p F')'

n C ( E ,F )

17. The Representation Theorem for Maximal Operator Ideals

221

for all Banach spaces E and F. In this case,

( E8 3 8 F)'

%(ElF')

.

Ex 17.3. Deduce directly from the representation theorem that T E Q if and only if T" E Q whenever Q is maximal. Hint: Embedding lemma and "9. Ex 17.4. Let (a,A) a and E, F be Banach spaces. Assume that a is totally accessible or: a accessible and E or F' has the metric approximation property or: E and F' have the metric approximation property. Then an operator T E C ( E ,F) is in the adjoint operator ideal Q* if and only if

-

Is

AA(T) := sup{lirE(S 0 T ) I

E S(F, E ) ,A(S) 5 1) < 00 .

In this case this number is A * ( T ) .Hint: Embedding theorem. Gordon-Lewis-Retherford [85] studied the ideal QA of all T with AA(T) < 00 under the name conjugate ideal of Q. Ex 17.5. Let (%,A)be a normed operator ideal. Show that T E C ( E ,F) is in Q* if and only if there is a c 3 0 such that for all M, N E FIN

for all R E C ( N ,M),S E C ( M , E) and U E C ( F ,N). Ex 17.6. Use the embedding theorem and

/IT' @ idE : F'

E

-

E'

@ r Ell

= A(T)

to show that: If (Q, A) a are associated, a is accessible, E and F are Banach spaces and E has the metric approximation property, then T E Q ( E ,F) if and only if there is a c 2 0 such that I(S o T ) 5 cA*(S) N

for all S E 5 ( F , E). In this case, A(") is the minimum of these constants c. See also Ex 17.15.. Ex 17.7. Show that !$Fa'(E,L p ) c ' J , ( E ,L p ) . Hint: Chevet-Persson-Saphar inequality. See 25.9. for more information about this relation. Ex 17.8. Use the result in 11.12. to show that: (a) There are no constants c1 and c2 with dz 5 clgz and 9 2 5 czdz. (b) A Banach space E is a Hilbert space if and only if $2 5 cgz on E @ & for some c > 1. Ex 17.9. Use Kwapieri's characterization 11.8. of absolutely p-summing operators to , if and only if show that T E V p ( E F)

idL,, 8 T I : ePl @dpl F'

-

epI@r

E'

222

17. The Representation Theorem for Mozimal Opemtor Ideals

is continuous.

-

Ex 17.10. If (a,A) a and a is accessible, then the following three statements are equivalent for each Banach space E and c 2 0 : E E space(%) and A ( i d ~5) c , (b) T 5 ca+ on G @ E for all Banach spaces G (or only G = El). (c) a 5 CE on G 8 E for all Banach spaces G (or only some Johnson space Cp). Ex 17.11. Let E or F have the approximation property and z E E&F for some finitely generated tensor norm, If (o&idi;.(r) = 0 for all (o 6 E l , then z = 0. Hint: (a)

17.20. and Ex 15.2..

Ex 17.12. Let E C(Ei,Fi) be two injective operators between Banach spaces and a a finitely generated tensor norm. If El or E2 has the approximation property, then T1&T2 E C(E1&,&, Fl&F2) is injective. Hint: 17.20. and this property for a = E . Ex 17.13. The canonical map L P ( p ) 8 * , E -, LP(p)&E is injective. Hint: Use 15.10. - or prove it directly for p < 00 using the fact that every f E L p ( E ) has separable range.

Ex 17.14. If a is a finitely generated tensor norm and a 5 Ap on L p ( p )@ El then LP(p)6~,E, LP(p)&E is injective. Also: if a 2 Ap on Lp(p) @ E , the mapping LP(p)&E + LP(p)&A,E is injective. Ex 17.15. Use the representation theorem and corollary 17.16. to prove the converse of proposition 1 in 17.19.: Let (a,A) be a maximal normed operator ideal the associated tensor norm of which is accessible. Then T E C ( E ,F ) is in 24 if and only if for all Banach spaces G and operators S E B'(F, G ) the composition S o T is integral. See also Ex 25.10.. Ex 17.16. Let S E C ( X ,Y)and T E C ( E ,F ) be continuous operators between Banach spaces and a,P two tensor norms such that

is continuous. Question: Is S @ T" : X E" -,Y @p F" also continuous? Unfortunately, we do not have a complete answer, even in the case where p = A. Assume a to be finitely generated. Show that the answer is positive under each of the following conditions: (a) T is weakly compact. Hint: [(S@ T)'(oJ'' = (o o (SQD T*)for all (o E (Y F)' if F : E" 4 F is the astriction of TI'. (b) p 5 cp' on Y 8 F" for some c 2 0 (the right-cofinite hull, see Ex 12.3.) Hint:

Proposition 17.4.. (c) O , is totally accessible. (d)

p is accessible and: Y or F" has the bounded approximation property.

(e) Y and F" have the bounded approximation property.

223

18. (p,q)-Factorable Operators

Ex 17.17. Density lemma for maximal normed operator ideals: Let (a,A) be a normed operator ideal, E and F Banach spaces, E, c E and Go C F' norm-dense subspaces, C C FIN(E,) and D c FIN(G,) cofinal subsets. Then AmaX (T

: E -+

F ) = SU~{A(Q;o T o

12)I M E C ,LO

E D) .

Hint: Representation theorem, density lemma 13.4. and cx' finitely generated.

18. (p,q)-Factorable Operators

The purpose of this section is to show that every ak,q-continuous functional on

E 8 F factors through the integrating functional on Lg1(p)8 Lp1(p) for some measure p. In terms of operators this means that every (p, q)-factorable operator T E E(E, F)

has the property that ICF o T factors through the embedding Lgt(p)4 Lp(p) . The most important special cases are the p-factorable and p-integral operators. In these cases, the proofs are actually simpler than in the general case. A central tool for the demonstrations will be the ultraproduct technique, the basic properties of which will be presented as well.

18.1. The maximal normed operator ideal (.Cp,q,Lp,g)of (p,9)-factorable operators is associated with the tensor norm aPIg. Using the definition of Q ~ directly, , ~ it was shown in 17.10. that for finite dimensional Banach spaces M and N the L,,,-norm of T E E ( M , N) is given by the factorizations

M-N

T

where Dx is a diagonal operator the entries of which can clearly be assumed to be non-negative: LP,,(T) = infllSll IlDXll IlRll * It is a simple matter of concentration (and this will be done in 18.10.) to show that Dx factors through an embedding Lq1(p)+ Lp(p)for a finite measure with control of the

224

18. (p,q)-Factorable Opemtors

norms. Since the operator ideal .CP,, is maximal, every T E Alp,,is determined by its finite dimensional components - and a tricky “fitting together” technique (ultraproducts), representation theorems for abstract Lp-spaces and a characterization of positive operators L , I ( ~ ) Lp(v)will give the desired characterization: T E .Cp,q(E,F) if and only if IEF o T factors through some embedding L,i(p) c* L p ( p ) (see 18.11.); recall from 17.8. that T E .Cp,q if and only if KF o T E .CP,(.

-

18.2. Not T : E -+ F but KF o T : E + F” can be factored. Therefore it is more F)’ rather than operators. reasonable to treat functionals in (E Using the fact that 6 = wp, on N‘@$,,it was shown in 15.8. that ?r = w; on L p ( p ) @F for every measure p (not necessarily finite), and hence the integrating functional P I : Lp(cL)

8 LP’W

fBg

--

K Jfgdp

(which is nothing but the trace) is wL-continuous. In other words, for every measure p the identity map on L p ( p ) is in the ideal .Cp of pfactorable operators associated with wp. For the ideal .Cp,q ap,qof ( p , q)-factorable operators the following result holds (the integrating functional is defined for l / p + l / q > 1 only if p is a finite measure!):

-

PROPOSITION: Let p , q, r E [l,001 with l/p + l / q = 1 + 1/r and p u finite measure on a set s1. Then the integrating functional P I : Lq’(P)

B LP4.4 fBg

--

IK Jfgdp

is ak,q-conlinuous. In other words, the embedding I : L q j ( p ) c* L p ( p ) is ( p , q ) -

factorable; momover,

Lp,q(I)

= IlIll = P ( W r = l l ~ I l l ( L q i @ ~ ; , q L , l ) ~

Clearly, p(R)l/OOis understood as being 1 if p(s1) > 0

*

.

PROOF: Since q’ 1 p , the embedding I is defined and continuous (the norm is p(s1)’jr) and 80 is the integrating functional /?I. Therefore, by the density lemma 13.4., it is enough to check the a;,,-continuity of PI on the dense subspace of step functions. Since a:,q is finitely generated, it is enough to take pairwise disjoint integrable sets Ci, ...,Cm C 0,the subspace

and M , := (M, 11 ll~,).Then it has to be shown that

18. (p,q)-Foctomble Opemtors

has norm

225

5 p(Q)'I'. For this define the functionals ' p k on M by

it follows that m

m

for all f i g E M and therefore

corresponds to , B , I M ~ M under the isometry (M,, it is immediate that

Mp/)' = Mi,

M i , . Since

it follows from the definition of ap,fthat

Thus, the first step towards the factorization theorem has been completed: the integrating functional (or the embedding), which will turn out to be the typical a;,,-continuous functional, actually is a;,,-continuous.

18.3. For finite dimensional M and N the typical factorization of p E ( M G q q N)' = MI

N'

goes as follows: the functional p has a; f-norm < c if and only if there exist operators R E C ( M , q t ) , S E C ( N , $ / ) and X E [0, oo[" such that

where 6~ is the bilinear form A k e k 8 e k and llRll llSll llbxll < c; this was already mentioned in 17.10.. The functional 6~ is positive (i.e. (ax f 8 g) 2 0 whenever f 2 0 and g 2 0). Since a;,( is finitely generated, cp E (E @ F)' is a;rq -continuous if and ~ uniformly bounded a;,,-norm (this is another only if the restrictions c p I ~ , g ,have

226

18. (p,q)-Factomble Opemtors

formulation of the maximality of Sp,().The ultraproduct technique (which will be explained now) allows one to study cp in terms of its finite dimensional parts. 18.4. Let I be a non-empty set. If 'XI is any filter on I, Zorn's lemma implies that there is an ultrafilter LL finer than 'XI. Recall that a topological Hausdorff space X is compact if and only if every ultrafilter on X converges; in particular, i f f : I -+ R is a bounded map, then the family (f(I)),€r converges along every ultrafilter U;the limit is usually denoted by limu f(6). Now fix an ultrafilter LL in 1. If (EL),Eris a family of Banach spaces, consider in the Banach space

the closed subspace c t ( E , ) of all (2,) with limu 11z411= 0

.

The ultraproduct of (E,),€r along LI is defined to be

(&)u

:=

foo(E')/C,U(EL)*

With the quotient norm this is a Banach space, the elements of which are denoted by ( 2 , ) ~(whenever ( % , ) , € I E &,(EL)), and after a moment's reflection one sees that

Il(2hll = 1p

112L11

In the case that each E, = E the ultraproduct (E,)uis called the ultrapower of E along U. Perhaps the reader is tempted to picture the ultraproduct as being something like tW/co - this is erroneous: The right analogue is c/co (where c is the space of convergent sequences). This indicates that ultraproducts are much smaller than one might at first guess; see in particular Ex 18.5. and Ex 18.10.. If (E,),E~ and (FL)&€r are two families of Banach spaces and T, E C ( E , , F L )with llT,II 5 c < 00 for all I E I, then

-

T' : &o(Ei) (2')

&o(J")

(T,d

is defined and llTrll 5 c. Since llZzL~~ 5 cllzcll holds, T I maps c f ( E , ) into c:(F,), hence there is a unique map T E f!((E,)u,(FL)u)such that

227

18. (p,q)-Factorable Opemtortr

commutes; T is called the ultraproduct (T,)u of the operators T, . It is clear from the description of the norm in the ultraproduct that

defines a continuous linear functional on iionr $ = limu 4'.

(E,)u

(F,)u with Il$ll = limu 11&11

. Nota-

PROOF: The mapping

is bilinear. With c := sup 11$,11,

I($', that

2' QD YJ

it follows from

- ($'YE' QD Y')I I 4 1 2 '

- 51 ' 1 IlV'li

+ 112;111 llsk - l7'11)

4 factors through the map $ of the statement. Since

one obtains

II$II 5 limu 11$,11

and even equality by choosing appropriate x' and y,

.0

If each E, is a Banach lattice, then .f!,(E,) is also a Banach lattice under componentwise ordering. Since c ! ( E , ) is a closed, solid subspace (since the norms are lattice norms), it follows (see Appendix A6.) that (E,)u is a Banach lattice under its natural quotient ordering. If each ELis an abstract Lp-spaces (1 I p < m), i.e.

1. + YllP = 1141"+ llYll"

if x A y = 0 ,

then their ultraproduct is also an abstract Lp-space: take z = ( c , ) ~and y = ( y r ) u with z A y = 0 ; then (2' A y') E c 2 ( E t ) and (2' - 2, A y') A (y' - 2' A y L ) = 0, hence

The same holds for abstract M-spaces

(112

+ yll = max{llzll, llyll} if c A y = 0).

228

18. (p,q)-Factorable Operators

The ultraproduct techniques in Banach space theory were developed around 1970 by various authors; recent references are Heinrich [lo41 and Sims [257].

18.5. To apply the ultraproduct technique for the description of a 4 E (E @a;,q F)’, with the help of its finite dimensional restrictions, define on the index set

I := FIN(E) x FIN(F)x]O, 11 an ordering by

(MI,NI,EI) I (M2,N2,&2) whenever MI C Mz, N1 C NZ and E I 5 ~2 , and take ?D the order filter which has as basis the sets { I E Ilk 2 I ~ and } LL any ultrafilter finer than ?D; note that L( is not unique. For z E E and I = (M, N , E ) define

x, := and E, := M. It follows that

{

z 0

--

ifzEM ifxeM

JE : E (E,)u z (ZJU is a linear map -which is an isometry by the definition of the norm in the ultraproduct. With F, := N for L = (M, N , e ) there is also an isometry JF : F + (F,)u Everything is now prepared for proving the

.

THEOREM: Lei l/p+ l/q 2 1 and p E ( E

F)’. Then there are strictly localizable measures p and v, operators R E 2(E,Lqt&>) , S E 2 ( F , and a positive functional 6 E ( L q I ( p )@* Lpl(v))’with (p, t @ y) = (6, Rx 8 Sy), i.e. Lpj(v))

-

It will follow from corollary 18.10. below that llvll.., = IIRll llSll llSll actually holds. (A linear functional p : E @ F K on the tensor product of two Banach lattices is called positive if (p,z @ y) 2 0 whenever z 2 0 and y 3 0.)

229

18. (p,q)-Factorable Opemtors

Now consider (see lemma 18.4.)

It follows, by the very definitions (recall the definition of 2, for t E E and similarly y,) that for z 8 y E E 8 F

and therefore

Since all 6, are positive, it is clear from the definition of the ordering in the ultraproducts and (.$:)u that 6 = limu6, is positive . By what was said above G, := (ce)u is an abstract L,-space if s < 00 and an abstract M-space if s = 00. The representation theorems (see Appendix A7.) show that G, = L , ( p ) (with norm and order) for some strictly localizable p if s < 00 and that for s = 00 the bidual GC is an L , ( p ) . Therefore, if p', q' E [1,00[, the theorem is completely proven. If, for example, q' = 00 then 6 factors

(q;)~

1

6 : G, BT Gpl

GL BrGpi = L,(p) 8rLp'(v)

A6

K

through its left canonical extension "6 which, clearly, is also positive and has the same norm as 6 . 0 The structure theorems imply that p and v can be chosen to be Borel-Radon measures on locally compact spaces. In terms of operators the theorem implies that for each T E f!p,q(E, F) there are strictly localizable measures p and v, operators R E C ( E ,L q i ( p ) ) ,S E C(Lp(v), F") and a positive operator U E f ! ( L q , ( p )Lp(v)) , such that

230

18. (p,q)-Factorable Operotors

18.6. Lp(id~,)= 1 by what was said at the beginning of 18.2.. Therefore this factorization and Lp(T) 2 IlRll IlUll llSll 2 IlRll IlU o Sll immediately imply:

COROLLARY 1: An operator T E C ( E ,F) is p-factorable if and only if IEF o T factors through some Lp(c() :

Moreover, Lp(T)= minllRll

llSll, the minimum taken over all such factorizations.

The measure can clearly be chosen to be a Borel-Radon measure. For p = 2 the fact that every subspace of a Hilbert space is l-complemented gives the following:

COROLLARY 2: T E C ( E ,F) is 2-factorable if and only if ii factors through a Hilbert space. Moreover, Lz(T) = min llRll IlSll, where the minimum is taken over all operators R E C ( E ,H) and S E C ( H ,F) with T = S o R (and H an arbitrary Htlbert space). In general, the factorization cannot be chosen to be through an L p ( p ) with a finite measure! Take T = idl,(q and a factorization

(t,(l?)is complemented in its bidual), so t,(I') is isomorphic to a subspace of L 1 ( p ) . If I' is uncountable,

c(

cannot be u-finite by a result of Pelczyliski [199], p.244.

18.7. For the ideal Jp N gp = ap,lof p-integral operators (1 5 p tion 18.5. reads

< 00) the factoriza-

231

18. (p,q)-Factorable Operators

Since, by 17.17., the positive operator U is even p-integral and absolutely p-summing with Pp(U) = I J U ) = IlUll, it follows from the Grothendieck-Pietsch factorization theorem (recall that L,(p) A C ( K ) for some compact K and use 11.3., corollary 2) that U factors

COROLLARY 3: For an operator T

E

E(E, F ) and 1 5 p <

00

the following three

statements are equivalent: ( a ) T is p-integral.

(b) There are a Borel-Radon measure p on a compact K, operators R E E ( E , C ( K ) ) and S E C(L,(p), F”) such that

commutes; I the canonical map C ( K )+ Lp(p).

(c) There are a finite measure p and operators R E E(E, L,(p)) and S E C(LP(p), F”) such that K F O T S =o I o R . I n this case, Ip(T)= minllSll IlI(( llRll, where the minimum is taken over all factorizations (IS in (b) o r as in (c). I

Thus, the natural map C(K)-L,(K, p ) is not only the typical absolutely p-summing but also the typical pintegral operator - only the form of the factorization is different: for absolutely p-summing operators into !,(BF,),for pintegral operators into F” (for p = 1 this is theorem 10.5.). For p = 00 the ideals , 3 and C , coincide. The factorization cannot be into F (instead of F“) since there are (l-)integral operators which are not Pietsch integral (see Appendix D9.).However, for p = 2, it follows as in 18.6. (corollary 2) that the operator S in the factorization can be chosen to have values in F, hence, T E &(ElF) if and only if there is a factorization

232

18. (p,q)-Foctomble Operators

Moreover, Iz(T) = minllRll llIll IlSll. But this is exactly the factorization of the absolutely 2-summing operators (see 11.3., corollary 2), hence

Since g ;

-

! & = 32

-

g2,

it follows that:

COROLLARY 4: 92 = g ; . In Ex 18.3. it will be shown that V p ( E ,F ) = J P ( E ,F) holds isometrically if E = C ( K ) or L,. If F is 1-complemented in F”, then obviously T E J p ( E ,F ) if and only if T R I factors E-C-L,&F and &(T) = minllRll llIll IlSll, the minimum taken over these factorizations. In particular, this holds if F is a dual space or if F has the metric extension property; example: F = 4, (r).It follows from the Grothendieck-Pietsch factorization theorem that

Vp(E,&Q(r)) J p ( E , 4 m ( W holds isometrically. Since (VP, Pp)is an injective operator ideal, this implies:

< 00 the injeciive hull of the Vp of absolutely p-summing operators:

COROLLARY 5: For 1 5 p is the ideal

(Vp,Pp)= (J?,

ideal Jp of p-integral operators

Iy).

For the associated tensor norm one obtains 1

1

N Bg;,e”, = V p ( N ’ , e ” , )= JP(N’,e”,) N

Bgp

.

Since g;, and g p are finitely generated, it follows that:

COROLLARY 6: g;, = g p on E 8 P , f o r all normed spaces E. We shall come back to this result in 20.14.. 18.8. For p-factorable operators the factorization theorem followed directly from theorem 18.5., for pintegral operators the Grothendieck-Pietsch factorization theorem was used. This suggests that the general case of (p,q)-factorable operators is even more complicated - and it is. Starting from the fact that for l/p l/q > 1 every

+

233

18. (p,q)-Factorable Operators

( p , 0)-factorable operator factors (into F“)through a positive operator L q , ( p )+ Lp(v) for some measures p and v (this is a consequence of theorem 18.5.), we shall proceed as follows: (1) Every positive L q t ( p ) + Lp(v)factors through a multiplication operator

-

Mg : Lq+)

-

Lp(v); Mg(f) := f g .

(2) Multiplication operators MB : Lq8(p) -+ L p ( p ) factor through an embedding L p ( p o ) for some finite measure pol hence are ( p , q)-factorable by what was said in 18.2.;moreover, Lp,q(MB) = llMBll= llgllr . Note that q’ 2 p . The fact (2) will be shown in 18.10. and (1) follows from Maurey’s factorization therem, which will be proven now. Lq8(po)

18.9. Every positive operator for all S E C ( E ,F )

ItT@

T : L q ( p i ) + Lp(p2) has the property (see 7.3.) that

-

: Lq(pl) @Aq

( p , q E [l,m] arbitrary). In particular, @

s : L q ( p l ) @A:

Ldv)

Lp(p2) @Ap

-

FII = llTll llsll

Lp(P2)@Ap Fll = IlTll llsll

for each measure v and each S E C ( L q ( v )F). , So T satisfies condition (a) of

MAUREY’SFACTORIZATION THEOREM: Let 1 5 p 5 q 5 00 and define r E [1,m] by l / p = l / q 1/r . For T E C ( E , L p ( p ) ) the following two statements are equivalent: (a) For every measure v and every S E C ( L q ( u ) F , ) (or only: S := idt,) the operator

+

T @ s : E @A:

Lq ( u )

-

Lp ( p ) @ A p F

i s continuous.

(b) There are g E L , ( p ) and R E C ( E ,L q ( p ) )with T = Mgo R. In this case, idt, : E @A; l q

-

Lp(p) @ A ~eqII = min IlRll IIgllr

1

where the minimum is taken over all factorizations as in (b).

PROOF: Assume first (b). Then it was proven in Ex 7.3. with the help of Holder’s inequality, that llMg

@

s : L q ( p ) @A:

Lq(’)

-

Lp(p) @A,,

Fll = 11~11rllSll

*

234

18. (p,g)-Factomble Opemtors

It follows that

has norm 5 llR 8 id : . * * 11 llMg 8 S : - 11 = llRll IlglirllS((. Conversely, assume (a). If p = q, the function g = 1 gives the desired factorization. So assume (a) with S = idt, for r < 00; this implies that c := llT 8 idd, : ..-I1 is a constant satisfying

...,Zn) in E . for all finite families (21, For q = 00 (and hence p = r < 00) the sum has to be replaced by a maximum, which means, in particular, that the increasing net ( f ~ ) fA := max{lTzl

12

E A } C &(p)

(where A C BE is finite) is norm-bounded by c. The type of order completeness which 5 c and

L , ( p ) has (see Appendix B5.) gives a non-negative g E L r ( p ) with llgllr lTzl 5 gllzll for all z E E. This shows that R E Z ( E , Lo3(p))defined by T Z

RX := 9

(5

:= 0)

has norm 1 and that T = Mgo R is a factorization as in (b) with IlRll llMsll 5 c. For 1 5 p < q < 00 (this is the remaining case) one has to find a g E L r ( p ) with 11gIlr I 1 and

Then Rx := 9-l . Tx and Mg again give the desired factorization T = Mg o R with llRll llMgll 5 cllgllr 5 c . For this, define s := q / p so s' = r/p . To apply Ky Fan's lemma (see Appendix A3.) observe that

K := {f E L&)

I f 1 0, 11fIld I 11

is weakly compact and convex. For every p-measurable h 2 0 the function [0, m] defined by if this integral exists if not

iph

:K

-,

235

18. (p,q)-Factomble Opemtora

is convex. The convex set {f E K I @h(f) 5 a} is norm-closed by Fatou's lemma from integration theory and hence weakly closed; this means that @ h is lower semiEE continuous. Therefore the set of functions @ z l , . . . , z n : K -+ [-m, 001 for 2 1 ,...,zn given by @=I,

...,sn(f):= cq

2

11Xk11'

-

12

ITzklqf-'@

k=l

k=1

is a convex set of concave, upper semi-continuous functions. For z1,...,X n E E the function f defined by

is in K and satisfies, by inequality (*),

(the case a = 0 is trivial). Ky Fan's lemma gives a function ...,z n E E the inequality a21,...,z n ( f o ) 2 0

fo

E

K such that for all

21,

holds. In particular,

aI(f0)2 0, which means that

J I T z l q f 7 d P 5 cql141q

1

for all x E E. The function g := f;lp satisfies llgllr 5 1 and (w). 0 The proof shows that T factors through a multiplication operator (as in (b)) if and only if (*) holds and this is how Maurey's factorization theorem was originally stated [186]. Recall from 7.9. that T C3 S is always continuous if q 5 p . Other factorization results in the spirit of this theorem will be given in Ex 18.14. and Ex 18.15..

18.10. For positive operators L,,

-

L, this implies:

COROLLARY: Let l/p+l/q 2 1. I f T : L q j ( p ) -+ Lp(v)i s a positive operator, then there are a finite measure po and operators R E 2 ( L q j ( p ) Lqj(po)) , and S E C(Lp(po), Lp(v)) such t h a t T = S o I o R

Ts

236

18. (p, q)- Factorable Operators

and IlTll

= llSll llIll IlRll . In particular, T

is ( p , q)-faciorable and L,,,(T)

=

PROOF: Since q' 2 p and T satisfies condition (a) of Maurey's factorization theorem (this was already noted at the beginning of 18.9.), T decomposes into T = M, o R, with llTll = IlgllrllRoll . It remains to factor the multiplication operator Mg : L,I (v) 4 Lp(v); recall that l/p = l/q' + 1/r . For this, take g1 := sign(g)lgl-'/q'

and

and the probability measure dp, := Igl'llgll;'dv liM9l Lq'W

-

l l K 2

and MB = Mgao [ I : Lq'(p,)

--

g2

:= Igl'/P

(recall 0/0 := 0). Then

L,'(clo>ll = l l ~ l l ~ r ~ q '

: LP(P0)

L P W I I = 11g11?"

Lp(p,)] o Mgl . It follows that

T=M,,oIoM,,oR,. Since llIll

= Lp,q(I)= 1 by 18.2., this factorization satisfies

LPAm 2 llTll = Il~ll'llRoll= l l ~ 9 a I IllW, I1 IlRoll 1 L IlMgaII I141 llM9i 0 R o l l = IIMg,ll LP,q(I) llM9i

0

Roll 2 Lp,q(T)

and everything has been proven. 0

18.11. Summarizing these results, one obtains the general factorization theorem for (p, 9)-factorable operators.

THEOREM: Let p , q E [l,oo[such that l / p + l/q the following two statements are equivalent:

> 1. For each operator T E E ( E , F )

(a>T E EPAK F ) * (b) There are u finite measure p , operators R E C(E,L q / ( p ) )and S E C(LP(p), F") such that K;F0 T = S OI 0 R E

T

F

F"

. In this case, LP,,JT)= minllSll 11111 IlRll over all such factorizations. Recall that the case l/p 18.6..

+ l/q

= 1 of pfactorable operators was already treated in

18. (p,q)-Factomble Operators

237

PROOF:If T has such a factorization, it follows from proposition 18.2. (and the regularity of maximal operator ideals, 17.8.) that

Conversely, if T E gP,,(E,F ) the ultraproduct theorem 18.5. gives a factorization

with some measures p and v, a positive operator U and

By corollary 18.10. the operator U can be decomposed into U = So o I o R, with LP,,(U)= llUll = IlS,ll 11111 llRoll and I : L q t ( p o )+ L p ( p o )for some finite measure p o . It follows that KFOT = S o S o 0 l o RooR and

LP,Q(T)L llsrl IlUll IlRIl

I tlS0 S o l 1 11I11 t w o 0 Rll 2 LP,,(T)

- the last inequality by the first part of the proof. 0 By examining the way in which the measure p is found it becomes clear that p can be chosen to be a finite Borel-Radon measure on a locally compact space R - and hence, even on a compact space (by passing, e.g., to the onepoint compactification of R).

If the range space F is 1-complemented in F” (or p = 2), then the factorization can be made into F. Simple manipulations show that:

COROLLARY: Take l / p + l / q > 1. For each

cp E ( E @r

F)’ the following statements

we equivalent:

(4 cp E ( E

@pa;,,

F)’

(b) There are a finite measure p , operators R1 E &(E,L , , ( p ) ) and Rz E f!(F,L,t(p)) such that (9,z @ Y) = L ( R i + ) . (RzY)~P .

In this case, l l c p l l ( ~ ~ F~ ), I = minllRlll l[R~Ilp(fl)’/~, the minimum taken over all such Psq factorizations.

238

18. (p,g)-Factomble Operotors

Exercises: Ex 18.1. Use the factorization 18.1. of T E C,,,(M,G> and the little Grothendieck theorem to show that for all 1 5 q 5 00

whenever H is a Hilbert space and E an arbitrary normed space. Ex 18.2. (a) An operator T E C(E, F) is in the injective hull f!? of the pfactorable operators if and only if T factors through a subspace of some L p ( p ) . Moreover,

where the minimum is taken over all R E C ( E ,G)and S E C(G,F) with T = S R and G C L p ( p ) for some measure p. (b) T E C ( E ,F) is in the surjective hull Crrof 2, if and only if ICFo T factors through a quotient of some L p ( p ) . Moreover,

the minimum taken over all such factorizations. Ex 18.3. If E = C(K) or L,(p), then P p ( E ,F) = 3,(E, F) holds isometrically and the operator S in the factorization 18.7. has necessarily values in F (instead of the bidual F"). Ex 18.4. (a) Let II be the ultrafilter of all sets containing a fixed point L* E I. What is the ultraproduct along II of a family (E,)u of Banach spaces? (b) Describe the relationship between ultrafilters on I and points of the Stone-Cechcompactification P I of I (discrete topology). Ex 18.5. If II is an ultrafilter on I and E, := K for all I E I, then (E,)u = K . Hint: limuf(L) defines a linear functional on &,(I). See also Ex 18.10.. Ex 18.6. (a) If T, E C(E,,F,) are metric injections, then (T,)u is a metric injection. (b) Show that the construction of (Z)u establishes an isometry

Ex 18.7. (a) The ultraproduct of C'-algebras is a C'-algebra. (b) The ultraproduct of C(K)-spaces is a C(K)-space. (c) For every Hilbert space H and 1 5 p < 00 there exist a strictly localizable measure p and an isometry I : H 4 LP(c().For 1 < p < 00 the Hilbert space is isomorphic to 1 a complemented subspace of some Lp(c().Hint: 6 4 L p ( p n ) for some pn via Gauss functions, see 8.7.; for the second part use Rademacher functions, see Ex 8.17..

18. (p,q)-Factomble Opemtors

239

E E: is a uniformly bounded family of functionals, then (cp, ( 2 , ) ~ := ) limu((p,, 2,) defines a linear functional cp on (E,)u with llcpll = limu 11cp,11. It follows that this defines an isometric injection

Ex 18.8. (a) If cp,

(b) Take an operator T E C ( E ,G) and a special ultrafilter U defined on the index set I := FIN(E) x FIN(G’) x ]0,1] as in 18.5.. For L = (M, N, 6) E I define E, := M, G, := GINo and T, := Q$ O T O : E, = M C , E -, G + GINO = G,

IZ

- the finite parts of T. Then there is a factorization as follows:

In the index set one can drop the interval ]0,1] for this construction, but (as one saw in the proof of 18.5.) it is sometimes quite useful to have it. The decomposition KG o

T = QG

0

(Z)u 0 J E

of T is sometimes called the canonical decomposition of T into its finite parts (with respect to U). Note that QG and JE are independent of T. (c) Use (b) to prove the factorization result 18.5. for operators T E $,,(I?, F). Ex 18.9. (a) Let F be a normed space and U an ultrafilter on FIN(F) finer than the order filter. Define Fu to be the ultraproduct of all N E FIN(F) along U. Then there is an isometry JF : F 4 Fu. (b) Let T E C(F’, E’) and (ON E (N BT El)’ satisfy

@T El)’ with and supN I I c p ~ l l< 00. Then there is a cp E (FL(

and IIcpII = limu I I c p ~ l l. If S E C(E’, Fh) is the operator associated with the functional then the relation Jk o S o T = i d p holds. In particular, im(T) is complemented in F’ and T : F’ 4 im(T) invertible. Hint: Lemma 18.4.. This result is just “Lindenstrauss’ compactness argument” treated in Ex 6.4. in its ultraproduct version. tp,

240

18. (p,q)-Factorable Operators

Ex 18.10. (a) If dim E, 5 n for a family (E,) of Banach spaces, then its ultraproduct (E,)u has dimension In as well. Hint: For dim E, = n take Auerbach bases (Z,,k,(p&,k)k1, ...,n and define (pk := limucp,,k (Ex 18.8.). Then ((pk,(z,)~)= 0 for all k implies ( t , )=~ 0. (b) If T,E C ( E , ,F,) all have rank 5 n, then ( q ) u has rank 5 n. Ex 18.11. Let T E C ( E , F ) be such that for all 6 > 0 there is an n E N such that for all M E FIN(E) there is an S E C ( M , F ) of rank 5 n with IlTlw - Sll 5 E . Use the factorization of T into its finite parts (Ex 18.8.(b)Land Ex 18.10.(b) to show that KF o T : E -+ F" is approximable. Since the ideal 3 of approximable operators is regular (Ex 9.12.(c)), it follows that T is approximable. Ex 18.12. Use the previous exercise to prove the following result of Heinrich [104]: Let T E C ( E , F ) be such that for all separable subspaces Eo c E the restriction TIE. E C(Eo,F) is approximable, then T is approximable. Hint: Negation and Ex 18.11.. Ex 18.13. For 1 5 p 5 q < 00 and T E C(E,t,) the following statements are equivalent (l/p = l / q + l/r): (a) For all tl,..., z nE E

-

(b) There are an R E C ( E , Q , )with llRll5 1 and A E Bf, such that T = Dx o R, where Dx : t4 Q, is the diagonal operator associated with A. Ex 18.14. Let 1 5 q 5 p 5 00 and l / q = l/p+ l/r. For T E 2(Lp(p),F) the following statements are equivalent: (a) For each S E C ( E ,L4(v))(or only S = idto)

T @ s : Lp(c()

@ A ~E

F @A; L4(v)

is continuous. (b) There are a g E Lr(p) and R E 2 ( L 4 ( p ) ,F") such that K F o T = R o Mp. Hint: Dualize Maurey's factorization theorem. Ex 18.15. The following variation of Maurey's factorization theorem 18.9. was given to us by Pisier; according to him, it "has been known in the neighbourhood of Maurey for a long time". The idea of proof is exactly as in 18.9.. Take 1 5 p < r < q < 00 and T E i ! ( L 4 ( p ) L,(v)) , and define s , t by 1/s + l / q = 1/r and l/t 1/r = l / p . (a) If there are non-negative functions go E L , ( p ) and h, E Lt(v)such that

+

for all f E L 4 ( p ) ,then there are multiplication operators M,, and Mh, and an operator E C ( L r ( p ) ,L,(v))such that T = Mh, o o Mgo and 11p 1 l 5 c . Hint: The closure of {go! I f E L 4 } is l-complemented in Lr .

? !

241

19. (p,q)-Dominated Opemtore

(b) Assume that T satisfies

for all

fi,

...,fn E L , ( p ) and define u := q / r and v := r / p . Use the function Ifkl'gu'dp -

@fl,...,jrn(9, h ) := c' k=l

12

ITfkI'h-"dv

k=l

(for non-negative g and h in the unit balls of L ( u ~ ) 2 ( pand ) L v # ( v ) respectively) , to deduce from the lemma of Ky Fan that there are functions go and h, as in (a). Hint: The first term is norm-continuous since

(c) Verify from these results the following theorem for 1 5 p

llT 8 idtV : Lq(p) 8~~t r

-

< r < q < 00

:

&(v) C ~ G AI1 ~5 c

if and only if there are functions g E L,(p) and h E Lt(v) and !i!E .C(Lr(p),Lr(V)) such that T = h!fh o !i!o Mg and Ilgll.ll!i!ll llhllt 5 c . Note that the case r = q is a special case of Maurey's factorization theorem and r = p a special case of Ex 18.14.. Ex 18.16. The ideal .CP,,of (p, q)-factorable operators is contained in the ideal 'LD of weakly compact operators if and only if q' < 00 or p < 00.

19. (p,q)-Dominated Operators

In this section the Grothendieck-Pietsch domination theorem for absolutely psumming operators will be appropriately generalized to the ideal ap,,= Ei8,q,of ( p , q)-dominated operators. The interpretation in terms of factorizations will give Kwapieri's factorization theorem: ap,q = g ~ a r o-~a result p which shows the central role of the absolutely p-summing operators in the theory. At the end of this section some comments about the so-called "non-commutative" Grothendieck inequality will be made.

242

19. (p,g)-Dominated Operators

+

+

19.1. For p , q E [l,001, with l/p l/q I 1 (or equivalently: l/p' l/q' 2 l ) , the of (p,q)-dominated operators was defined to be the ideal associated with ideal 9p,q the tensor norm = In other words (see 17.11.), T E 9 P , q ( EF) , if and only if

+

lr((d,%)) I c ~ p ( z k E; ) w q ( d ; F ' )

where l / p + l / q 11.' = 1, the norm D,,,(T) being the minimum of these c. This estimate w u a consequence of the obvious characterization of the aqt,pt-continuous bilinear forms: Take l/p l / q = 1 l/r, then p E (E @ F)' is aplq-continuous if and only if &f((P,zk @ Yk)) 5 C wq'(zk;E)wp'(Yk; F )

+

-

19.2. Since Qp g;f = dLf = theorem implies that I(cp9

z €9 Y)I I IIL9.ll

+

(see 17.12.), the Grothendieck-Pietsch domination

IlYll I PP(L9)

(J

I(%',4 I P P ( W ) l/pllPll

BE#

for each p E (E @d,f F)' = V p ( E F') , and vice-versa. For arbitrary ap,q-continuous forms (or, equivalently, for ( p , q)-dominated operators, see the corollary below) there is also a powerful characterization of this type - a result which is essentially due to Kwapieli [160]:

+

THEOREM: Let p, q E [I, m] with l/p l/q 2 1 and K C BE' and L C B p w e a k compact norming sets for E and F , respectively. For p E ( E @ F)' the following two statements are equivalent:

( 4 9 E ( E @a,,* F)'. (b) There are a constant c 2 0 and normalized Borel-Radon measures p on K and on L such that for all z E E and y E F

Y

(where the integral must be replaced by 11 11 if the ezponent is m). In $his case, ~ ~ p ~ I ( ~ g =,min{c ~ , , ~I -c )as~ in (b)}.

PROOF:For p' = m (or, by symmetry, for q' = m) this is just the GrothendieckPietsch domination theorem 11.3.; so it may be assumed that p and q are > 1, hence r' < 00. If (b) is satisfied, take 21,...,z,, E E and yl, ...,y,, E F. Since r'/q'+ r'/p' = 1, it follows from Holder's inequality that

243

19. (p,q)-Dominated Operators

The converse will be proven using Ky Fan’s lemma (see Appendix A3.). For this assume ~ ~ y ~ ~ ~= ~ 1 ,8take ~ the p , convex q ~ yweak-*-compact set Mf(K)c C(K)‘ of probability measures on K and define

c := M f ( K ) x M f ( L ) c C(K)’ x C ( t ) ‘ Consider the set 5 of all functions yl, ...,gnE F such that

I

f : C 4R for which there exist 2 1 , ...,zn E E and

for a11 ( p , v) E C. It is clear that all such f are continuous and aEne and that the set 5 is a convex cone. Since K and L are weak-*-compact and norming, there exist for f E 5 (coming from ( z k ) in E and (yk) in F) elements zb E K and y‘, E L such that n

The inequality a / s measures

n

+ b/s’ 2 al/abl/d’(for s €11,co[and a, b 2 0), implies for the Dirac

since ~ ~ y ~ = ~ 1 ~: All ~ the ~ assumptions ~ p , qfor ~Ky )Fan’s ~ lemma are satisfied! It follows that there is a ( p , v) E C with f(p, v) 3 0 for all f E 5;in particular,

:= (Jq 1(~’,z)1q’p(dz’))’/Q‘and p := (J, 1(y’,y)lP’~(d3/))~/P’ (if a = 0, then I((o, Xz @ y)l‘ 5 r’/p’sL l(y’, y)IP’v(dy’) for all X and hence (9,z @ y) = 0). This is the

for

(Y

inequality in (b). 0

-

Since 9p,qair,pland

244

19. (p,q)-Dominated Opemtors

(isometrically) by the representation theorem for maximal operator ideals, the theorem has the following consequence:

COROLLARY: Take p , q E [l,001 with l/p+ l / q 5 1 and K C BE' and L c Bpi weak*-compact norming sets. For T € C ( E ,F ) the following statements are equivalent: ( 4 T E3P,9WlF) * (b) There are a c 2 0 and probability measures p and Y such that

holds for all z E E and y' E F', (replace the integral by In this case, D p , q ( T )= min{c I c as in (b)}.

11 11 if the ezponent

is

00).

19.3. The proof will show that the following result is just a reformulation of the characterization of the ( p , q)-dominated operators which was just obtained.

K W A P I E ~FACTORIZATION ~S THEOREM: Let p , q E [l,001 such that l/p Then Dp,q

+ l/q

5 1.

= %ya' 0 P JP

holds isometrically. In other words, T E C ( E ,F ) is ( p , 9)-dominated if and only if T admits a factorization

E-F

T

G such that R E V P and S' E Pq . Moreover, D p , q ( T )= min{Ppa'(S)Pp(R) I T = S o R } Recall that

I

qm:= C,so the result is nothing new if p or q is 00.

PROOF: If T has such a factorization and 1/r = l/p W Y i l

T 4 ) = [r((S'Yi,

RZk))

+ l/q,

then

5 tp(RZk)tq(S'ylk) L

L pP(R)wP(.k)p~(s').lq(Yi) and therefore, D P , q ( T )5 P,d""'(S)P,(R) (by what was repeated in 19.1.). Conversely, take T E 3 p , q ( EF, ) . Then, by the corollary above, there are probability measures p B p such that

on BE' and Y on

245

19. (p,q)-Dominated Operotors

(the casa p or p = 00 need not be treated). Define R,x := (., x) E C(Bp)and consider the diagram T

E

where Rz := I(R,z). Since Pp(Z) = 1 and Vp is injective (see 11.1. and 11.2.), it follows that Pp(R)5 1. The operator S is defined on R(E) by

SRz := Tx

- and this definition makes sense because

(1 l(d',d)l'"&''))

l(d,SRz)I IDp,q(T)IIRx11G

l"

.

Bprr

It follows that S is continuous on R(E) and has a unique extension to R ( E ) = G; moreover, the inequality implies that

which means that S' is absolutely q-summing and

Pdua'(S) I = P,(S') 5 Dp,,(T). This ends the proof. 0

-

-

19.4. Since .CP,' aP,' and !Di,rq'a;,,,it follows that the ideals .Cp,, of (p, 9)factorable operators and Dp',q~ of (p', q')-dominated operators are adjoint to each other (see 17.9.). Anticipating the fact that ap,' and a;,' are accessible tensor norms (to be proven in section 21), it follows from the inclusion a* o Q c J (see 17.19.) that

CP,' 0 BP',,' c J and I(TS) I L P , ' ( ~ P P ' , & ) 9p"q' 0 Cp,, c J and I(TS) IDP',Q'(TNP,Q(S) (for l/p

+ l/q

PROPOSITION

2 1). The composition is even nuclear in most cases: if 1/p + l / q 2 1 and (p, q ) 4 ((1, I), (1, oo),(00, I)}, then B)p',q' 0 .Cp,e

c 3 and

N(TS) IDP',q'(T)LP,q(S) .

246

19. (pfq)-Dominated Opemtors

This relation i s false for the excluded pairs ( p , q ) .

For the nuclearity of operators in

cp,qo

9pf,qi

see Ex 19.3..

PROOF:The best way to check that an integral operator is nuclear, is to use the Radon-Nikodjmproperty. Take S E 2 p , q ( EF) , and T E 9p~,qt(Fl G ) and assume that 1 < q < 00; then

9pf,q' c a,;1,

cm

(weakly compact operators)

by Kwapieri's factorization theorem (or more simply: since di 5 a;,q). Therefore T " ( P ) C G which implies that the astriction T" : F" -+ G of T" is also (p',q')dominated and D P ~ , 4 ~ (= T "DPt,qf(T) ) (since TI' is, and maximal ideals are regular, mix 17.8.). The factorization theorem 18.11. for ( p , q)-factorable operators gives

with Lp,q(S)= Lp,q(Sf')= llUl1 llIll llVll . Since I E C p , q ,it follows that T" o V o I is integral and hence nuclear because L,I is reflexive and has the approximation property (see Appendix D8., corollary); moreover,

N(T* o V o I ) = I(T" o V 0 I ) 5 D p ~ , q ~ ( TllIll ) ~.~ V ~ ~ It follows that T o S is nuclear with the desired estimate. If q = 1 and 1 < p < 00, observe first that the Grothendieck-Pietsch factorization theorem (11.3., corollary 1, (c)) implies that PP# = ?~DO?J?~~ holds isometrically; hence,

Dpl,oo0 cp,l= 9pf,m 0 9;,?= Pp10 Pi,c m 0 3 c n and N 5 11 11 o I, where the latter is one of Grothendieck's results related to the Radon-Nikodjm property (see Appendix C8.). To see the negative results, first take (PId = (131): a,,, 0 c1,1= c 0 3 = 3 @ n . For the remaining two cases ( p , q ) = ( 1 , ~ or ) (00,l) recall that co does not have the Radon-Nikodjm property (Appendix D3.), hence there is a T : C[O,11 -, co which is absolutely summing but not nuclear (Appendix D7.). Since C[O,11' has the approximation property (it is an abstract L-space), TI is also not nuclear by 5.9.; since, clearly, idcpl] E 2 , and idt, E 21 (by 18.6., for example), it follows that

T E PIO 2 , = 91,aO 0 em,1 T' E PPa1 o2 1 = !Doo,l o 2 1 , ~

247

19. (p,q)-Dominated Operutors

- and this completes the proof. 0 The following important result of Grothendieck is a special case:

COROLLARY: The composition of two absolutely 2-summing operators is nuclear; moreover, N(TS) 5 P2(T)P2(S). Since 32 = rlp2 (see 18.7.), this follows from

19.5. If K and L are two compact topological spaces, then T

_< K C W ~ on C ( K )c3 C ( L )

by Grothendieck’s inequality (see 14.5.). Since ( C ( K ) theorem 19.2. implies:

C(L))’= % i [ ( C ( K )C, ( L ) ) ,

PROPOSITION: For each (p E Bif(C(K),C ( L ) )there are normalized Borel-Radon meaon K and Y on L such that for all ( f , g ) E C(K ) x C ( L )

sures p

Obviously, the smallest constant having this property (for all K and L) is the Grothendieck constant (recall proposition 14.6.).

19.6. A state X on a C*-algebra A is a linear functional A E A’ with 11X11 = 1 which is positive, i.e. X(zz*) 2 0 for all x E A. Since every commutative C+-algebra is isometric (as a C’-algebra) to some C ( K ) and the probability measures on K are exactly the states of C ( K ) , it follows: Let A and B be commutative C*-algebras and cp !Bil(A, B). Then there are states XI on A and A2 on B such that I d t t

Y)l

5 K:llVII

2 112

M142)112~2(lYl 1

*

This is another formulation of Grothendieck’s inequality which (with another constant) also holds for arbitrary C*-algebras A and B if (different from the usual definition of the modulus) /.I2 := 1/2(22’ z*z);this deep result was conjectured by Grothendieck (R6sum6, problkme 4), first proven by Pisier [220], assuming the approximation property, and then by Haagerup [97] in the general case. The result is sometimes called the Haagerup-Pisier-Grothendieck inequality or the non-commutative Grothendieck inequality. If K , denotes the best possible constant, then a simple example in A = B = R(t?2,&) shows that K , 2 2 (see Pisier [225], p.119), which implies, in particular, that K,$ < K, . Actually, Haagerup [97] proved the following stronger version, which easily implies that K , 5 4, and in which the constant 1 cannot be improved:

+

248

19. (p,q)-Dominated Operators

If A and B are C*-algebras, I,O E %it(A, B ) , then there are states A',

A2

on A and

p1,pz on B such that lcP(z1

Y)l

I l l ~ l l ( ~ l ( = *+) ~ 2 ( z * ~ ) ) ' / 2 ( ~ 1 ( Y+v *P2(Y*Y))'/2 )

*

Unfortunately, it is not true (as it is in the commutative case) that the tensor norms x and w2 are equivalent on the tensor product A @ B of two C*-algebras: Since 1 2 is l-complemented in the C*-algebra E(&,!2), this would imply that x and w2 were equivalent on 42 @ f a , but, clearly, idt, is not 2-dominated, i.e. tr $$ ( t 2 BW,4)'; this observation is due to Pisier. Since w2 5 w: (see Ex 19.5.), the following result is weaker; recall that w2 22, the ideal of operators factoring through a Hilbert space (18.6.).

-

COROLLARY (Haagerup [97]): If A and B are C*-algebras, then every continuous operator T : A + B' factors through a Hilbert space; moreover, L2(T) other words, s52w: onAc3B.

I 211Tll.

In

PROOF: The non-commutative Grothendieck inequality implies that

+ A2(z*z))11~2IIIYY*II + llY*Yll11~2 5 5 2llTll[2-'(~1(+x*) + .x2(z*41]1/2 IlYll

I(TZ1Y)lS IlTllJZ [2"(Al(ZZ*)

*

The definition (.I%) := 2-1(A'(z%*)

+ A2(%*.))

gives a non-negative sesquilinear form on A x A such that the associated seminorm p satisfiea p ( z ) := ( z ~ z ) ' 5 / ~11~11. It follows that A / ker p (with its natural norm coming from p) is a preHi1bert space and hence its completion H is a Hilbert space. The canonical map R : A -+ H has norm I 1. The inequality at the beginning implies that llTZ11 I 2lITlI l1WH *

If So : R(A) -+ B' is defined by So(Rz) := Tx, then So is well-defined, has norm 5 211Tll and extends to an operator S : H -+ B' with llSll = llSoll. Clearly, T = S o R and therefore,

LdT) I IlSll IPll I 211Tll Since

( A @r B)'

a

A f!(A,B') = &(A, B') A ( A

B)'

by the representation theorem for maximal operator ideals, the result is proven. 0 For a nice application of this see Ex 19.9. - Ex 19.11.. The reader will find more information about the Haagerup-Pisier-Grothendieck inequality in Pisier's book [225]. Using the ideas of Pisier and Haagerup, Grothendieck's inequality for C*-algebras was

19. (p,q)-Dominated Operators

249

extended by Barton-F'riedman [6] to continuous bilinear forms on the larger class of so-called JB*-triples (see also Chu-Iochum-Loupias 1291).

Exercises: (where p or q are different from oo),then E is finite dimensional. Ex 19.1. If i d E E gp,q Ex 19.2. Using the Grothendieck-Pietsch domination theorem, show directly that every operator T E o ppsatisfies an integral estimate a8 in corollary 19.2.. Ex 19.3. If (p,q ) {(I, l), (1, oo),(oo,l)} and T E ( f ! p , q o DPtIqt)(E, F), then T' is whenever T = R o S is such a factorization. nuclear and N(T') 5 Lp,q(R)Dpt,qt(S) In particular, K;Fo T is nuclear and, if E' has the approximation property, then T is nuclear as well. If ( p , q ) E {(l,l),(l,m),oo,l)}, then it may happen that none of the operators T, T' or K F o T is nuclear. Hint for the negative part: T at the end of 19.4. has a bidual which is not nuclear. Ex 19.4. Deduce from 12.5. and 12.8. the following results: (a) If PI 5 PZ and q1 i q z , then ( 9 p l ,919 Dp1,ql) C (Bpa,q:, Dpa,ga)(b) If p,q €11,a[, then go,,, C PZ. (c) If p , q E 12, oo[,then 9p,q = DZ. Ex 19.5. Show that w2 5 gz _< w; . Hint: Operator ideals. Ex 19.6. If p or q is < 00, then 9p,q C !Zl . Ex 19.7. (a) Every operator C ( K )+ C(L)' is in 9 2 . Hint: Grothendieck's inequality. (b) 9 2 (t ff . Hint: Consider the operator T : L,[O, 11 3 f-fdA E L,[O, 11' for the Lebesgue measure A. Ex 19.8. (a) Show with the little Grothendieck theorem that P$'"'(E, F) C %(E, F ) whenever E is an 2!-space. Hint: Prove this first for l , and then use the CP-local technique lemma. (b) There is a set I' and a Banach space F such that

!J?ral

!Z are injective and '432 $t ff, but '732 o Z !D C R . Hint: (.p2 and P2 o ? (c) Deduce from (a) and (b) that 9 2 !$ ! Z l o D z . Ex 19.9. Use corollary 19.6. and the fact that w; and w2 are totally accessible (to be proven in section 21) to show the following results for arbitrary C*-algebras A and B. (a) The canonical map A&B --+ A&B is injective. (b) w z 5 2e on A' 43, B'. (c) The natural embedding A & B (A' & B')' is an isomorphic embedding; more precisely, n 5 2% on A 8 B .

-

250

20. Projective and Injective Tensor Norma

Clearly, (c) improves (a). Recall that the C'-algebra 2(&,&) does not have the approximation property. E x 19.10. Let H be a Hilbert space. Then every operator C ( H , H ) --+ 'YZ(H,H) factors through a Hilbert space. E x 19.11. If A is a C'-algebra such that A' is isomorphic to a C'-algebra, then A is finite dimensional. Hint: Use the fact that infinite dimensional C'-algebras are not reflexive (this follows, e.g., from Exercise 4.6.12 in Kadison-Ringrose [139]).

20. Projective and Injective Tensor Norms

The projective norm 7r respects metric surjections and the injective norm E respects metric injections. In this section these properties are studied systematically for arbitrary tensor norms. In particular, injective and projective norms (i.e. those respecting metric injections or surjections, respectively) closest to a given tensor norm are constructed. The associated operator ideals will be described. Finally, some more aspects of Grothendieck's inequality will be treated.

20.1. A tensor norm Q on NORM (or on FIN, or other classes) is called right-injective on NORM (or on FIN, ...) if for all metric injections I : F L* G the operator

is a metric injection for E , F, G E NORM (or E FIN, ...) and right-projective on NORM (or on FIN, ...) if for all metric surjections Q : F --y G the operator

is a metric surjection for E, F, G E NORM (or E FIN, ...). If a*is right-injective (resp. right-projective), then the tensor norm Q is called left-injective (resp. left-projeciive) on NORM/FIN/ ...; if Q is right- and left-injective (resp. right- and left-projective), it is called injective (resp. projective) on NORM/FIN/ ... . Clearly, E is injective and I is projective. Observe that i d E 6 ) I : E&,F Q EgaG is a metric injection whenever i d E @ I is and it follows from the quotient lemma 7.4. that : E g a F --y E&,G is a metric surjection whenever i d E €4 Q is.

251

20. Projective and Injective Tensor Norms

The duality implies the

REMARK:a is right-injective on FIN if and only if a' is right-projective on FIN. 20.2. This remark will be extended to tensor norms on NORM. The right-injective

tensor norms are relatively easy to handle, but when treating the right-projective ones some problems occur when passing from Banach to arbitrary normed spaces. So their study will be prepared by a precise investigation of their behaviour with respect to dense subspaces. For this, let ,D be a tensor norm on NORM x C (where C is either the class of all Banach spaces or the class of all normed spaces), and recall from Ex 12.3. the definition of the right-finite hull 'p

p'(z; E l F ) = inf{p(z; El N) 1 N /3+ is a tensor norm and ,f?I/I'

E FIN(F), z

E E 8N }

.

+

I/I .

LEMMA: (1) Ifp is right-projective on NORM x C , then p = 'p on NORM x C . (2) I f p is a tensor norm on NORM such that p = /I' on NORM x BAN, then /3 = 'p on NORM x NORM and

E ~ ~ F C - L E ~ ~ F all ( E lF ) E NORM x NORM. (3) I f p is a tensor norm on NORM, right-projective on NORM x BAN, then it is right-projective on NORM x NORM.

for

PROOF:(1) If G E C, then there is a metric surjection 1

Q:F-G such that F has the metric approximation property (if G is complete, take F := l l ( B ~ ) and in the general case an appropriate dense subspace of t , ( B & )which necessarily has the metric approximation property by 16.3., for example); then, for each normed space E the approximation lemma 13.1. implies that

It follows that for each element z E E 8 G there is an N E FIN(F) and a 2 E E 8 N with idE €3 Q ( i = ) L and P(2; E , N ) I(1

+ c)P(t.; El G )

20. Projective and Injective Tensor Norms

252

and therefore ,

P(z;El G ) I

P-'(z;E, G )I

P ( t ; E , Q N )I P ( i ;

El

N )5 (1 + E)P(z;E , G )

*

(2) Take z E E 63 F. Then the metric mapping property gives r ( z ; E , P) = P ( z ; E l P)

5 P(z; El F ) 5 p-'(t;

El F )

.

For N E FIN(F), with t E E 63 N and P ( z ; El N ) 5 (1

+ &)P-(z; El E )

1

use the principle of local reflexivity 6.6. to find an operator R : N 4 F with IlRll 5 1 + ~ and Ry = y for all y E N n F. The operator T : E' 4 F associated with t has values in N n F, which implies that

Consequently,

P-*(t; El F ) I P ( i d ~63 R(z);E l R N ) I IIRIIP(t;E , N ) I I (1 + &)'P-'(z; El P) , which proves (2). To see (3), recall again the quotient lemma in 7.4. and take Q : F --y G a metric surjection between normed spaces; then the completion Q : -+ G is a metric surjection with ker Q = and, by assumption,

idE @ Q : E @p

P

-

E @p 6

is also a metric surjection. Since by (1) and (2)

E@pF-E@pF

and E @ p G - E @ p e

are metric injections and, clearly (idE@Q)(EQDF) = E @ G ,it follows from the quotient lemma that idE @ Q : E @ p F E QDp G

-

is a metric surjection if

id^ €3 0) = E €3 ker Q But this is obvious since ker 0 = ker

(see 2.7.).

EC3d

ker( idE @ Q)

.0

20.3. Now the announced duality between right-injective and right-projective tensor norms can be proved. At the same time, and this is in some sense natural, a preliminary

253

20. Projective and Injective Tensor Norms

observation concerning the accessibility of these tensor norms will be made (a more careful investigation will be given in section 21).

PROPOSITION: Let a be a tensor norm on NORM. (1) If a is right-injective on FIN, then % and (2) If a is right-projective on FIN, then

d

d

are right-injective on NORM.

is rightprojective on NORM.

( 3 ) If a is finitely or cofinitely generated, then: a is right-injective on NORM if and only if a‘ is right-projective on NORM. (4) If a is right-injective or right-projective on FIN, then a i s right-accessible. 1

PROOF:(1) and (4): If a is right-injective on FIN, then for F cr G and z E E 8 F d ( z ; E, G) 5 Z ( z ; E, F ) = = inf{a(r; M, N n F)lM E FIN(E), N E FIN(G), z E M @ N} =

= inf{a(z; M, N ) [...} = d ( z ; E , G),

d is right-injective. To treat the cofinite hull, first (4) will be shown: For this, take (N, F) E FIN x NORM and o E N 8 F and assume a is right-injective on FIN. Then, by what was already shown and by the approximation lemma, it follows that

so

hence a is right-accessible. Now recall that a is right-accessible if a’ is (see 15.6.). Hence, if Q is right-projective on FIN, the dual a’ is right-injective on FIN, therefore a’ is right-accessible and so is a. Now it is possible to show that % is right-injective on NORM if a is right-injective 1 on FIN: For F L,G and z E E 8 F the following holds by the two results which were already shown:

254

20. Projective and Injective Tensor Norms

1

(2) Using lemma 20.2.(3), it is enough to consider a metric surjection Q : F 4 G between Banach spaces. By (4) the tensor norm a' is right-accessible, hence for every N E FIN result (1) implies that

and therefore idN 8 Q : N 8F a

-

N 8; G

is a metric surjection. Now take E an arbitrary normed space: ~ ( zE ;, G)

{ N , G ) I N E FIN(E), z E N B G } = = inf {Z'(w; N , F) I N E FIN(E), i d N €3 Q(w) = z } = = inf

z(%;

= inf { s ( w ; E, F ) I i d E 8 Q(w) = z }

.

Finally, statement (3) follows from (l),(2) and the remark at the end of 20.1.. 0 It is not true that the cofinite hull % is right-projective on BAN if a is right-projective 1 on FIN; to see an example take a = A and . t l ( B ~--y) F for a Banach space F without the metric approximation property. Then 1

F'@+~~(BF = )F ' ~ , ~ ~ ( B F ) - - Y F ' € ~ ~ FF .# F ' ~ ~ A

Since there is no Hahn-Banach theorem for general operators between Banach spaces (1.5.), A is neither right- nor left-injective. Dually (this means using (3) of the last proposition), E is neither right- nor left-projective; see also 20.20.. 20.4. For the a,,,-tensor

norms the following result holds:

PROPOSITION: Let 1 5 p 5 00. Then ( 1 ) d, is right-projective; consequently, gp is left-projective and g; = d; is rightinjective. ( 2 ) az,, is right-injective, C Y , , ~left-injective and right-projective. In particular, w2 is injective and w; = wi i s projective.

PROOF:Since

20. Projective and Injective Tensor Norms

255

result (1) is a direct consequence of the following observation: If Q : F --y G is a metric surjection, E > 0 and y1, ...,y, E G, then there are & E F with Q($i) = yi and

To ax that z E E @ F and

E

is right-injective, take a metric injection I : F of z in E @ G with

> 0. Choose a representation

c, G,

Then (by what was explained in 17.10.) the associated operator T, : E' factorization

an element

+F

admits a

z ; G). The subspace S - l ( I ( F ) ) is 1-complemented with llRll110~11IlS((5 ( l + ~ ) a z , ~ (E, in 6 ,hence there is an So : 6 -+ F with llSoll5 llSll and T, = Soo D Ao R. It follows that n

z

=

CX~CL @ S O e h E E @F k=l

and

which shows that 0 2 , ~is right-injective. The other statements in (1) and (2) follow easily by dualization and transposition. 20.5. To see an application of this, recall from 15.10. the Chevet-Persson-Saphar

inequalities dp

5 Ap Ig;*

on L P W @ E

and also that dp and g;.' are the closest tensor norms to Ap (see 15.11.). I t was shown in 15.10. that dp = Ap = g i l if E = L p ( v ) .More generally,

PROPOSITION: If E i s a Banach space which can be obtained by starting with some Lp(v) and then taking metric quotients and metric subspaces finitely many times, then

for every measure p .

256

20. Projective and

Injective Tensor Norms

PROOF: Fix p and consider the maximal normed operator ideal ,!A such that L p ( T ) := I I i d ~ , ( p )@ T : Lp(cI) @d, E-Lp(p)

@A,

of all T E &(E,F )

F)ll < 00

(see 17.4.). Since d, is right-projective and Ap right-injective (7.4.), this operator ideal is injective and surjective. This impLes that the class of spaces E such that L,(idE) = 1 is stable under forming metric subspaces and metric quotients (both # {0}, clearly). Since L , , ( i d q v l ) = 1 by what was said earlier, it follows that dp = Ap for those spaces satisfying the assumption. For gp, consider the ideal of those T such that

IlidL,(p) @ T : Lp(p) @A, E-Lpb)

@g;f

Fll< 00

.

Observe that it is again injective and surjective since g;, is right-injective and A, is right-projective (7.4.), and argue in exactly the same way. 0

In Ex 20.8. and EX 20.9. it will be shown that the spaces in the assumption of the proposition are precisely the subspaces of quotients of some L, (equivalently: quotients of subspaces of some L p ) . It will turn out in 25.10. that actually these are exactly the spaces E with dp = Ap = g;’ on L, @ E . For Ap = d;f and Ap = g , see also 25.10.. 20.6. Every tensor norm a is less than or equal to 7r and 7r is projective. Hence it is reasonable to search for a smallest tensor norm /3 2 a which is projective.

THEOREM: Let a be a tensor norm on NORM. Then there is a unique right-projective tensor norm a/ 2 a on NORM with the following property: If /3 2 a is right-projective, then /3 2 a/.Moreover, a/ is finitely generated from the right, i.e. a/ = (a/)+. The right-projective associate

of a can be calculated using the following property:

(Y/

If E is normed and F a Banach space, then

i s a metric surjection. If E and F are arbitrary normed spaces and z E E @I F, then

(*I

I

a/(%; E , F ) = inf{a/(z; E l N) N E FIN(F), z E E @ N }

.

1

The fact that a/ respects quotient mappings F-FIG - is of course responsible for the use of the symbol a/.

PROOF:Uniqueness presents no problem. Existence: a/ will be constructed first on NORM x BAN and then extended, using the introductory lemma 20.2..

257

80. Projective and Injective Tensor Norms

(a) If (E, F ) E NORM x BAN, define a/ to be the quotient seminorm on E @ F given by the mapping E @a ti (BF)-E @F . Since E a 5 rn and x is right-projective, it follows that E 5 a/ 5 rn on the class NORM x BAN. To see the metric mapping property on NORM x BAN, take operators Z E E(Ei, Fi) and use the lifting property of the spaces t,(I')(see 3.12.) to obtain a T 2 with

5 11Z;11Il!&i;l It follows that llT1@af";II of a/)that llTl@ T2 : El

@a/

E2-F1

5 ~ ~ T 1 ~ ~ ( l +and ~ )hence ~ ~ T(by2 the ~ ~definition , @a/

F2ll

5 111;11(1+

w 2 1 1

for all E > 0. Thus, a/ is a tensor norm on NORM x BAN. (b) To show that a/ is right-projective on NORM x BAN take a metric surjection Q : F --y G between Banach spaces and again use the lifting property:

The diagram

implies that i d s @ Q is a metric surjection. (c) Lemma 20.2.(1) now implies that a/ = (a/)+on NORM x BAN. This means that the definition on NORM x NORM a/ := (a/)+ gives a tensor norm which extends a/ from NORM x BAN to NORM x NORM. Lemma 20.2.(3) shows that a/ is right-projective on NORM; moreover, by the definition a 5 a/ on NORM x FIN, and therefore, a 5 a+ 5 (a/)- = a/.

20. Projective and Injective Tensor Norms

258

<

(d) Finally, take any right-projective tensor norm ,8 2 a;it follows that a/ p on NORM x FIN by the definition of a/.Since p = p’ on NORM by lemma 20.2.(1), it follows that a/ = (a/)+5 8 ,‘ = ,8. This ends the proof. 0 Consider the natural metric surjection

Q : ei(Bil(r))A el(r) ; then there is a lifting T : 41(I’) + 41(Bcl(r)) of idt,(r) (i.e. Q llTll 5 1 E . Therefore, for any normed space E, the diagram

+

o

T = idtl(r)) with

+

shows that a/ 5 (1 €)aon E 8 tl(r).Since a 5 a/ always holds, it follows:

COROLLARY 1: If E is a nonned space, then a =a/

on E@ll(I’)

for every set I’.

For obvious reasons \a := ( ( a t ) / ) t

is called the lefl-projective associate of a .

COROLLARY 2: Let a be a tensor norm. Then

is called the projective associate of a; it is the unique smallest projective tensor norm

2 a, is finitely generated and

1

f l ( B E ) @a L l ( B F )

E

@\a/

F

is a meiric surjection whenever E and F are Banach spaces.

The proof follows easily from the “transitivity of metric surjections” and the theorem. 20.7. Fortunately, the injective case is simpler.

120. Projective and Injective Tensor Norms

259

THEOREM: Let a be a tensor norm on NORM. Then there is a unique right-injective tensor norm a\ 5 a on NORM such that /3 a\ for all right-injective tensor norms /3 5 a. For all n o m e d spaces E and F

<

is a metric injection. a\ is finitely generated from the right.

The tensor norm a\ is called the right-injective associate of a .

PROOF:Define a\ on E @ F to be the norm induced by E@FLE@utrn(BF'). Then & 5 a\ _< n on NORM since E is right-injective. To see the metric mapping property, take S E C(E1, E2) and T E C(F1,F2). By the metric extension property of t,(I') there is an extension T of T such that

commutes. This easily implies the conclusion. a\ is right-injective: If I : F c, G is a metric injection, then the metric extension property again gives a mapping i such that

The definition of a\ immediately implies that /3 5 a\ whenever p Clearly, /3 = /3' for every right-injective tensor norm. 0

As in the projective case, / a := ( ( d ) \ ) t

5 a is right-injective.

260

20. Projective and Injective Tensor Norms

is called the left-injective associate of a and

is the injective associate which is the unique largest injective tensor norm smaller than a. It follows that E F &(BE#)& ~W(BF') .

r'.

Clearly, every injective tensor norm is finitely generated. The notation, with all the various slashes, may seem confusing at first, but it is really quite natural and not that difficult to memorize: Start with the right-projective associate a/ (coming from quotients FIG), - then take its counterpart a\ for the rightinjective associate - and finally the rest by symmetry.

COROLLARY: If the Banach space F has the A-extension property, then a\sa
onE@F

for all normed spaces E.

PROOF:F is A-complemented in . t w ( B p ) .0 20.8. The following is a special case.

PROPOSITION: For every tensor norm a , normed space E and n E N

Now the 2p-local technique lemma 13.5. can be applied.

COROLLARY: Let a be a tensor norm and E a normed space. (1) If F is an 2:,x-space, then asa/sAa'

on E @ F .

( 2 ) If F is an 2&,x-space, then a\
on E B F .

Note that a- 5 p a on E @ F if F has the p-bounded approximation property by the approximation lemma (C;-spaces have the bounded approximation property, see 21.6.) and a = ' a if a is finitely generated.

20. Projective and Injective Tensor Norms

261

PROOF:The local technique lemma showed, in case (l), that the proposition implies that a/ 5 (a/)* _< Xa’ on E @ F (see the remark after the proof in 13.5.). In case (2) it follows that a- 5 X(a\)+ = Xa\ . 0 20.9. The following gives a simple test for recognizing whether or not a tensor norm

p is the projective or injective associate of a given tensor norm a : PROPOSITION: Let a and p be tensor norms. (1) If

P

is right-projeciive, then the following staiemenis are equivalent:

(4 P = a/

*

(b) For all E E NORM and n E N

E @p (2)

= E @a

q

isometrically.

If P is right-injeciive, ihen the following siatements are equivalent: (4 P = a\ *

(b) For all E E NORM and n E N

E @p P , = E @& P,

isometrically.

(3) If a and p are finiiely generated, then in both cases it is enough t o tesi only for finite dimensional E. (4) If P is projective, then ,B = \a/ if and only if f o r all n E N

q @p q

q

=

isometrically.

( 5 ) If p is injective, then p = /a\ i f and only if for all n E N P, @ p P , =P ,

Pm

isometrically.

PROOF: That (a) implies (b) (in (1) and (2)) follows from the last proposition. Assume (l)(b). Then by the local technique lemma, ‘p = a+ on E @ l l ( r )and hence p = a on E @ l , ( r )by the approximation lemma: the fact that P is right-projective along with property (*) in theorem 20.6. gives (a). The remaining implication in (2), as well as (4) and (5), follow in the same fashion. Statement (3) is obvious. 0 It is worth noting that these characterizations (with the same proofs) also hold if constants are involved - and this will be used. For example: I f a and p are tensor norms, P injective, then c p L /a\ 5 d p if and only if for all n E N cp

Q

5 dp

on P,

@P, .

20.10. The following formulas contain many of the fundamental properties of projective/injective associates and finitelcofinite hulls; they create a type of “calculus” which will be helpful when dealing with accessibility in the next section.

20. Projective and Injective Tensor Norms

262

PROPOSITION: Let a be a tensor norm on NORM. -c

(1) (Z)\= a\

and (S)/=

3.

c

(2) (=)\ = a\ , but, in general, (Z)/# (3) (a/)’ = (a’)\ and (a\)‘ = (a’)/ . (4) ( a / ) *= /(a*)and (a\)* = \(a*) .

a’/ .

PROOF: (1) By 20.8. it follows that

+

Since p := a\ is right-injective by proposition 20.3.(1), the test 20.9. gives

The same argument applies to the right-projective associate. (3) and (4) also follow from the preceding test: a’and (a/)‘ are finitely generated and clearly, N @ u ~ -c (N’ @a = (N’ mu/q)‘= N @(a/)#e”,

e)’

hence, (a’)\= (a/)’, which implies all the formulas in (3) and (4). c

(2) Note first that a\ is right-injective by proposition 20.3.(1). Since (by (3) and 20.6., corollary 1)

E’&# J ! ~ ( B = F ~E’@(a\y ) ti(BF’) and (by the duality theorem 15.5.)

F d+E @z~,(BF’) @(3\ [email protected] F a\

E @ - !,(BF#)

(E‘ @at 1

J!~(BF~))’

( E ’ @ ( a \ ) f t i ( B ~ f ,) ) ’

a\

c

one obtains (=)\ = a\. The related formula for the right-projective associate fails to hold: In fact, f = T on E @ .tl(I’) and hence, c-

(%)/ = ?f/ = s # f = ( T / ) since u is right-projective. The first statement of the proposition implies the

COROLLARY: If a is a finitely generated tensor norm on NORM, then all of its ciates a / ,a\, \a, /a,/a\ and \a/ are finitely generated as well.

asso-

80. Projective and Injective Tensor Norms

263

20.11. If a is a finitely generated tensor norm and (%,A) its associated maximal operator ideal, what do the ideals associated with a/,a\,/a,\a,... look like? Finding an answer to this question will be very fruitful for the study of both tensor norms and ideals. Start with the right-injective associate a\ of a,denote by (93,B)the ideal associated with a\ and fix T E Z ( E , F). Since (&,(Bp))'is an L l ( p ) (by Kakutani's representation theorem, Appendix A7.) and hence an X;,l-space, it follows that

If /&- and P I ~ Tare the functionals associated with T and I o T (where I : F r-, t O o ( B p ) ) , then

commutes. Since id @ I' is a metric surjection, pT is continuous if and only if PI,,T is continuous (with the same norms); the representation theorem for maximal operator ideals yields that T E 93 if and only if I o T E %; moreover, B ( T ) = A ( I o T). Hence, the definition 9.7. of the injective hull % j n j of an operator ideal gives the first part of the

THEOREM: Let a

-

( % , A ) be associated. ( 1 ) The injective hull (gi"j,A'"') of (%, A) is maximal and associated with a\. In particular, a is right-injective if and only if (a,A) is an injective operator ideal. ( 2 ) The surjective hull (a8"', AbUr)of (%, A) is macimal and associated with /a.In particular, a is left-injective if and only if (a,A) is a surjective operator ideal. 1

PROOF: To see (2) take the canonical quotient map Q :~ ~ ( B E ) - and E T E Z ( E , F); look at the diagram PToO

to obtain the result, as before. I 3

To quickly draw one consequence from these relationships:

K

264

20.

Projective and Injective Tensor Norms

COROLLARY: If (9,A) is a maximal normed operator ideal, then

PROOF: This is just (at)\ = (/a)'.0

20.12. The projective associates give rise to factorization theorems for the corresponding operator ideals. Denote by (a/,A/) the ideal associated with a/ and take T E C ( E ,F ) . Then

If a-I. is continuous, then the Hahn-Banach theorem gives an extension p with the same norm; it follows that K F o T = I' o L , and, by the representation theorem for maximal operator ideals, A(&) = A / ( T ) .So part of the following result has already been shown.

-

-

-

(a,A) and denote by (a/,A/) a/ and (\a,\A) \a the maximal operator ideals associated with the projective associates; take T E f!(E,F ) :

PROPOSITION: Let a

(1) T E !Z/(E,F ) i f and only if there exist a measure p , operators R E % and S E 2 such that

E

T

F

F"

I n this case, A/(T) = minA(R)IIS)), where the minimum is taken over all such factorizations and i s attained for a metric surjection S : L1(p) -* F" for a B o d - R a d o n measure p. (2) T E \Q(E, F ) if and only if there exist a compact set K , operators R E C and S E !2l such that

20. Projective and Injective Tensor Norms

E

T

F

i=-+

265

F"

In this case, \A(T) = minIIRIIA(S), where the minimvm is taken over all such factorizations and is attained for a metric injection R.

The space C ( K ) in (2) can be replaced by L , ( p ) .

PROOF:For T E Q l / ( E F), , II~'IIA(LIp)= A / V )

,

and it remains only to use Kakutani's representation theorem Appendix A7., which gives that &,(Bp)' = L1 ( p ) for some Borel-Radon measure. 1

For the converse, note first that L1(p) = L1(po)for some Borel-Radon measure po (by Kakutani's representation theorem), therefore, Ll(p)' L,(po). Now 20.8. and 20.10. imply that E @(?I)/ Li = E L i and hence A(R) = A/(R) by the representation theorem for maximal operator ideals. To see (2) apply the same type of argumentation using the diagram

A

teT

E

F'

In Ex 20.7. the associated operator ideal for the projective associate \a/ is described. Ex 20.6. discusses extension and lifting properties related to 2l/ and \%.

-

-

20.13. The characterization of T E QlZL'"J a\ and T E /a in terms of idG @ T given in 17.15. and 17.16. can be modified to be in terms of a (instead of a\ and /a). We omit stating and proving this now: In 29.10. it will be a very easy consequence of the calculus of traced tensor norms. 20.14. The simple relationship between the injective associates of a and surjective and injective operator ideals provides nice information about the associates of ap,p:

PROPOSITION: Let 1 5 p 5 00. Then (1) gp\ = g;, = 4, . (2) \gi = gpt and d i / = dpt . (3) \(gp\) = gp and W P ) / = dP * (4) T\ = g& = w k = wi = d&, and

E/

= d , = w1 .

20. Projective and Injective Tensor Norma

266

-

Jy

-

PROOF: Corollary 5 in 18.7. states that = ppholds isometrically whenever 1 5 p < 00; since gp Jp and g;# '&, theorem 20.11. (31'"j a\) implies that gp\ = g;# . For p = 00 it is known from 12.9. that

-

gm=woo=E

onE@P,

and therefore 20.9. implies that goo\ = E . Since g2 = a2,1 is right-injective (20.4.), the last statement follows from the first (note that 92 = g2f was already shown in 18.7.). All the other statements come from (1) as well, using transposition and dualization (see 20.10.). 0 20.15. The behaviour of the aP,* is particularly interesting when one of the involved indices is 2: in such a case, the discussion is about tensor norms/operator ideals con-

nected with Hilbert spaces. The following list exhibits some of the most important examples:

The symbol

-

indicates that the norms are equivalent.

PROOF:Only the statements for 9 2 and w2 will be proven - the others follow by transposition, dualization, and from g2 = g; . Since g2 is left-projective and right-injective, it follows that g2 = \g2 = g2\. Moreover, w2 is injective, hence w2 = /w2\ = w2\ = 1.12. The remaining statements are a consequence of the little Grothendieck theorem: In Ex 20.10. the reader is asked to show (with the factorizations coming from the characterization of \%, %/ and \%/) that

S 92 5 KLG\% \w2/ 5 w; 5 K:G(\w2/) 9 2 / 5 w; 5 KLGg2/ \w2

Moreover, this implies that

= (\W2)t 5 d2 5 KLGW2/ = 192 2 w2 KLGW2 = KLGw;* 2 ( S Z / ) * = / & )

w2/

20. Projective and Injective Tensor Norms

Since \g2/ = g2/ and

/g2\

267

= 192, all the relations have been demonstrated. 0

n Tp,q(Z;

F ) := inf{ (I(ak,L) : qj

-t

$lllqt(zL)lpt(Yk)

Z

=

a k , L z L @ gk}

k ,L=

1

(for l/p+ l/q 3 1); as in 20.4., it is easily seen that -yp,c is projective (Ex 20.2.) and it turns out that b;,q/

= Y9,P

- in particular, wz = y2,2. Everything has been prepared for giving a proof of this result but we prefer to postpone it to section 28. 20.17. Certainly the most exciting relationship between tensor norms is GROTHENDIECK'S INEQUALITY (tensor form):

PROOF: Since w2 is injective, it follows from the test 20.9. that the first relation is nothing but Grothendieck's inequality 14.4.. The inequality with the projective associate \e/ follows by duality. 0

-

-

-

Since r 3 (the integral operators), w2 2 2 (the operators factoring through a Hilbert space, see 18.6.),ir\ J'"J 91 (the absolutely summing operators, see 11.3. or 18.7.) and /r\ = /(r\) = (IT)\, this can be reformulated as

A

GROTHENDIECK 'S INEQUALITY (operator form): 9;ur

= ( j s u + ) i n j = gZ

L2(T) 5 P;"'(T) = (IaU')inj(T) 5 K G L ~ ( T. ) Moreover, Grothendieck's constant KG is the best constant i n this inequality. Or - as Grothendieck formulated it orginally in the Usumk:

COROLLARY: If H i s a Hilberl space, then P f ' ' ( i d ~ ) 5 K G . Since for T E Z ( t , , l , ) the relation

268

20. Projective and Injective Tensor Norma

holds, it follows what was called "Grothendieck's theorem" in 17.14.:

<

In particular, the best constant c in this inequality is PIu'(idf,) I KG; for an operator T E 2 2 factorization shows that Pi"'(T) 5 cL2(T) and hence KG 5 c. It follows that c = Pd,u'(idf,)= KG.Moreover, the maximality of the operator ideal ?J3:"' (and an obvious complementation argument) implies that

KG = Pfu'(idt,)= sup Piu'(idp) n

and Pfu'(id~) = KG for all infinite dimensional Hilbert spaces H (see also Ex 20.12.).

PROPOSITION: The number KG(n) := Pf"'(idp) is the smallest constant c 2 1 such that Grothendieck's inequality an matrix form is satisfied for m x m matn'z (m E N arbitrary) such that

5: Let

(ai,j)yj=i be a

Then

PROOF: For the moment, call the smallest constant c in the matrix inequality MG(n). As in the proof of 14.8., it follows that H 5 MG(n)dm on el 8 6 ,which means that PI(T:l l + 5)5 MG(n)IITII. In particular, KG(n) = P y ( i d q ) _< MG(n) Conversely, define for n E M and z E M 8 N = L(M', N) (finite dimensional Banach spaces)

(this is not a tensor norm!). Since

20. Projective and Injective Tensor Norms

269

holds isometrically, it follows that r 5 Ko(n)wp) on em,@E for all m E N.Moreover, if 2; and yj are in the unit ball of 4, then m

m

have norm

5 1 and factor z, := C G = l ( x i l pj)ei @I e j

E

@

If (ai,,) is an m x m matrix satisfying the assumption, then

has norm _< 1 in

em,, which means that

‘p

defined by

(em, @Ir em,)’, and hence

In the real case Krivine [154] proved:

KE(2) = K z ( 3 ) < 1.517

Recall that r / 2 5 K$ from 14.7.. Using Krivine’s method and its complex modification by Haagerup (see 14.7.), Konig [148] estimated some “finite dimensional” Grothendieck constants in the complex case

1.1526 1.2108 1.2413 1.2600 1.2984 1.3181 1.3300 1.3381

5 K,$(2) 5 Kg(3) 5 K (4) 5 K %(5) 5 KG(lO) 8 5 Ks(20) 5 Ks(50) 5 Ks

5 5 5

1.2157 1.2744 1.3048 5 1.3236 5 1.3628 5 1.3834 5 1.3962 < 1.4049

and conjectured slightly better upper bounds. For other descriptions of K G ( ~see ) Ex 20.16., Ex 28.6., Ex 28.13. and Ex 28.14..

2 70

20. Projective and Injective Tensor Norms

20.18. There are more Banach spaces E which satisfy Grothendieck’s theorem

(they are sometimes called G.T. spaces). In fact, in 23.10. it will be shown that all C;-spacea do - this is rather simple. For a long time it was not known whether or not there are any other spaces satisfying Grothendieck’s theorem. In 1976,Kisliakov [142] and Pisier [219]independently constructed spaces which do: for example, quotients L1(p)/R where R is a reflexive subspace. Bourgain [16]showed that L I / H ’ (where H’ is the Hardy space) satisfies Grothendieck’s theorem. We refer the reader to Pisier’s book [225],ch.6, for more details on this topic and give only the following characterization, which appeared first in [225]in a slightly different form (see Ex 20.14.), but is in some sense also contained in Lindenstrauss-Pelczyriski’s famous 1968 analysis of Grothendieck’s inequality; see also Ex 20.13.and Ex 20.14..

PROPOSITION: For every Banach space E the following two statements are equivalent: (a) q E ’ , l 2 ) = Pl(E’,t 2 ) (b) C ( E ,41) = 9 2 ( E ,ti) . The constants in (a) and (b) are the same. That (b) holds for E = co can be shown as in 17.14.using also section 23 on Xg-spaces.

?r

5 K ~ w 2on

c, @ c,; see

PROOF: Observe first (by the representation theorem 17.5.for maximal ideals and the &-local technique lemma 13.5.) that (a) means that there is a best constant c1 < 00 such that P1(T) 5 c1llTll for all T E C(E’,G), and (b) that there is a best c2 < 00 such that D2(T) 5 c211Tll for all T E C ( E , e ) . Assume (a) and take T E C ( E , e ) . Then, by 17.9. and since 9: = 2 2 ,

-

Hence the factorization of L2 from 18.1.,T = T o QferT, the fact that w i 2; and (a) give

-

ppa’

gg

=

since Jtr(Ro S)l 5 A*(R)A(S) for operators between finite dimensional spaces. Conversely, take T € C(E’,G) and, in order to apply Kwapieri’s characterization 11.8., S E 2(l&,,El). Then 2* = 3 implies that

20. Projective and Injective Tensor Norms

271

In Ex 20.14. it will be seen that P2 5 D2 5 K L G P on ~ 92(E,t,) = q2(E,tl). Therefore, one can replace 9 2 by ' 3 2 in this proposition - at the price of an additional constant in one implication.

COROLLARY: Let E be a Banach space and c 2 1. Then the following are equivalent: (a) ?r 5 c d , on E ' @ & . (b) 7 r 5 c w 2 o n E @ c c , . 20.19. There is a quite useful result due to Saphar [242] which is in the same spirit as

the previous one:

PROPOSITION: Let E be a Banach space, c >_ 0 and 1 5 p three statements are equivalent: (a) For every M E FIN and T E C ( E ,M )

5 00.

Then the following

P I ( T ) 5 CIp#(T). (b) For every M E FIN and T E C ( E ,M )

PROOF: Observe first, that (by 15.7. and the embedding theorem 17.6.)

since g;/ is right-injective and hence right-accessible by 20.3.. This implies that (c) is equivalent to 5 CE on ti' @ E and therefore, by 20.9. (recall the remark at the end of 20.9. concerning constants) and 20.14.

\$ I c(\c) = cg,

on M @ E.

272

20. Projective and Injective Tensor Norma

By the representation theorem for maximal ideals, (a) is equivalent to $1

on M 8 E.

Icgo3

Since goo is left-projective, it follows again from 20.9. that \g;, 5 cgw on M @ E. Hence, the equivalence of (a) and (c) is shown. Since gp = \gi, by 20.14., the same arguments give the equivalence with (b). 0 In 25.6. a much more general statement of this type will be given. For p = 2 the proposition shows that P l ( T ) Ic P z ( T ) for every T E E(E, F) whenever E satisfies (c) and F is an arbitrary Banach space. The little Grothendieck theorem 11.11. shows the following:

COROLLARY: E v e y absolutely 2-summing operator T : 12 t r a y Banach space) is absolutely l-summing and

+F

(where F is an arbi-

P l ( T ) 5 KLGPZ(T) * In 11.9. it was shown that Pz(idl;) = fi.It follows that n-'12P1(idL;)

5 KLG

for all n E N.But lim n-llzPl(idm;) =

n-w

6 2

lim n-'12P1(idq) = -

fi

n-+w

by 11.10., hence the best constants in the little Grothendieck theorem are actually those stated in 11.11.. 20.20. One may ask whether or not there exists a tensor norm Q which is at the same time injective and projective! Existence would imply, by 20.17.,

g;

5 wz'

-

\6/

5 5 /.\ Q

w2

I

hence 2 2 C f ) ? ~which is impossible since id12 is not power compact. More generally, and at a much deeper level:

PROPOSITION: There is no tensor norm which is right-injective and right-projective. PROOF:This would imply that

20. Projective and Injective Tensor Norms

273

and hence that c 21.But this is not true by a result of Gordon and Lewis [84] which solved an old problem from Grothendieck's R&umC ([93],p.72, question 2). 0

Exercises: Ex 20.1. Given a normed space E and a tensor norm a such that for all G E NORM and F C G the space E aU F is an isomorphic subspace of E @u G, then there is a constant A 3 1 such that a ( z ;E , G)5 a(%;E , F ) 5 A a ( z ; El G)

for all F c G and z E E @ F. Ex 20.2. Use the method of proof from 20.4. to show that the tensor norms 5,,,from are right-injective. Ex 12.8. are projective and the tensor norms Ex 20.3. Show that a = a\ = /a = /a\ on C ( K )@ C ( L ) for all compact K and L and all tensor norms a. Ex 20.4. (a) Show that the ideal J of integral operators is neither injective nor surjective. (b) Show that = (Jpdua')surholds isometrically. Ex 20.5. (a) (a/)\3 a if a is right-injective. In particular, ((T\)/)\ = T\. (b) Show that (\a)\ 5 \(a\) for every tensor norm a. Hint: Use the description in terms of L ~ ( B Eand ) l,(Bp). Ex 20.6. Assume (%,A) a. (a) a is left-projective if and only if for all ElG E BAN, each subspace F C E and each T E %(F,G) there exists an ezlension f' E %(ElG")of n c o T with A(T) = A@). (b) a is right-projective if and only if for all Banach spaces El metric surjections Q : F -+ G and all T E B(E, G) there exists a lifting T E g ( E , F") of KG o T (i.e. Q" o T = KG o T) such that A(?') = A(T). Hint: It is enough to show that a is right-projective on FIN. Ex 20.7. For a given tensor norm a denote by (\%/,\A/) the maximal normed operator ideal associated with \a/.Show that T E C ( E , F) is in \%/ if and only if there are a compact K ,a strictly localizable measure /A on some set 52, operators R,S E C and U E % such that

!J3ra'

-

T

E - F .

FN

20. Projective and Injective Tensor Norms

274

Moreover, \A/(T) = min I1RIIA(V)llSll,the minimum being taken over all such factorizations.

Ex 20.8. Let E be a Banach space and 1 5 p 5

The following three statements

00.

are equivalent: (a) idE E inJ.

xrr

(b) E is isomorphic to a subspace of a quotient of some L p ( p ) . (c) E is isomorphic to a quotient of a subspace of some L p ( p ) .Hint: cZLSur

= %'"j 'w.

Ly

Ex 20.9. The class of all Banach spaces E such that (idE) = 1 is stable under formation of isometric subspaces and quotients. Hint: (/wp\)\ = /wp\ ... Ex 20.10. Use the little Grothendieck theorem to show that \.C2 = 32,\Cz/ = ,pa' o ?2(= 9 2 ) and 321 = 9 2 with equivalent norms. More precisely,

E x 20.11. Show that (\w2)\ = \w2 and /

( w 4 = w2/. Hint: Consider the associate ideals, 20.11. and 20.12.. E x 20.12. Show that Iltrt2 :12 @ ( /&q ~Kll = K G .

-

Ex 20.13. (a) If a is left-accessible, E and N are normed spaces, N finite dimensional, then

(E

N)" = E"

@a

N

holds isometrically. (b) Use (a) and the fact that g& is accessible (to be proven in the next section) to verify that the bidual E" of a Banach space satisfies Grothendieck's theorem if and only if E does.

E x 20.14. (a) Show that P2 5 D2 5 KLGP2 on I)z(E,tl) = % ( E , e l ) for every Banach space E. (b) Formulate (a) in terms of tensor norms on E @ c,,. Ex 20.15. (a) Show that fi 5 P l ( i d p ) 5 K ~ f iHint: . 11.9.. (b) Deduce from this that llid@ id : 6 @c e;l

E x 20.16. Let Sn-lbe the unit sphere of S,-l x

If a

N

-

6 @ r qll 5 K

G ~ . and sn the scalar product restricted to

(%,A), then

Hint: Use the isometries z-(z 1): and z-(2 C(Sn-l)). case K G ( ~=) ~ ( 8 , ;C(Sn-i),

I .) of t?j into C(Sn-l). Note the special

21. Accessible Tensor Norms and Operator Ideals

275

21. Accessible Tensor Norms and Operator Ideals

The natural mapping

is a metric injection if E and F have the metric approximation property (see section 15). In this section other conditions are investigated under which I is an isomorphic injection - or at least injective. Some criteria will be presented in order to show that a given tensor norm is totally accessible. With the help of certain “accessibility” notions for operator ideals it will turn out that all a& are totally accessible. In the second part of this section some more general approximation properties will be analyzed.

21.1. The duality theorem 15.5. for a tensor norm a,which is nothing else but the embedding theorem 17.6. for the associated maximal operator ideal a, states that

Since Z 5 a 5 d always holds for the finite and cofinite hull a,the problem under investigation is to find conditions which ensure that Q 5 ?F on E C3 F for a given finitely generated tensor norm and Banach spaces E and F. Recall from 15.6. that a tensor norm Q is called right-accessible if M, F) = %(-;MIF) for all M E FIN and F E NORM (or only E BAN by 13.4.), left-accessible if its transpose a‘ is rightaccessible, accessible if it is both right- and left-accessible and totally accessible if c a =+ a. With the notions of injectivity and projectivity of tensor norms it is possible to verify accessibility in many cases. The following result is basic to what will be done; it is due to Saphar [241].

z(.;

PROPOSITION: Let a be a tensor norm. (1) The right-projective associate a/ and the right-injective associate a\ are right-

accessible. ( 2 ) If a is left-accessible, then a\ is tofally accessible. ( 3 ) The tensor norms (\a)\,/(a/), and the injective associate /a\ are totally accessible. In particular, injective tensor norms are totally accessible.

PROOF:The first statement was already proven in 20.3.(4). For the proof of (2) take normed spaces E and F. Since & , ( B p ) has the metric approximation property and a

276

21. Accessible Tensor Norms and Operator Ideals

is left-accessible, the approximation lemmashows that 'Z = d on E @.te,(Bp). Now the formulas from 20.10. imply that for z E E 8 F 4

a\(%;E , F ) = + a \(z; E , F ) = Z ( ZE,&,(Bp)) ; = c

= a (%; E , & ( B p ) ) =

t

\(Z;

t

E , F ) = a\(%;E , F ) .

(3) is a simple consequence of (1) and (2). 0

There is no statement like (2) for the projective associate a/:Just take the projective norm ir = T / as an example. But even if Q is totally accessible, a/ need not be; this follows from the following two facts which will be shown in a moment: w i is totally accessible (see 21.5.) but w i / is not (see 21.6.).

COROLLARY: The tensor n o m s g;,

w2

and 92 are totally accessible.

PROOF: Since gpJis left-projective and g; = gpJ\ (see 20.14.), it follows that g;

= gpJ\ = (\99')\

is totally accessible. Moreover, g i = gp and wp is injective. 0

21.2. It turns out that it is sometimes easier to check the accessibility of a given finitely generated tensor norm through its associated maximal operator ideal. A quasi-Banach operator ideal ( 0 ,A) is called right-accessible, if for all (M, F) E FIN x BAN, operators T E C ( M , F) and E > 0 there are N E FIN(F) and S E C ( M ,N ) such that

M-F

T

\ J'5

and A(S) 5 (1

+ E ) A ( T ).

N

It is called left-accessible, if for all (E, N ) E BAN x FIN, operators T E C ( E ,N ) and E > 0 there are L E COFIN(E) and S E C ( E / L , N ) such that T

E-N 0;

1 /$

and A(S)

5 (1 + &)A(T).

A left- and right-accessible ideal is called accessible. Moreover, ( 0 ,A) is totally accessible, if for every finite rank operator T E S ( E ,F) between Banach spaces and E > 0 there are ( L ,N ) E COFIN(E)xFIN(F) and S E 2 ( E / L ,N ) such that

21. Accessible Tensor Norms and Operator Ideals

E-F

277

T

-

and A(S)

5 (1 + E ) A ( T .)

JI:

EIL

N

S

Obviously, every injective quasi-Banach ideal is right-accessible and every surjective ideal is left-accessible. The canonical factorization T

E-F

I -

im T

ElkerT

shows that a surjective and injective quasi-Banach ideal is even totally accessible. For example, the injective and surjective ideal 2, of type p operators (see Ex 9.15.) is totally accessible and the injective ideal C9 of cotype q operators is right-accessible. We do not know whether or not C9 is left-accessible (being injective it would even be totally accessible - by 21.1.(2) and the following proposition, or by a direct proof). In Ex 21.1. it will be checked that the factorizations in the definitions of accessibility also hold when only the metric approximation property is assumed. Reisner [230] called accessible operator ideals “semi-tensorial”.

21.3. For a maximal Banach operator ideal associated with a (finitely generated) tensor norm a the notions of accessibility coincide; this will follow from the embedding theorem E‘Q-a F ~(E,F).

c ‘ .

PROPOSITION: A finitely generated tensor norm is right-accessible (resp. left-accessible, accessible, totally accessible) if and only if its associated maximal Banach ideal is.

PROOF:It will be shown that a is totally accessible if and only if (%,A)has this property; all other proofs are similar. Assume that a is totally accessible and let T E 5 ( E , F ) . Then

which implies that there are (M, N) E FIN(E’) x FIN(F) and u E M 8 N with a(u;M , N ) 5 (1

+ &)A(T)

and

+

1%’€3 I s ( u ) = ZT

.

Hence, T,, E C ( E / M o ,N) satisfies A(T,,) 5: (1 E)A(T) and 1; o T,,o Q&,,= T . Conversely, let (a,A) be totally accessible. By the embedding lemma 13.3. it suffices to check that a(*; E‘, F) 5 Z(*; E‘, F )

21. Accessible Tensor Norms and Operator Ideals

278

for all Banach spaces E and F. For z E E' 8 F and E > 0 there are L E COFIN(E) and N E FIN(F) as well a8 an operator S E C ( E / L ,N) such that

-

A(S) 5 (1 + E)A(T,) and 1; o S o Qf = T, . For 2s E Lo @a

1 N = % ( E / L ,N) it follows that

N = (E/L)'

L (1 + E)%(z; E', F )

cr(zs; Lo, N ) = A(S)

and If: 8 IE(zs) = z

,

which completes the proof. 0

-

Since 21dUa' a* and a* a* (see 17.8. and 17.9.) for. the dual and adjoint operator ideals, the proposition 15.6. gives the N

COROLLARY: Let ( Q , A ) be a maximal normed operator ideal. (1) (adu0', Ad"'') is right-accessible (resp. left-accessible, totally accessible) if and only if (a,A) is left-accessible (resp. right-accessible, totally accessible). ( 2 ) (%*,A*)is right-accessible (resp. lejl-accessible) if and only if (%,A) is leftaccessible (resp. right-accessible).

21.4. The following result will be quite useful:

Let (a,A) and (%, B) be quasi-Banach ideals, (a,A) injective and PROPOSITION: left-accessible, (%, B) totally accessible. Then (% o a,B o A) is totally accessible.

PROOF: Take T E $(El F) and E > 0. Then there are R E a ( E ,G) and S E %(G,F ) such that T = S o R and B(S)A(R) _ < e)(B ( 1 o A)(T). +Since is injective, one can choose this factorization such that R(E) = G, hence S(G) c T ( E ) and S is also a finite rank operator. Since % is totally accessible and left-accessible, T factors as follows: T

E

F

F

Consequently,

B o A(So 0 R,)

I B(S,)A(R,) I( 1 + c)B(S)(l+ E)A(R)I (1 + E ) ~ 0BA(T) ,

which proves the result. 17

21. Accessible Tensor Norms and Opemtor Ideals

279

For more results on the accessibility of composition ideals see Ex 21.2. and 28.6.. 21.5. Everything is prepared for giving an easy proof of the following important result, apparently due to Gilbert-Leih [Sl]:

THEOREM: Let p, q E [l,001 satisfy l / p + l / q 2 1. (1) ap,qand (.CPlq,Lp,g) are accessible. ( 2 ) a;,q and ( 9 p 8 , q t , DP,,(t)are totally accessible. PROOF:Since the tensor norms and operator ideals in question are associated and a is accessible if a* is (by 15.6.), it is enough to show that 9p,,q~ is totally accessible. Kwapieli's factorization theorem 19.3. states that

-

-

9pr,ql

=

!py!ppl. 0

Since '&, g; and pdu'" g;' are totally accessible (by corollary 21.1. and 21.3.) and the operator ideal &,I is injective, the preceding proposition shows that is totally accessible. 0 21.6. A structurally interesting application of the total accessibility of certain tensor norms is the

PROPOSITION: Let (a,A) be a maximal normed operator ideal with associated tensor norm a. If a* is totally accessible, then every Banach space E with idE E 24 has the bounded approzimation property with constant A ( i d E ) . PROOF: In order to use the characterization theorem 16.2. (concerning ?r 5 X f ) for * E'@ E by 17.15.. the bounded approximation property, observe that ?r 5 A ( i d ~ ) aon Consequently, ?r

5 A ( i d ~ ) a=* A ( i d E ) z 5 A ( i d E ) f

on E' 8 E

,

which gives the result by 16.2.. 0

COROLLARY 1: For 1 5 p 5 00 and X 2 1 every .C:,x-space E has the bounded approzimation property with constant A.

-

PROOF: Since .Cp wp is accessible and E 5 wp 5 X E on G@ E for every Banach space G by the local technique lemma (see 13.5.), it follows from the characterization 17.16. that L p ( i d E ) 5 A. The fact that wp' is totally accessible implies the result. 0 This was a "positive" application of the proposition. But the proposition also has a negative flavour: if there is an E E space()#) (this means, by definition i d E E Q)

280

21.

Accessible Tensor Norms and Operator Ideals

without the bounded approximation property, then a* is not totally accessible! An example: Each subspace E c 4, is contained in space(2y); since there are subspaces of tpwithout the approximation property (if p # 2, see 5.2.), the adjoint of wp\ il? cannot be totally accessible:

-

COROLLARY 2: For 1 5 p 5 00 and p

# 2 the tensor norm wL/

is not totally accessible.

21.7. Let a be a finitely generated tensor norm. A Banach space E is said to have the a-approximation property if for all Banach spaces F the natural mapping

is injective. E has the bounded a-approximation property with constant X 2 1 if for all Banach spaces F the natural mapping

) ~A ~= 1 one speaks of the metric a-approximation satisfies “ ( 2 ; F, E) 5 X ~ ~ I (...;z if property. For a = T the usual approximation properties are obtained. For a = gp this notation was introduced by Saphar [243] under the name “approximation property of order p” (and generalized to ap,pby Diaz, L6pez Molina and Rivera Ortun [52]).

PROPOSITION: (1) A Banach space with the approximation property has .the a-approximation property for all tensor norms a. (2) E has the bounded a-approximation property with constant X i f and only if Q 5 A% on F €3E for all Banach spaces F. I n particular, i f a is totally accessible, then each Banach space has the metric a-approximation property. (3) If a is accessible, then E has the bounded a-approximation property i f it has the bounded approximation property (same constant). (4) It is enough t o check the definitions for the (bounded) a-approximation property for dual spaces F. (5) If El’ has the (bounded/metric) a-approximation property, then E has it. PROOF: (1) is a reformulation of 17.20, and (2) follows from the duality theorem 15.5.. The approximation lemma and (2) imply (3) and the rest is a consequence of the embedding lemma. 0 Thus, every Banach space has the metric g2-approximation property since 92 = g; is totally accessible. It will be shown in 31.7. that Banach spaces E of cotype 2 satisfy

F

E = F &ig,E

for all p E [2,m]

,

21. Accessible Tensor Norms and Operator Ideals

281

hence they have the gp-approximation property for these p. There are spaces of cotype 2 without the approximation property: Szankowski’ssubspaces oft, (where 1 5 q < 2, see 8.6. and 5.2.(1)). In 1982 Reinov [228] showed for 1 5 p < 00 and p # 2 that E -, El& E is not injective. In particular, (1) There is a Banach E such that E1hgv E does not have the gp-approximation property and the tensor norm gp is not totally accessible. (2) There is a Banach space with the gp-approximation property but failing the bounded gp-approximation property. (3) There is a Banach space E with the approximation property which does not have the bounded gp-approximation property for any p as above.

21.8. That the (bounded) a-approximation property is really a property of “approximation’’ of certain operators can be seen as follows. If !2l is the maximal Banach operator ideal associated with a,the adjoint operator ideal satisfies

?Z*(E,F”) = ( E @ a t F’)’ = ( E hat F’)‘ isometrically. For the weak topologies u := u(%*(E,F”), E @ F’) and

8 := u(?Z*(E,F “ ) , E hatF’)

lemma 16.2. reads as foIIows:

PROPOSITION: Let E and F be Banach spaces, a a tensor norm and A 2 1. Then on F ‘ @ E is equivalent to each of the following two statements about the unit

a 5 A= balls:

(4

B n * ( E , F J Jc ) ABEI&.F*

.

(b) B ~ * ( E , F Jc J) . In particular, E has the bounded a-approzimation property (with constant A) if and only if (a) or (b) is satisfied f o r all Banach spaces F. The latter follows from proposition 21.7.(4).

21.9. For the bounded gp-approximation property (1 < p < 00) it is enough to check the definition for reflexive spaces. This will now be shown.

LEMMA: Let 1 5 p 5 00 and E , F arbitrary Banach spaces. For every z E F6bg,E there are a separable refletive Banach space GI an injective operator S E 2(G,F ) and 20 E G 6 g v Ewith S @I idE(W) = z .

PROOF: For p = 00 read c, instead of looin this proof. By 12.7. the element z E FGgrE has a representation m

282

21. Accessible Tensor Norms and Operator Ideals

with (An) E t,, a zero sequence (2,) in F and wpr(yn;E) 5 1. Define K to be the absolutely convex, closed hull of {zn}.Then [ K ] is a Banach space such that the embedding [ K ] ~1F is compact, and hence factors injectively through a reflexive space G1 by the Davis-Figiel-Johnson-Pelczyliski factorization theorem (see 9.6.); take G to be the closed linear span of {z,} in GI. Since {z,} is bounded in [ K ] ,it is bounded in G, and w := C,"==l Anzn@ yn E G&, E gives the result. 0

In Ex 21.5. there is a metric version of this lemma.

PROPOSITION 1: For 1 5 p 5 00 a Banach space E has the g,-approzimation property if (and only i f ) IF : F 6 g ,E - F ~ L E is injective for all separable, reflexive spaces F. Recall that the gl-approximation property is the usual approximation property.

PROOF: Take F to be an arbitrary Banach space, z E FGgVE and choose G, S and E G&,, E as in the lemma. Then the diagram

w

F & ~E, S@g,idE

IF

F& E

-T

S@,.idE

IQ

Ggg, E

G& E

implies the result since IG and S & i d ~ are injective. 0 For the bounded gp-approximation property the analogous result holds at least for l
PROPOSITION 2: For 1 < p < 00 a Banach space E has the bounded gp-approzimation if (and only i f ) gp 5 A& holds on F @ E for all reflexive Banach spaces F. property PROOF: Take an arbitrary Banach space F and z E F @ffv E. One has to show that for every cp E (FQffVE)' = v p r ( E ,F') of Ppr-norm one the inequality I(cp0, .)I

IG

( z ;F, E )

holds. The Grothendieck-Pietsch factorization theorem 11.3. shows that the operator T E V p , ( EF') , associated with cp factors through operators R E Vp1(E,G) and S E L!(G, F'), where G is a subspace of some Lpt, hence reflexive, llSll = 1 and PP1(T) = PPr(R).It follows that there is a yo E (G' @gv E)' of norm one with (9, z ) = ( P o ,

[(s' K F ) @ i d E ]

(2))

*

21. Accessible Tensor Norma and Operator Ideala

283

Therefore, I(Pp,Z)l

s g p ( [ ( s ’ O K F ) ~ i d E ] ( % ) ; G ’ , EA) & s (’*’;G’,E)s

5 xlls’ 0 KFll

llidE11&(Z; F, E ) 5 A&(%; F, E )

by the metric mapping property. The proof works neither for p = 1 nor for p = 00; it is not clear to us whether the statement holds for p = 1, which is the usual bounded approximation property (probably not, by the foliowing.corollary).

21.10. For 1 5 p

< 00 an operator T E C ( E ,F) is called Pietsch p-integral if it factors T

E-F

Ts I

C(K)

L, (PI

, on K. With for some compact K and Borel-Radon measure u

the class ‘$333, of all Pietsch pintegral operators becomes a Banach operator ideal. Clearly, ‘$333, = q33 and ‘$3& = 332 = ‘$32 hold isometrically (see 18.7.). Moreover,

F’)

A 3,(E, F‘)

by the factorization theorem for p-integral operators. As for p = 1 (see Appendix D9.),the fact that there are (for p # 2 ) Banach spaces without the gp-approximation property was used by Reinov [228] in order to show that there are p-integral operators which are not Pietsch p-integral. It will be shown in 33.6. that

F’c!&, E = ‘WP( F, E ) holds isometrically if F‘ has the Radon-Nikodym property and F‘638,E injective. This will be used to prove the

F’& E is

COROLLARY: Let 1 < p < 00 and E a Banach space which i s A-complemented in its bidual. If E has ihe gp-approximation property, then it has the bounded gp-approxirnation property with constant A. Note that for p = 1 only the weaker statement 16.4. is known.

284

21. Accessible Tensor Norms and Opemtor Ideats

PROOF: By proposition 2 it is enough to show that gp 5 A& on F' 8 E for reflexive -+ E the associated operator and let P : E" E be a projection. Since, as a reflexive space, F has the Radon-Nikodym property, it follows that

spaces F. Take z E F' 8 E , denote by T, : F

by the regularity of Jp and the embedding theorem. 0

21.11. It is good to be able to verify the a-approxim'ation property with the help of the coincidence of operator ideals: PROPOSITION: Let E be a Banach space and a and /3 finitely generated, accessible tensor norms with associated operator ideals 24 and % such that %(ElF ) = %(ElF ) for all reflexive Banach spaces F. (1) Then for all even duals E(2k)of E and all Banach spaces F 24(E(2k),F ) = % ( E ( 2 k )F, )

.

(2) If p' is totally accessible,, then the space E and all its even dual spaces E ( 2 k )have the a' -approximation property.

( 3 ) If a is totally accessible, then all the odd dvals of E have the p'-appro;cimation property.

PROOF: Since (EGa,F)' = % ( E ,F') = % ( E ,F') = ( E g p F ) ' , it follows that a' and p' are equivalent norms on E 8F for all reflexive spaces F. Taking the Johnson space C2 for F, lemma 17.16.shows that there are c1 2 0 and cp 2 0 satisfying c1a' 5 p' c2a' on E @ M for all M E FIN. Since ( E @-, N)" = E" @y N if 7 is accessible (see Ex 20.13.), the fact that the tensor norms are finitely generated implies that E ( ~ ~ ) &F, = , , E ( ~ ~ )F & ~ , for all Banach spaces F ; in particular, (2) holds. The regularity of 24 and B ! implies c

(1). To see (3), observe first that the embedding theorem implies that % and p are equivalent norms on all E' @ F' for reflexive F. Since a and p are accessible, it follows c that a = and ,O = p on E' 8 M for all M E FIN - and the same arguments as before show that E ( 2 k t 1 ) & a F= E(2kt+')&3pF for all Banach spaces F; this implies (3).

21. Accessible Tensor Norma and Opemtor Ideals

285

Taking Q = g; and 0 = g p i , the fact that g;l is totally accessible, corollary 21.10. and proposition 21.7.(5) give the THEOREM:Let 1 < p < 00 and E a Banach space such that

for all reflexive Banach spaces F . Then: (1) For all Banach spaces F and all even duals of E

(2) E and all its even duals have the metric g,-approximation property. (3) The odd duals of E have the d,#-approximation property.

(1) is due to Kisliakov [143] and (2) to Bourgain-Reinov [17]. H” satisfies the assumption of the theorem for all 1 < p < 00 : This was shown by Kisliakov [143] and Gordon-Reisner [86] - both using the fact that H m is complemented in the bidual of the disk algebra A and a result of Mityagin-Pelczyriski (see [202], p.16) stating that %$(A’ *) = JP(A’ -1. It follows in particular that H” has the metric gp-approximation property (BourgainReinov); the reader is reminded that it is not known whether or not H a has the approximation property.

21.12. Here is a summary of the results which were obtained in this section about the accessibility of Laprestb’s tensor norms Q ~ : , ~ (1) All Q ~ and , ~all a;,q as well as all their projective and injective associates are accessible (21.5. and 21.1.). (2) Totally accessible are: w2, d 2 , g 2 , qq and ap,2(by 21.5. and 21.1. since ~ 2 is , right-inj ect ive) . (3) Not totally accessible are: d p and g p for 1 5 p < 00 and p # 2 (by Reinov’s counterexample, 21.7.)’ all w i / for p # 2 (see 21.6.). We do not know whether or not goo and wp (for 1 _< p 5 00 and p # 2) are totally accessible.

Exercises:

Ex 21.1. Let (a,A) be a quasi-Banach operator ideal. Use the factorization of finite rank operators proved in 16.9. to show the following results:

~

21. Accessible Tensor Norms and Operator Ideals

286

(a) If M E FIN and F has the bounded approximation property with constant A 2 1, then for every T E 2 l ( M , F ) and E > 0 there is a subspace N E FIN(F) such that A(T: M + N ) 5 X(l+E)A(T: M + F ) . (b) If N E FIN and E' has the bounded approximation property with constant X 2 1, then there is an L E COFIN(E) and To E Q ( E / L , N ) such that T = To o Qf and A(To) IX(1 E)A(T). (c) If 2l is left-accessible and F has the bounded approximation property with constant A 2 1 (or: rz( is right-accessible and E' has the bounded approximation property with constant A 2 l), then for every T E $ ( E , F ) and 6 > 0 there are subspaces L E COFIN(E) and N E FIN(F) and To E C ( E / L ,N) such that T = If,, o T o o Q f and A(T') 5 A(1+ e)A(T). (d) If Q is left-accessible, then Q'"j is totally accessible. (e) If 2l is right-accessible, then 2l'"' is totally accessible.

+

If (%,A) is maximal normed, note that all these statements follow from 21.3., the embedding theorem, the approximation lemma and the respective properties for tensor norms. Ex 21.2. (a) If ('2, A) and (B,B)are both right-accessible (resp: both left-accessible), then ('13 o IZL, B o A) is right-accessible (resp: left-accessible). be two quasi-Banach operator ideals with 2l C '13 and (b) Let (%,A) and ('13,B) B 5 A such that (%,A) = ('13,B) on FIN. Then (%,A)is right-accessible (resp. left-accessible, totally accessible) whenever ('13, B) is. Ex 21.3. Principle of local reflexivity for operator ideals: Let (%,A) be a rightaccessible quasi-Banach operator ideal, F a Banach space, M € FIN and N € FIN(F'). Then for every T E C ( M , F") and E > 0 there is an operator R E C ( M ,F) such that

(1) I (1 + W T ) , (2) ( 3 / , R z )= ( T z , d )for all ( 2 , ~ ' )E M x N, (3) Rz = Tz whenever Tz E E, (4) If T is injective, R can be chosen to be injective and satisfy [[R-lIlI (l+~)llT-'[[. Hint: Use 6.6.. If (a,A) is maximal normed, the weak principle (i.e. only (1) and (2)) follows from Q ( M , E)" = %(M, E") (see Ex 20.13.), by Helly's lemma. Ex 21.4. (a) Given E > 0, every finite rank operator T : E --+ &(I?) has the form T = C:=, &Byrn , with tP(z&)5 (1 &)Pp(T) and wpj(ym) I 1. Hint: gp+, = gp\ and accessibility. (b) For each T E 5 ( E ,F) with Pp(T) < 1 there are z;,...,z; E E' with l,(zA) 5 1 and

+

for all z E E.

Ex 21.5. Check that in lemma 21.9. for every E > 0 the operator S E C(G,F) and w E G&,, E can be chosen 80 that llSll 5 1 and gp(w; G, E) 5 (1 + E)gp(z; F, E ) .

287

22. Minimal Operator Ideals

E % ( E , F ) and E > 0, then there are reflexive spaces G I and G2, operators R E 2 ( E , GI),To E %(GI,G2) and S E .C(G2,F ) such that T = S o To o R and IISllN(To)llRII5 (1 s)N(T).Hint: Ex 21.5.. Ex 21.7. (a) Take l / p + l / q 2 1. Then for each operator T E Pqt(F,G)and each Banach space E

Ex 21.6. If T

+

5 Pq#(T).

llT@ id^ : F6a,,qE-G6g,Ell

(b) Use Kwapie6's factorization theorem to show that a Banach space E has the Q

~ , ~ -

approximation property if it has the gp-approximation property. Hint: For each functional cp E ( F E)' and L , = U o S one has (cp, z) = tr,@ 8 i d ~ ) ( @ S idE)(z). This result is due to Dim, L6pez Molina and Rivera Ortun [52].

22. Minimal Operator Ideals

The smallest Banach operator ideal which coincides with a given Banach operator The main result ideal for finite dimensional spaces is called its minimal kernel amin. of this section is the representation theorem of minimal Banach operator ideals in terms of the tensor norm which is associated with the maximal hull am'" of a.

22.1. If (a,A) is a quasi-Banach operator ideal, then its minimal kernel is defined to be the quasi-Banach operator ideal

(a,A)min:= (amin, Amin):= (5,II 11) o (a,A) o

11) ,

where f is the ideal of all approximable operators; in other words (see 9.10. for the definition of the composition of operator ideals),

I

Amifl(T) = inf{ IISllA(To)llRIIT = S o Too R with S,R E 3 and To E

a} .

(a,A) is called minimal if (a,A) = (a,A)min. - REMARK:5 o 5 = 3 holds isomeln'cally. PROOF:Take T E E((E,F) and choose T,

C,"=lT,

(with respect to the operator norm

E 5 ( E , F ) and 1

11 11)

and

5 A, t

00

with T =

c,"=l LllTnll 5 llTll(1+ e). If

288

22. Minimal Opemtor Ideals

Nn := im(Tn), then

This implies that 5 = 505 = 50C o 5, hence 5 is the minimal kernel of all operators: Cmin = 8. Moreover,

(isometrically) - in other words,

amin is a minimal operator ideal.

Since, obviously,

amin c Bminif a c %, these facts show that 8 is the largest minimal operator ideal.

If is pBanach, then aminis q-Banach for some q by 9.10.. To avoid the quasinorm catastrophe the following result will be formulated only for p-Banach operator ideals; but recall from 9.3. that every quasi-Banach ideal has an equivalent p-norm.

PROPOSITION: Let (a,A) be a p-Eunuch operator ideal. (1) amin(M,N ) = a"""(M, N ) = % ( M ,N ) holds isometrically for all MIN E FIN. (2) For each T E amin there are T, E 5 o 9l o 5 with Amin(Tn- T) 40. min = (a,A)min . (3) = (a,A)""" . (4) (a,A)min (5) If (%, B) is a q-Banach operator ideal with B 5 cA on FIN for some c 2 1, then a m i n c B m i n and Bmin5 cAmin. In particular, ( 5 ) shows that (a,A)min is the smallest q-Banach operator ideal (where q is arbitrary) which induces the norm A on FIN.

PROOF: The relation for amax in (1) is immediate from the definition 17.2. of maximal operator ideals. A 5 Amin is always true and Amin 5 A on C ( M , N ) follows from T = idN o T o id,+,. Clearly, (1) implies (4). To 8ee (2), take T E amin(E, F ) , an E > 0 and afactorization T = SoTooR E f o a o g with IlSllA(T')llRII 5 A m i n ( T ) ( l + ~ )For . Sn,R, E 5 with IIR-&l( -+ 0 and IIS-Snll + 0 define the operators Tn := SnoTooR,. Since Amin is an r-norm for some r, (2) follows from

To see (3), take T E am'" gives the factorization

. Then this construction (applied to a"""instead of 9l)

22. Minimal Opemtor Ideals

where To,nE ama"(Mn,Nn)

289

A %(MnlNn). It follows that

AGn) I llsnIIAmat(~o)li~ll

A(Tn

- Tm) I((IIsn - Sm(IAmaZ(To)I(R,IOP + (IISmIIAma"(To)IIh - RnII)p]l'p

1

hence (T,) is an A-Cauchy sequence. It converges in g ( E ,F) and its limit is clearly T. Since A is a p n o r m , it follows that

A(T) = n-oo lim A(Tn) 5 n-m lim IJSnllAmaC(To)llhll 5 Am"" This means that

%ma' min

min (T)(1+ E )

*

c !2l and A 5 AmaZ amin

g m a z min

which implies that and Amin 5 Amaz min

The reverse, however, is obvious. The remaining statement ( 5 ) follows from (1) and (3). 0 The proposition implies that for the study of the maximal and minimal operator ideals QmaZ and Jllmin one may always assume that % is maximal. 22.2. Let (%,A) be a maximal n o m e d operator ideal and (Y its associated finitely generated tensor norm, i.e. % ( M ,N) = M' @a N for all finite dimensional M, N (see section 17), and denote by

J : E'

@F

-

Qmin(E, F)

the natural embedding J ( z ) := T, . For M E FIN(E') and N E FIN(F) the diagram

El @a F

I

Win(E,F ) 3 I:

o T o Q&,

obviously commutes. If z E E'@ F and u E M @ N with Is' @ I;(.)

= z, then

Amin(J(t))= Amin(I{ o T, o QZ0)I A(T,) = a ( u ; M , N ) ,

290

22. Minimal Operator Ideals

which implies that Am'"(J(z)) 5 a(%;E', F ) . Even more is true:

THEOREM FORMINIMAL OPERATOR IDEALS:Let (%,A) be a BaREPRESENTATION nach operator ideal and a the tensor norm associated with (%,A)mas. Then the canonical map J : E'@,F 2 P i n ( E ,F )

-

is a metric surjection for all Banach spaces E and F. In particular, the minimal kernel of a Banach operator ideal i s normed.

PROOF:One may assume that % is maximal. By what was said above J is defined and has norm 5 1. By definition, M' @, N = %(MIN ) . holds isometrically whenever M ,N E FIN. Take operators R E S ( E , G 1 ) , T o E O ( G 1 , G z ) and S E S ( G z , F ) ; if Z S ~ T , ~:= R J-'(S o To o R), then a ( z S o T o o R ; E',

F ) 5 IlsllA(To)llRII :

Indeed, take (as in the proof of 22.1.) the factorization SoT,oR

E - F

Then a(zSoT.oR;

CZI 1z(zT1);E', F)IIIKIIIII~II~(~T,;M', N )=

E', F ) =

= Il~llA(T1)IllKll Il~lllA(~0)ll~ll = II~IIA(T0)llSll *

This inequality is the key to the following reasoning. Take T E W'"'"(E, F ) , factor T = S o To o R with ((S((A(T,)((RII5 (1 + &)Am'"(T) and choose &, S, E 5 with llR -, 0,IlS - Snll -+ 0. The proof of 22.1.(2) showed that T,, := S, o Too & converges to T in Qm'"(E, F ) . Denote r, := J-'(S,, o Too &) E E' @I F ; then Z,

- zm = J-'((Sn

- S m ) o T o o Rn) + J-'(Sm

and therefore by the inequality,

- &))

o To o (R,

291

22. Minimal Opemtor Ideals

It follows that (zn) is a Cauchy sequence in E'&,F. Its limit z E E'&F j ( z ) = lim ~ ( z , ,= ) lim T, = T n+oo

n+W

in

satisfies

amin

since J" is continuous. Moreover,

a ( z ;E', F ) = lim

n+w

This, together with

cy(zn;E',

F)

+E ) .

lim llSnJJA(To)ll%ll5 Amin(T)(l

f8-w

ll.fll 5 1, shows that .f is a metric surjection.

0

A special case: The projective norm 1 is associated with the ideal 3 of integral operators, hence the description 3.6. of nuclear operators gives ( 3 ,I)min= (%, N) .

This is the reason why the operators in 21mi"(E,F ) are called a-nuclear if 21ma"N a Moreover, it follows that the ideal % is the smallest normed minimal operator ideal.

.

COROLLARY 1: Let a be associated with the mazimal hull of the Banach operator ideal

a. If E'

or F has the approcimation property or if a is totally accessible, then

E'&QF = 21min(E,F ) holds isometrically.

This follows from corollary 17.20.; clearly, it would be enough to assume that F had the a-approximation property or E' the a'-approximation property (see 21.7.).

22.3. Examples: KP4) := (.cP,P, f.'p,4)min Np) := V p , (Hp, Kp) := (Cp, Lp)min (fiP,41 ( q 7 ,

the ideal of ( p , q ) - compact operators. the ideal of p - nuclear operators. the ideal of p - compact operators.

The associated tensor norms are a p , 4gp , = ap,l and wp = aPs~. The representation theorems 12.6. and 12.7. for E'ga,,,F and the representation theorem for minimal operator ideals give

292

22. Minimal Opemtor Ideals

The formula for Kp,q also shows that the (p,q)-compact operators are those which factor

E - F

T

.1eql -Ts

DA a diagonal operator (e, can be replaced by c,) ,

D X

eP

.

-

and KP,Ji?) = inf llSl1 ~IDAIIlIRl1 Since Qmi" = 5 o !2lmi" o 5,the operators R and S can be taken to be compact. Note the special cases %, = Rp,l and fip = ffp,pr; it follows that the p-compact operators are just those which factor compactly through the sequence space 4,. 22.4. The following consequence of the representation theorem for minimal operator ideals is trivial, but useful:

COROLLARY 2: If E and F are Banach spaces, a on E' @ F, then (Ijmin(E,F ) C )2Lmi"(E,F ) and

-

2l and p

-

% such that a

5

cp

Ami"(T)5 c Bmifl(T) for all T E %mi"(E, F). By the embedding theorem (and 15.7.) the next result (again for maximal normed ideals and fixed Banach spaces E and F ) appears as a special case of this corollary: If %(E, F ) C %(E,F ) and !2l is totally accessible (or: 2l accessible and E' or F has the bounded approximation property or: El and F have the bounded approximation property), then

!Bmin(E,F ) c !21min(E,F ) .

Since I 5 &ga0

on

63

(see 14.8.), the local technique lemma implies that

5 AKGg, 5 AKGgp for every Hilbert space H,every 2!,A-pace rJmin - %,, it follows that:

,

on H @ F

F and all 1 5 p 5

00.

Since g p

-

3, and

PROPOSITION: %(H, F ) = %,(H, F ) and N ( T ) 5 A K G N ~ ( T for ) every Hilbert space H , 2f,A-space F and p E [l,001. This is in some sense a "nuclear" form of Grothendieck's inequality. 22.5. The representation theorem for minimal operator ideals represents the third of the three basic links between the metric theory of tensor products and the theory of

22. Minimal Opemtor Ideals

Banach operator ideals: If the maximal Banach operator ideal generated tensor norm (Y are associated, i.e. if

M'

293

(a,A) and the finitely

N = %(MlN )

@a

holds isometrically for all M, N E FIN, then for all Banach spaces E and F the following theorems are valid (17.5., 17.6. and 22.2.): (1) The representation theorem for maximal Banach operator ideals:

%(E,F')

A (E

and

F)'

F ~ )nI q

% ( E ,F ) A ( E

~F ) , .

(2) The embedding theorem:

6q ~ F ) .,

E I ~a - F

( 3 ) The representation theorem for minimal Banach operator ideals: E1gaF

1

21min(E,F ) .

This troika of results is a powerful tool for investigating maximal and minimal n o me d operator ideals. A first example is the

COROLLARY: Let (a,A) be an accessible maximal Banach operator ideal. If E' or F has the metric approximation property or i f

(a,A) Is totally accessible, then

amin(E, F ) = 5(E,F)*

% ( E ,F )

The proof follows from the fact that, in this case, (Y = 'Z on E' @ F (see 15.7. and 21.3.). As before, it would be enough to assume that, for example, F has the metric a-approximation property. A consequence of this corollary is:

since g,;

- Vp

P F n ( E ,F ) =

mP' d+P p ( E ,F )

is totally accessible.

22.6. Another nice example of the interplay between the three basic theorems is the following alight extension of a result due to Schwarz [254].

Let (%,A)be a maximal Banaeh operator ideal. If (%,A)is totally PROPOSITION: accessible or i f E or F' has the approximation property, then

(Bmin(F,E))' = %*(El F") holds isometrically.

294

22. Minimal Opemtor Ideala

-

PROOF: If a is the tensor norm associated with Q, then 3L* Q*(E,F I I )

(E

FI)'

' a and therefore

A ( F I ~ ~ E. ) '

Since F'g3,E = amin(F,E ) by 22.2., corollary 1, the result follows. 0 The duality bracket can be calculated by using the trace: Use 17.15. to see (first for elementary tensors) that for T E 2l*(E,F") Qmin ( F ,E ) = F'&

S-

E

- 1 idFr&T

-

F'& F"-

%(F', F')

S'

O

T'

0

KFI

trFl

(T,S) E K

and

where S" : F"

4

E is the astriction of S"; it follows that P,S) =

{

trp(S' 0 T' 0 K trE(S" o T)

p )

if F' has a.p. if E has a.p. .

In the case of a being totally accessible, the duality bracket cannot always be calculated with the trace on operators. This can be seen as follows: The diagram shows that S' o T' o KFI and S" o T are nuclear operators - and it may happen that all nuclear operators in %(F', F') [or % ( E ,E)] appear this way; therefore, if F' [ or E ] does not have the approximation property, the trace is not defined. For an example, take a = E , hence Q* = 3 and amin = 3, and G a reflexive space without-the approximation property. Then %(G, G) = %(co,G ) o T(G, c,) and %(G, G) = $(el, G ) o %(GI41). Since 41 and G' have the Radon-Nikodfm property, it follows from Appendix D8. that

%(G,G)= ~ ( C O , G ) ~ ~ ( G= ,~% ( ~)i , G ) o J ( G , 4.i ) Therefore ( E , F ) = (c,,G') and ( E ,F ) = (GI!,) show that for a = E none of the two operator trace formulas holds for all pairs ( E ,F ) of Banach spaces. Even more: Taking the direct sum of these two counterexamples one can verify that there is no quasi-Banach operator ideal a with a continuous trace such that for each ( E ,F ) either all S' o T' o KFI E %(F', F') or all S" o T E % ( E ,E ) ; see Ex 22.9. for the definition of such ideals and the main argument for this.

22. Minimal Opemtor Ideals

295

22.7. If T is nuclear, then T’ is - but the converse is not true by the example given in 16.8.. For other nonned minimal operator ideals the question of duality reads as follows: If (a,A ) is a maximal normed operator ideal with associated tensor norm a, then the transposed tensor norm at is associated to 2Idua1,the ideal of all T such that TI E !2l (see 17.8.). The diagram

shows that Ti is a-nuclear if T is &‘-nuclear and Amin(T’)5 Adua’min(T)holds; see also Ex 22.5.. Under which circumstances is the converse true? In other words, when does a m i n dual ( E , F ) %dual min ( E lF ) c !2lmin ( E ,F )

t

hold? If F is a dual space, then it is easy to see that this is true (Ex 22.5.). As for further situations where the answer is positive an appropriate description of the a(F‘, F ) - a ( E , El)-continuous operators in amin( F‘, E) will be helpful:

PROPOSITION: Let % mapping

F6aE

-

a be associated and El F Banach spaces. Then the canonical

-

I

{T E Qmin(F’, E ) T u ( F i ,F ) - a ( E , El)-continuous}

is defined and has norm

5

1. It is a metric surjection under each of the following

conditions:

( 1 ) F is 1-complemented in F“. (2) F”/F has the approzimation property.

( 3 ) E has the approximation property.

PROOF:The first statement is clear since, by 4.2., all operators coming from F&E are a(F’, F ) - .(El El) continuous. The representation theorem also gives that, for a given a(F‘, F ) - a ( E , E’) continuous To E amin(Fi,E ) , there is an zo E F”@QEwith j ( z o ) = To and F”, E ) 5 ( 1

&(to;

It remains to prove that ( 1 ) If P : F”

---+

tois

+ &)Amin(To) .

(or can be taken to be) in F&E.

F is a projection with IlPll = 1, then the diagram

296

22. Minimal Operutor Ideal8

commutes. Since ICF o P o Ti = Ti by assumption, it follows that

To= J’ ( ( I C F 8 ~ d E ) ( P ~ i d E ) ( Z o ) ) and hence P&dE(Zo) E FGaE represents To. (3) If the space E has the approximation property, it will be shown that z, is already an element of F&E. For this, take cp E (F’16aE)twith c p p 8 ~= 0; it has to be shown that (cp,~,) = 0. If L, E C(F”, E’) is the operator associated with cp, then, by 17.15., L,&dE

: F”&E

is continuous and (9,z,) = trE(L,&dE(z,))

E‘& E

-

E’&E

:

-

C(El, El) 3 L ,

3 S’ . 0

Since L,IF = 0 and Ti(E’) C F, it follows that L, o Ti = 0. The space E has the approximation property, hence the lower map I is injective; therefore L , o Ti = 0 implies that L,@lidE(Z,) = 0 which gives (cp, z,) = 0. (2) Finally, assume that F“/F has the approximation property. As before, take a functional p E (F’t&aE)’ with c p l =~ 0.~ Denote ~ by U E $ ( E l FI’I) the operator associated with cpt E (E&tF”)’ = %*(E,F’“). Since im(U) C Fo

F’”

(the polar taken in F”’) and F o is complemented in F”‘ (by Z ” ’ - ~ ’ ’ ’ - I C F I ( ~ ’ ’ ‘ I,)), it follows that the astriction U, : E + F o of U is in a* as well. Note that F o = (F”/F)’. With the canonical surjection Q : F” --* F”/F, it follows from 17.15. that Q&Jo

F”8aE

-

F’t&(F”/F)t -F”/F&(F”/F)’

is continuous. A check on elementary tensors shows that the diagram

297

22. Minimal Operator Ideals

commutes. Since Ti(E') c F, it follows that Q o Ti o 17:= 0 and the same argument as before gives the result. 0 The space (F"/F)' is complemented in F"' (this was just shown in the proof), hence F"/F has the approximation property if F"' has it. It will be shown in a moment (22.9.) that the statement of the proposition does not hold if only F" is required to have the approximation property. 22.8. Coming back to the question under which conditions

amin c

%dual min

holds, the proposition gives the

(a,A) be a Banach operator ideal and El F Banach spaces. If F is l-complemented in F" or if E' or F"/F has the approximation property, then

COROLLARY 1: Let

g m i n duel ( E ,F

) = %dual

min

-

(El F )

holds isometrically. If a (Qmazl Amax)this means that: T E f ! ( E ,F ) is at-nuclear if and only if T' is a-nuclear; the respective norms coincide.

= ( Q l A)dua1 , one may assume (a,A) to be PROOF: Since clearly, (a,A),,= maximal. A weakly compact operator S E f!(F', E') is u(F', F ) - u(E', E") continuous are compact, the if and only if it is a dual operator. Since all operators in amin proposition implies that the canonical map

-

~ 6 3 ~ ~ (TI ' E lZLmin(F',E') IT E 2 ( E l F ) } is a metric surjection under each of the hypotheses of the corollary. Since

FgaE'

l

-

commutes, this (and El&+ F a-nuclear; moreover,

-t+

C(F', E') 3 T'

T

21dua1

C ( E lF ) ) shows that T is a'-nuclear if T' is

The converse is always true. 0 The fact that S' is p-nuclear if S is Pt-nuclear easily implies the following: The dual T' of T E f ! ( E ,F ) is at-nuclear if and only if t c ~ o T= T"otc~: E + F" i s a-nuclear. Therefore, corollary 1 (applied to a t ) gives the

298

22. Minimal Opemtor Ideal8

COROLLARY 2: Let (a,A) be a Banach operator ideal and El F Banach spaces. If F is I-complemented in F" or if El or F " / F has the approzimation propedp, then T E amin(E, F ) if and only if KF o T E Bmin(E,F"); moreover, Ami"(T) = A m i " (o~T~)

.

In other words, T is a-nuclear if and only if K F o T is a-nuclear (where a is the tensor norm associated with am,.).Clearly, the result is trivial under the first assumption that F is complemented in F"; for more information about these questions, see 25.11., Ex 22.6. and Ex 25.12.. 22.9. For a =

these results were already proven if F is complemented in F" (see Ex 3.34.) and if El has the approximation property (see 5.9.) - but not yet under the hypotheses on F"/F. The example in 16.8. showed that it is not enough to assume that E has the approximation property. It is also not enough that F" has the approximation property; the key to a counterexample is the following result due to Lindenstrauss [172]: If G is a separable Banach space, then there is a Banach space F such that F" has a basis and F"/F is isomorphic to G. If one takes G to be a space without the approximation property, one obtains a space F" with a basis such that F"/F does not have the approximation property.

PROPOSITION: There are Banach spaces E and F such that F" has a basis and there is a non-nuclear T E G ( E , F ) with nuclear dual. PROOF: For E := F"/F, as before, it is enough to find a non-nuclear T E C ( E ,F )such that KFOT is nuclear. For the quotient map Q : Fit -+ F"/F consider the commutative diagram

(F"/ F)'& F"/ F

A %(F"/ F, F"/ F ) 3 Q o S

T

The construction of F implies that 52 and 53 are injective, but J1 is not! Therefore there is a z, E (F"/F)'& F" such that w, := id&Q(z,) # 0, but its associated operator JI(w,) is zero. If To := Ja(z,), then Q o To = 0, hence im(T,) C F and To = K F o T for some T E f!(F"/F, F ) . If T would be nuclear, then the diagram would imply that To = 0. 0

299

22. Minimal Operator Ideals

The results of 22.7.-22.9. are taken from Losert-Michor's exposition [179] of Grothendieck's %sum& A counterexample similar to the above one was also recently constructed by Oya-Reinov [196]. The investigation of 24."' min and m i n i min ill be for accespostponed to 25.11. since it requires some more flexible descriptions of amin sible operator ideals 24.

Exercises: Ex 22.1. Use the Davis-Figiel-Johnson-Pelczyliski-factorization theorem (see 9.6.) to show that for every T E 21min(E, F) Amin(,) = inf IISllA(To)llRII ,

%(GI, G2)for reflexive GI and Gz and, obviously, R, S E 5 and T = S O T ,o R. Ex 22.2. Let E and F be Banach spaces, one of which has the approximation property. where To E

Then

I

E& F = {T E ff(E',F ) T a(E', E)-a(F, F')-continuous}

.

Hint: 22.7. if F has the approximation property; dualize and approximate T' in the other case. Ex 22.3. Give conditions (like the approximation property, accessibility) which imply that Amin(,) = AmaZ(T)on $ ( E , F). Check this for the ideals: ff,%,gp,q,2p,Dp, q - l , q , and v p . Ex 22.4. If ('2, A) is a maximal Banach operator ideal, F an CL-space, then a i n j min (E, F) = )2Lmin(E, F)

for all Banach spaces E. What about the norm? Hint: 20.8.. For nuclear operators the converse is true, see 23.7.. Ex 22.5. Use factorization to show that %dual min

min dual

C%

and

5 Adual

~min(Tt)

min

(T)

for all quasi-Banach operator ideals. Moreover, %dual min (E, F')

=amin

dual

(E,FO

with equal norms. Hint: Ex 22.1.. Ex 22.6. Let (a,A) be a quasi-Banach operator ideal. The regular hull (a,A)reg of (a,A) is defined to be the class of all T E t ( E ,F) such that K F o T E and A r e 9 ( T ) := A ( K Fo T). is a quasi-Banach operator ideal containing a. (a) Show that (a,

23. 2;-Spaces

300

(b) (a,A)dua'reg = (Q, A)dua';in other words, (Q, A)dua'is regular. (c) (Q, A)dua'min reg = (a,A)mindual, Hint: Ex 22.5.. Ex 22.7. Eztension and lifting properties of minimal operator ideals: Let a be a tensor norm with associated maximal normed operator ideal (Q, A). 1

(a) a is left-projective if and only if: Whenever I : G-E and T E Qmin(G,F), then for every E > 0 there is an ectension f' E 21min(E, F) with Amin@) 5 (1 +&)Amin(T) and T = ? o I. (b) a is right-projective if and only if: Whenever Q : G A F and T E Qmin(E, F),then for every E > 0 there is a lifting f' E IZLmi"(E,G) with Amin@) 5 (1 &)Amin(T)and T = Q o F. Hint: Representation theorem and 20.3.(1) and (2). Ex 22.8. Let (Q, A) be a maximal normed operator ideal with associated tensor norm a. Use the characterization 17.15. to show that

+

- in

other words, T o S is nuclear if S is a-nuclear and T is a*-integral. Conversely,

T E Q*(E,F ) if for all S E !21min(C2,E) the operator T o S is nuclear. Hint: Lemma 17.16. and 22.2., corollary 1. Ex 22.9. A continuous trace T on a quasi-Banach operator ideal (%,A)assigns to each T E Q a number T(T) E K such that (1) T restricted to each Q(E,E) is linear and continuous. (2) T ( z ' @ ) = (z',z) for t' E E' and z E E. As for the existence of ideals with a continuous trace see the books of Konig [147] and Pietsch [216]. Show the following: If the Banach space G does not have the approximation property, then 'JI(G, G) 6 Q(G,G) for each quasi-Banach operator ideal which admits a continuous trace. Hint: Closed graph theorem, 5.6. and t r ( z ) = T ( J Q ( z ) ) for all z E G'&G.

23. 2;-Spaces

This section gives a systematic presentation of the class of all 2;apaces. These spaces were introduced in Chapter I and have already been used at various times. Locally they look like and hence share many properties oft,. They are very closely related to the class of 2,apaces investigated by Lindenstrauss and Pelczyliski in their famous 1968-paper; in order to make them easier to use we modified their definition

23. i!; -Space8

301

slightly and as a result they behave better under dualization and complementation. The main results about .C,- and hence E:-spaces presented in this section are, at least in spirit, due to Lindenstrauss, Pekzyriski and Rosenthal.

23.1. Recall from 3.13. that for 1 5 p _< 00 and 1 5 A < 00 a normed space E is called an Ci,.+-space, if for each M E FIN(E) and E > 0 there are R E C ( M , q ) and S E C ( q ,E) for some rn E N factoring the embedding 15 such that llSll llRll 5 A+& :

E is called an t:-space if it is an Zip,.+ for some A 2 1. It has already been shown that all C ( K ) and L,(p) are C&,,-spaces (see lemma 4.4.) and that all L , ( p ) are .C;,l-spacea (Ex 4.7.). It is easy to see (Ex 4.7. and Ex 6.6.) that E is an 2;,A-space if and only if its completion?!, is an ,Ci,.+-space. The local techniques for .2;-spaces, i.e. passing from statements about the el: to all t;-spaces, are summed-up in - the 2p-local technique lemma about inequalities between tensor norms (see 13.5.). - its generalization to the continuity of operators ide; @ T (see Ex 13.7.). - the &-local technique for minimal operator ideals (Ex23.13.). - and the following lemma for maximal operator ideals (which by corollary 2 of 22.3. also has a consequence for minimal ideals):

.Cp--Loc~~ TECHNIQUE LEMMAFOROPERATOR IDEALS:Lei (%,A)and (%, B) lie quasi-Banach operator ideals, (23, B) maximal, c 2 0 and E a Banach space. (1) If B 5 cA on C($, E ) for all n , then %(F,E ) C B(F,E ) and

B(T)5 cXA(T) for all Z;,.+-spaces F and T E %(F, E ) . (2) If B 5 CA on C(E,%) for all n, then %(E,F ) C %(ElF ) and

B(T)5 cAA(T) for all 2;,,-spaces F and T E %(ElF ) . (3) If%($,I?,) C %(tP,lq) and B 5 cA on %(I?,,t?,),then %(ElF ) C %(ElF ) and

B(T)5 c b A ( T )

302

23. .f!;-Sjmces

for all Ci,x-spaces El C:,,-spaces F and T E %(ElF).

PROOF: Since % is maximal, consider for M E FIN(F) T

M C ' . F - E

A/.

IlRlI

IlSll i A + &

9

c

which gives ( 1 ) . For (2) anticipate the dual characterization in corollary 4 below (23.4.):

E-F

T

(2'" c

IlRll IlSll 5 A + E

for L E COFIN(F) - and again maximality gives the result. (3) is a consequence of (1) and (2). 0 The ideal (Cp, Lp) of p-factorable operators turned out to be the class of all operators T E C(E, F ) such that K F o T factors through some L p ( p ) ;moreover,

L p ( T )= min{llRII

IlS(( I

KF

R S o T : E-Lp-F")

.

This was obtained in 18.6. by ultraproduct techniques.

COROLLARY: The assumptions of (3) imply the formulas C , O % O ~ ~ C and %

B~c(L,oAoLp).

PROOF:If U E C, , then U" factors through some L,(p). Take T o S o R E C, o 2l o Cp; then

Since B(TlS"R2) 5 cA(TlS"R2) by ( 3 ) of the proposition, it follows that (TSR)", and hence T S R is in '13 - with the desired norm estimate. In the spirit of this corollary, statement ( 1 ) of the proposition reads E) and (2) 2, o %(El.) c % ( E , .) with the norm estimates. %(a,

0

ep(-, E) C

23.

2;-Spaces

303

23.2. The ideal (XP,Lp)of p-factorable operators is associated with the accessible tensor norm w,, the adjoint w; of which is even totally accessible (21.5.). This implied that every 2:,Aapaces has the X-bounded approximation property (see 21.6. and also Ex 23.1.). Recall from Ex 9.8. that the space ideal space($) is the class of all Banach spaces such that idE E 2,.

THEOREM: Let 1 5 p 5 co and 1 5 X statements are equivalent: (a) E i s an 2i,A-space.

< 00. For each normed space E the following

(b) idg is p-factorable, i.e. 6 , E space(2,), and L,(idE) 5 A. (c) For all Banach spaces G (or only G = E’ or G some predual of E )

w; 5 a 5 Xw;

on

G @ E.

on

G@ E.

(d) For all Banach spaces G (or only G some Johnson space C,) E

_< wp _< X E

(e) There as a factorization idE” = S o R through some L p ( p ) (where p is strictly localizable) with llSll IlRll 5 A.

PROOF: By what was already said in 23.1. and 13.4. one may assume E to be complete. . , in section 17 (see 17.15, and The characterization of the accessible operator ideal C 17.16.) applied to T = idE gives the equivalences between (b), (c) and (d). The last statement means that L,(idEv) _< A (note that Lp(T)= min ...), so (e) is equivalent to (b) since T E I# if and only if T” E (21 for each maximal normed operator ideal I# (see 17.8.). Since G @c A$ = G (see 12.9.), the tp-local technique lemma 13.5. shows that (a) implies (d). To see the converse take M E FIN(E) and denote by ZM E M’ 61E the element associated with :1 E S ( M , E). Then

which, by the definition of w, (see 17.10.), gives the desired factorization 1; = S o R through some with IlS(l llRllL X + E . 0 The last calculation was in some sense the crucial point of the proof of the theorem. 1: (1) A normed space E is an 2:,A-space if and only i f i t s dual E’ is an COROLLARY .Ci, -space. (2) Complemented subspaces of 2;-spaces are L!;-spaces; more precisely, i fG c E i s p-complemented and E an 2iaA-space,then G is an L!i,pA-space.

304

29. 2F-Spaces

(3) E is an 2;-space if and only if its bidual E" is isomorphic to a complemented subspace of some L p ( p ) . The bidual of an i!;,l-space is isometric to a l-complemenied subspoce of some L p ( p ) .

-

-

PROOF: The relation 2pdY' w;, = w, 2, means: idE E i!, if and only if id^)' = id~~pl E 2,t (with the same norms, see 17.8.). Statement (2) is obvious from (c) or (d) of the theorem and the last statement follows from this and (e). 0 In particular, the class of 2:-spaces is exactly the class of spaces isomorphic to a Hilbert space (this is already known from 11.11. where this result was shown using the little Grothendieck theorem). Since wP 5 cPwz for all 1 < p < 00 for some cp 2 1 (see 12.8.), it follows that

COROLLARY 2: Halbert spaces are C;-spaces

for all 1 < p

< 00.

However, they are neither 2&- nor Ef-spaces (if they are infinite dimensional) by the little Grothendieck theorem (11.11., corollary).

23.3. Lindenstrauss-Pelczyliski [173] called a Banach space E an Zp,x-space if for each M E FIN(E) there is an N E FIN(E) with M c N and the Banach-Mazur distance

Obviously, g,,x-spaces are i!i,A-spaces. Infinite dimensional Hilbert spaces fail to be l$,,x-apaces (for p # 2 ) since d ( 4 , q )= n11/2-1/PI + 00 for n 4 00 (see Ex 26.4. for a proof of this well-known fact); but this is the only difference: Lindenstrauss-Rosenthal [174], theorem 4.3., showed exactly the non-trivial implications of the following two statements: 1 < p < 00 : A Banach space is an 2i-space if and only if it is an CP-space or isomorphic to a Hilbert space. p = 1 or 00 : A Banach space i s an 2!-space if and only if ii is an 2,-space. However, the constants may vary! Cg-spaces are much handier than 2,p-spaces; for example, it is not true that the 2,Cpl-constant of the dual of an Cp,A-space is 5 A since there are infinite dimensional Cm,l-spaces, but no infinite dimensional 21,1-spaces! See Lindenstrauss-Tzafriri [176], p.199. It seems to be unknown whether or not the is an 2,~pl,~+~ for every E > 0. dual of an 2 , ~ The Lindenstrauss-Rosenthal characterization of 2;-spaces implies that 2;-spaces E which are not isomorphic to a Hilbert space contain the $ uniformly: There are M,,E FIN(E) with sup d ( M n , $ ) < 00. For p = 1,00 this can be seen as follows: By local reflexivity (see explicitly corollary 6.6.) and theorem 23.2.(e) one may assume that E is complemented in some L p ( p ) .If p = 00, it follows from a result of Pelczyliski

23.

2;-Spaces

305

[197] that E contains a subspace isomorphic to c,, which implies the result. But this observation is helpful also when p = 1 : In this case it follows that the Ck-space E‘ contains colhence there is a surjection E” -W el; this easily implies (by lifting the unit vectors) that E” has a subspace isomorphic to L1.

COROLLARY 3: (1) Infinite dimensional 2& -spaces have neither proper type n o r proper cotype.

( 2 ) Infinite dimensional 2f-spaces have no proper type, but do have cotype 2.

( 3 ) C;-spaces have type min{p, 2 ) and cotype max{p, 2 ) for 1 5 p < 00. (4) space(q3) # space(&) i f P

#q

and PI q E [I, 4.

PROOF: By what was said above (1) follows from Ex 7.14. and (2) from Ex 7.13. and 8.6. - this latter reference also gives (3). These statements and Ex 8.13. give (4). 0 In particular, no infinite dimensional space can be simultaneously an CS,- and an Cf-space. 23.4. A dual characterization of C;-spaces is the

COROLLARY 4: A Banach space E is an C;,x-space if and only if for every subspace L E COFIN(E) and E > 0 there i s a factorization Qf = S o R through some with IiSIl IlRll I + E E

9,”

EIL

PROOF: If E is an 2;,x, then its dual is an C;,,x. Thus, if L E COFIN(E), there is a factorization

hence Qf = Rb o Sb o K E . Conversely, and in essentially the same way, the condition implies that E‘ is an 2pB,,x+pace. 0

23. 2 ;-Spaces

306

23.5. In 20.14. it was shown that r\ /T

= w;

= w& and E / = wl; transposition yields and

\E=

wco

.

Thus, theorem 23.2. gives a characterization of 2 :- and .C&-spaces in terms of respecting subspaces and quotients; for 2 : the result is due to Stegall-Retherford [259].

COROLLARY 5: Let E be a Banach space. (1) E is an 2:-space if and only i f E & respects subspaces isomorphically. More 1 precisely, E is an 2;,A-space if and only i f for each G- F and z E E 8 G ~ ( zEl ; G) 5 A r ( z ; El F )

.

(2) E is an Sg-space i f and only i f E C& respects quotients isomorphically. More 1 precisely, E is an 2k,A-space if and only i f for each Q : F + G

PROOF: That the conditions are necessary follows from the properties of r\ and E / . Conversely, observe that if E respects quotients, then, using !l(Bc) --y G (recall theorem 20.6.), there is a AG with E / 5 AGE on E@G- and a simple lp-sum-argument shows that the constant can be chosen independently from G. The inclusion about the unit balls gives as well E / 5 AE on E 8 G for all G. Since E / = w k , this implies (2). Similar arguments give the result for (1). 0

-

Grothendieck’s theorem 3.11. says that the 2f,l-spaces are exactly the spaces of the form L1(p) for some strictly localizable measure p. Moreover, it follows that Lf-spaces share the interesting lifting property expressed in 3.12., corollary 1 - in particular, the lifting of compact operators, see Ex 3.31.. The result in 3.10. now reads that E i s an 2:-space (resp. an 2:,A-space) if and only i f E’ is an injective Banach space (resp. has the A-extension properly). For 2f,,,-spaces this means (by the duality result in corollary 1) that a dual space is A!&, if and only if it has the A-extension property. Moreover, as in 4 . 5 , the fact that for an 2&,A-spaceE ff (G,E ) = G’& E = G’& E (L&+paces have the approximation property) shows that these spaces E have the compact extension properly with constant X : For every isometric embedding G L+ F, every operator T E R(G, E ) and E > 0 there i s an extension ? E R(F, E ) of T with 11p1 l 5 (A e)llTII. This, as well as its converse (see Ex 23.4.), is - in its spirit - due to Lindenstrauss [171]. Going to the completion, the corollary gives the following result due to Kaballo [134], which can be used for the lifting of vector valued functions, see also 4.5., corollary 1.

+

23.

.Ci-Spaces

COROLLARY 6: A Banach space E is an C",space

307

if and only if the completion E&*

respects quotients isomorphically.

PROOF:The quotient lemma 7.4. implies that E&,* respects quotients; 80 the condition is necessary. Conversely, take a metric surjection !l(r) --y C2 onto a Johnson space Cz. Then the condition implies that E&,,,C2 = E&C2 holds isomorphically and hence E / 5 X E for all E 8 G by the very properties of C2, see lemma 17.16.. 0 23.6. It was shown that dual C&-spaces are injective. This is not true for arbitrary Lsapaces; take the Cs-space co which is not complemented in tm.Every injective space E has the X-extension property with X = IlPll, where P : !,(BE') 4 E is a projection (Ex 1.7.). Define the projection constant of a Banach space E to be

I

X(E) := inf{llPII P E c(&(BEf), E) projection onto E} E [0, m] . PROPOSITION:

Lm(idEf) = >(El)for each Bonach space E .

'

PROOF: If P is such a projection, then i d E f = f ' o IET ( B E f f ) is a factorization through an Lm(p), hence L , ( i d p ) 5 llf'll. Conversely, if X = L,(idEf), then, by what was said above, EJ has the X-extension property and therefore X(E') 2 A. 0

Note that it was shown that El has the X(E')-extension property, in particular, X(E') =

*in{ llPll I ...}.

-

Since the adjoint operator ideal C& w& = g& is the ideal 'J31 of all absolutely l-summing operators, it follows from 17.19. that for each M E FIN

X ( M ) P l ( i d u )3 dim M and there is even equality if the norm on M satisfies certain invariance conditions (e.g. M = 5).This gives a possibility of calculating projection constants; for the euclidean space 4 see 11.10,. See also Ex 23.8. for the study of the extension of operators.

-

23.7. In order to obtain some operator characterizations of C&-spaces recall J x for the integral operators, Pin = % and the representation theorem for minimal operator ideals. The diagram

308

29. C;-Spacea

commutes, and also the left vertical map is injective since r\ is totally accessible by 21.1.(2). Arguing as in 5.9. or Ex 5.13. with the Hahn-Banach theorem, one obtains

(.3inj)min(E, F) = %(E,L,(Bp)) n 2 ( E , F ) = %'"j(E, F ) if E' has the approximation property. In 25.11. a much more general result will be obtained, showing that this also holds if F has the approximation property. By the characterization of quasinuclear operators (see Ex 9.13.) this means exactly that:

LEMMA:If E' or F has the approximation property, then E'&\ F = U%(E, F ) holds isometrically.

The following characterization is due to Stegall-Retherford [259].

PROPOSITION: For each Banach space E the following statements are equivalent: (a) E is an 2%-space. (b) QZ'JI(E,F) = %(ElF ) f o r all Banach spaces F (or only C2). (c) u f R ( F ,E) = %(F, E ) f o r all Banach spaces F (or only Cz).

PROOF:If E is an C&, its dual E' is an Cf-space; since x\ = wit, theorem 23.2. implies that r\ and 17 are equivalent on E' 8 F, which implies (b) by the lemma. Conversely, (b) gives ?r\ M x on E' 8 C2, which shows (17.16.) that ?r 5 Ax\ for some X on all E' 8 F. This implies that E' is an Cf-space (by the theorem) and E is an C& -space. That (a) implies (c) is a consequence of the fact that a NN a\ on F' 8 E for all tensor norms a if E is an C&-space (see 20.8.). For the converse, it follows that (again by the lemma) ?r M x\ on E' 8 E. The theorem implies that E' is an Zf-space. 0 In Ex 23.9. the reader will find a characterization of C$-spaces in terms of pcompact operators Rp = CPn.

23.8. The preceding characterizations were in terms of minimal operator ideals. To obtain a characterization with p'-dominated operators (recall 2; = !Dp/) the following lemma, which is a special case of the much more general quotient formula to be proven in section 25, will help. LEMMA:Let 2 be an accessible maximal normed operator ideal. Then f o r each Banach space E the following statements are equivalent: (a) E E space(%).

23. 52;-Spaces

309

(b) W ( E ,F ) = 3(E,F ) f o r all Banach spaces F . ( c ) %'(F, E ) = 3(F,E ) f o r all Banach spaces F. Q of 2l is accessible. Since a* o !2i C 3 and 17.19., statement (a) implies the others. Assume (b), then

PROOF:The associated tensor norm Q o 'ZL'

c J by

is continuous and hence G @c E + G BU E, which implies that idE E %, see 17.16.. If (c) is satisfied, use the fact that a* is accessible and CZhas the metric approximation property, to see that the diagram

I proves the continuity of C2 and theorem 17.15..

E

1 +

CZ

E, which implies idE E 2l by lemma 17.16.

It is clear from 17.19. that, for example, I(T) 5 A ( i d E ) A ' ( T ) in (b). Since 2; = Pp,, the p'-dominated operators, it follows that:

PROPOSITION: For a Banach space E and 1 5 p 5 00 the following

are equivalent:

(a) E is an 52;-space.

(b) !D,t(E, F ) = 3 ( E ,F ) f o r all Banach spaces F. (c) PDp#(F, E) = J(F,E ) f o r all Banach spaces F. Recall that 91 = !$JI and Pm= Vdua' 1 . 23.9. A!;-spaces

enjoying the Radon-Nikodjrm property are rare:

PROPOSITION: Let the 52:-space E be complemented an EJJ.Then E has the RadonNihodim property if and only if id is isomorphic to some ll(r). PROOF:Observe first that ll(r) has the Radon-Nikodim property (see Appendix D3.). Conversely, theorem 23.2.(e) implies that E is isomorphic to a complemented subspace G of some L1(p). By proposition 2 of Appendix D2.(Lewis-Stegall theorem), the projection P : L 1 ( p ) 4 G factors through some l1(r0):

310

This shows that G (and hence E) is isomorphic to a complemented subspace of tl(I',) and, as a consequence, has the lifting property (see 3.12.). It follows that E is isomorby Kothe's result mentioned in 3.12.. Cl phic to some This result and the following interesting consequence are due to Lewis and Stegall[170].

COROLLARY 1: The dual E' of a Banach space E is isomorphic t o some

t,(r) if and

only if

Pl(E,F ) =

F)

for all Banach spaces F.

PROOF: It follows from 23.8. and 23.2. that P l ( E , .) = J ( E , if and only if E' is an Ciapace. Appendix D8. gives that J(E, F') = % ( E ,F') for all F if and only if E' has the Radon-Nikodjm property. This, together with the proposition and 5.9., easily implies the result. 0 0

)

The Hardy spaces HP are $;-spaces for 1 < p < 00 since they are complemented in Lp[O,274 (see Ex 4.7.). Pelczyliski [201] showed that H" is not a quotient of any C ( K ) . Since H" is a dual space ([202], p.ll), it cannot be 20, by theorem 23.2.(e).

COROLLARY 2: The Hardy space H w is not an ZCg,-space, the Hardy space H' i s not an 2:-space.

PROOF:It is known that H' is a dual space ([202], p.12) and separable, and hence has the Radon-Nikodjm property. If it were an Ciapace, it would be isomorphic to 41 (by the proposition), hence weak and strong convergence of sequences would coincide: The sequence (exp(in.)) converges weakly to xero by the Riemann-Lebesgue lemma, but each function has norm 1, a contradiction. 0

23.10. Recall from 17.14. Grothendieck's theorem

2(tiIt2)= Pi(el,tz) and P l ( T ) 5 KGllTll . By the Cp-local technique lemma 23.1. for operator ideals this can easily be extended: GROTHENDIECK'S THEOREM: If E is an C:,A-space and H a Hilbert space, then every T E C ( E , H ) as absolutely I-summing and

Pl(T) 5 KGAllTII

23. 'cp"-Spaces

311

This improves the little Grothendieck theorem, which stated that T E V 2 ( E ,H ) . The same argument as before shows that the first theorem in 17.14. reads as follows for C+paces:

THEOREM: Let E be an Ck,A- and F an Cf,l,-space. Then every T E C ( E ,F ) is 2-dominated and (hence) absolutely 2-summing; moreover,

In the next section it will be shown that for 1 I p 5 2 the sequence space lp is isometrically contained in some L t ( p ) - the Lkvy embedding. Since V2 is injective,

T : &, implies that C(L,, .fp) = !&(4',

-

l p L L i ( p )

,tp)and hence, by the Cp-local technique for operators:

1: If E is an C&,A- and F an C;,,-space for 1 I p COROLLARY

5 2, then

-

Clearly, the result also has a tensor norm formulation, see Ex 23.15.. Note that = 9 2 (with equivalent norms) since w$\ 92 = g ; (see the table in 20.15.). See Ex 25.5. and Ex 25.6. for more information about this result. Proposition 20.19. gives the "adjoint" form:

COROLLARY 2: If E and F are Banach spaces, E an C;,A-space for 1 5 p 'Y2(E,F ) = V i ( E ,F )

and

5 2, then

P i ( T ) I KcAP2(T) .

<

Corollary 1 implies, in particular, that every operator CS, -+ Cp"(for 1 p 5 2 ) factors through a Hilbert space. More generally, it will be shown in 26.1. that every operator C: -+ C; (with 1 _< p 5 2 5 q I 00) has this property. A nice consequence of this result is a characterization, due to Grothendieck, of Hilbert spaces.

3: Let 1 5 p 5 2 5 q 5 00. A Banach space E is isomorphic t o a Hilberi COROLLARY space if and only if it is isomorphic to a quotient of an 2:-space and isomorphic t o a subspace of an 2;-space. PROOF: If T : 2 : -W E 4 C;, then T factors through a Hilbert space; this easily implies that E is isomorphic to a Hilbert-space. Conversely, since pz' is an isometric subspace of some LP(pn) (for example with Gauss functions), the ultraproduct technique gives that every Hilbert space H is isometric to a subspace of an ultraproduct of L,, which is an L, by 18.4. (this was Ex 18.7.(c)). Dualizing the embedding H 4 L 4 t ( p ) gives L h ) H*0 +

312

23. Cg-Spaces

Exercises: Ex 23.1. C;-spaces have the metric approximation property if 1 < p < 00. Hint: Reflexivity. Ex 23.2. There is no constant c such that ww 5 cw2. Hint: End of 23.2.. Ex 23.3. A Banach space is an ZG-space if and only if its bidual is isomorphic to a complemented subspace of some C ( K ) ,K compact. Ex 23.4. A Banach space E is an CB,,,-apace if (and only if) it has the compact extension property with the constant A (see 23.5.). Hint: Cz cr &(I'), proof of corollary 6. Ex 23.5. (a) Every injective Banach space is an Cs-space. Hint: Ex 1.7.. (b) A Banach space is an ECQapace if and only if its bidual is injective. 1

(c) E is an CCQ-space if and only if for every metric injection G c-* F and T E C(G,E) there is an extension ? E C ( F ,El') of K E o T.See also Ex 3.20.. Ex 23.6. Let E and F be normed spaces and

the infimum being taken over representations t = Chi Z k @ yk. Show that 6 1 is a finitely generated tensor norm 5 wl and that u1 5 A& on E @ F if E is an CLIA-space. This tensor norm is due to Matos (see also [2]). The associated maximal operator ideal consists of those operators T : E F such that K F o T factors through a Banach lattice (see Pietsch [214], 23.2.9. and 23.3.4.). Ex 23.7. E is an C;.-space if and only if $ ( E , .) = C ( E , -) and if and only if C p ( . E) , = C(*, E l . Ex 23.8. Take E and F finite dimensional Banach spaces and I : E .fw(r)a metric injection. For T E C ( E ,F ) = E' 8 F define the extension noma A(T) of T by

-

-

? E C(&(I'), F),il'o I = T} . (a) A(T) is independent of the special injection Z and I?; moreover,

llTll 5 A(T) 5 A(idE)IITII. The number A(E) := A ( i d ~is) the projection constant of E , defined in 23.6.. (b) A(T) = \ E ( Z T ; E', F) if ZT E E' @ F represents T. (c) The following trace duality holds: A(T)=sup{Itr(SoT)(IP1(S:F+E)
I

P 1 ( T ) = sup { )tr ( S o T)l A(S : F

+E)

5 1) .

313

23. f!;-Spces

These results are due to Garling-Gordon [79]. Ex 23.9. Recall from 22.3. that E'&,,F * &(El F ) = .C,"'"(E, F ) is 8 representation of the p-compact operators. Show that for each Banach space E the following statements are equivalent: (a) E is Zz-space. (b) %(ElF) = Z ( E , F ) for all Banach spaces F. (c) &(F, E ) = Z(F, E ) for all Banach spaces F. In both statements 3 can be replaced by a. Hint: w i , = wp and the ideas of 23.7.. The result is due to Johnson [129]. Ex 23.10. Every separable dual space isomorphic to a complemented subspace of an L 1 ( p ) is isomorphic to el. Hint: 23.9.. Ex 23.11. The dual E' of a Banach space E is isomorphic to some lI(I') if and only if the following two conditions are satisfied: (a) For all metric surjections Q : F -+ G the mapping i d E C 3 Q : E@& F

-

EC3c G

is a quotient mapping. (b) For all Banach spaces F the canonical mapping

El& F'

-

(E

F)'

is surjective. Hint: \E = woo, 16.5., and 23.9.. Ex 23.12. Apply the Chevet-Persson-Saphar inequalities 15.10. to show that for Banach spaces E and F the following holds: (4

p p I ( E ,F ) C J$'"(E, F )

if E is an i!;-space

(b)

p F a ' ( E lF ) c J p ( E ,F )

if F is an 2;-space.

For a converse see 25.9.. Ex 23.13. f!p--local technique for minimal operator ideals: Let (24, A) and ('13, B) be quasi-Banach operator ideals, (a,A) minimal and normed. Then the statements ( l ) , (2) and (3) of the ZP-local technique lemma 23.1. hold. Note that '13 was not supposed to be maximal. Hint: Use the representation theorem for minimal operator ideals with Q

N

am'=.

Ex 23.14. For S E i 2 ( L P ( p ) , L p ( v ) (with ) llSll = 1) consider the maximal operator ideal 52s := C t r B A rsee ; 17.4.. Show that every Zs-space is in space(&). In particular, if 1 c p c 00, each Hilbert space is in space(&). Hint: L p ( p ) E space(&), see Ex 7.8.. For a more general result see 26.3.. Ex 23.15. Show that r 5 K G A p d z on E C3 F if E is an C&,A- and F an 2;,,,-space, 2 5 p 5 00. Hint: 23.10.. Ex 23.16. Let a be a tensor norm.

314

24. Stable Meaaurea

(a) If E and F are C!-spaces

with constants X and p, then a

I \a/I 4Ja

onE8F.

(b) If E and F are C&-spaces with constants X and p , then /a\ I a

onE@F.

L Ada\

-

(c) If Ei are 2 : - and Fi are CL-spaces, Ti E C(Ei,Fi), then TI @T2 :El @Dr E2

FI @ r F2

is continuous. Hint: w2 5 w;. Ex 23.17. (a) L p ( p )@ A ~Lp(v)is an Ci,l-space. Hint: p @ v. (b) E is an C;-space if and only if E c* E” factors through an C;-space. (c) El E2 is an 2;-space if the Ei are. Hint: Use Ap and 13.2..See also Ex 34.2.. (d) If E is an 2;)-and F an Xitapace, then the space !&(E,F’) of absolutely p summing operators is an C;-space. Ex 23.18. Show that w$ 5 K G X ~on E the tensor product E @ F of an &,A-space and : into an CL-space an C!,p-space. Deduce that every 2-factorable operator from an C is integral; for this, see also the operator form 20.17. of Grothendieck’s inequality.

24. Stable Measures

In this section embeddings A?, c* L 4 ( p ) for an appropriate mewure p will be studied. These embeddings exist only if p = q (trivial) or p = 2 or 1 5 q < p < 2.The p-stable measures/distributions due to P. Lkvy, which are so important for the central limit theorems in probability theory, will be used for the embeddings; MarcinkiewiczZygmund in 1939, and later on Kadec, noted that patable measures may serve for the construction of the desired embeddings. As was already seen in the improvement of Grothendieck’s inequality in 23.10.,these Ldvy embeddings may be useful; in the final part of this section their importance will become self-evident through results of Saphar, Kwapieri and Maurey about absolutely q-summing maps.

24.1. If 7 is the Gaussian measure on K w , studied in 8.7.,the “Gauss-Khintchine” equality in proposition 8.7.says that for all 0 < q < 00 (the same proof for 0 < q 5 1)

24. Stable Measures

315

and all n

This means that I ( c u k ) := Cr=la k g k defines a map I from e2 into the y-measurable functions such that llW,(7) = C 2 , , l l 4 l ~ a 15q<m for all z E 12. In particular, for every q E [1,m[there is a measure p (a multiple of 7) and an isometric embedding .f2 4 L , ( p ) . Note that the Rademacher functions give an isomorphic (not isometric) embedding of ez into L, ( p ) via the Khintchine inequality. What about the embedding of e, into L , ( p ) for some measure p if p E [I, m]\{2, q}? If the separable space eP is isomorphic to a subspace of L , ( p ) , one may clearly assume that p is separable and a-finite -and, hence, even finite. Using the Carathhdory-Halmosvon Neumann theorem (every non-atomic separable finite meamre is isomorphic to the Lebesgue measure on some interval, see Brown-Pearcy 1181, p.181), one obtains that 1 1, is isomorphic to a subspace of L,[O, 11$, t,-L,[O, 11.

# 2. If q > p o r p > 2, then the sequence is not isomorphic to a subspace of L , ( p ) for any measure p.

PROPOSITION: Let p , q E [l,m[ and q # p space

e,

PROOF: Since type and cotype respect subspaces and 1, does not have type min(2, q ) if p < 2 and p < q, and does not have cotype max{2,q} if p > 2 and q < p (see 8.6. and Ex 8.13.), only the case 2 < p < q remains. An old result of Banach ([4], p.185) says that t, is not isomorphic to a subspace of l, for different r, 8 €11, co[.But KadecPelczyriski [137] showed for 2 < q < 00 that a closed subspace E of L,[O, 11 is either isomorphic to t 2 or contains .fq isomorphically - so the last case is also impossible. 0 Clearly, this proposition concerns the comparability of the linear dimension (in the sense of Banach) of t, and L,[O, 11. 24.2. In the remaining case 1 5 q < p < 2 there is even an isometric embedding of ep into some L , ( p ) ! It seems that the only way to show this is using Ldvy’s stable measures, which can be obtained as Fourier transforms of certain functions. The result which will be presented is implicitly due to Ldvy; the interpretation of Ldvy’s result in terms of embeddings of l, into L, was observed by Marcinkiewicz-Zygmund [182] in 1939 and, more explicitly, by Kadec [136] in 1958 (see also the survey of Pekzyriski-Bessaga [205], 512, for more information about this type of question). A continuous function f : Rm + C i s called positive definite if

i,j=1

24. Stable Measures

316

for all 4 E N,scalars E Cand Z j E R". If f w , w E 52, are positive definite functions and v is a (positive) measure on R,then it is obvious that the function f defined by

f(z) =

In

fw((z)v(d w)

for z E Rm

is positive definite if these integrals exist. In particular, the Fourier transform of a finite measure p on R"

is positive definite (as usual in this context, we omit the constant (2x)-"12 in the Fourier transform). Using Choquet's representation theory one can show the converse (see Choquet [28], vol.11, p.260):

BOCHNER'S THEOREM: Every positive definite function on form of a finite Borel-Radon measure on Rm.

I P is the Fourier trans-

The set !J? of positive definite functions on Rm has excellent stability properties: (1) As was already mentioned: fn f w d v E !J? if all f w E !J?; in particular, Q is stable under forming positive linear combinations. (2) If fn E !J? and fn -+ f pointwise, then f E !J?. (3)Iff,gE!J?,thenf'gE!J?. The last statement followsfrom the fact that the Schur product (ai,jbi,j) of two positive definite matrices (ai,j) and ( b i j ) is positive definite (see Ex 24.1.). Series expansion and (1) - (3) imply: (4) I f f E !J?,then expof E '43. The aim is to show that the function

on R" is positive definite for 0 < p 5 2, where 11 112 is the euclidean norm on R"; Bochner's theorem will then supply the measures to be used. For p = 2

so

exp(-ll

. 1);

is the Fourier transform of the measure having Lebesgue density

(4~)-"/~ exp(-II. 11;/4), which is (up to scaling) the Gaussian measure. For p

the argument is not as direct: First observe that cos(w, z)=

+

-1 [exp(i(w, z)) exp(-i(w, z))]= 5 2

(

y - w >

[z]

<

2

24. Stable Measures

317

(Dirac measures) implies that cos(w, .) E q for all w E Rm.

LEMMA1: For all

2E

Rm and 0 < q < 2

where

and, form 2 2 ,

(um-2

i s the area of the unit sphere in Rm-').

This is an easy calculation which begins with an isometry T satisfying x = IlxllTel. Note that the formula does not hold for q = 2. The result (it will also be used for the next lemma) is crucial for the

PROPOSITION: The function

fp(.)

:= exp(-ll

ell;)

on

IR" is positive definite for each

O
imply the result - the last term just gives a nonproperties (l), (4) and (2) of negative multiplicative constant. 0 Obviously the function

-11 - 1;

fails to be positive definite.

on < p 5 2, there is a finite measure with S(vr) = fp. Since fp and all its products with polynomials are Lebesgue integrable, it follows that S-l( fp) is an infinitely differentiable function - the density of v r with respect to the Lebesgue measure on Rm;moreover, the density S-'(fp) is symmetric and satisfies 24.3. Bochner's theorem gives that, for 0

IR"

1 = fp(0) =

1

S-'(fp)(w)dw = vp"(R") .

R"

It follows that vp" is a symmetric probability measure on

R" .

318

24. Stable Mea8ure.e~

LEMMA2: For 0 < q < p

<2

the q-th moment of v;

etists; more precisely,

PROOF: By lemma 1

since vp" is a probability measure and exp(-llwll;) standard calculation. 0

= Re(S(v;)[a!]).

The rest is a

Note that the calculation shows that the p-th moment of vp" does not exist! 24.4. Only the cases m = 1,2 will be needed from now on (but see Ex 24.3.) and R2will be interpreted as C. In this way the real and complex case will be treated jointly: for 0 < p < 2 there is a probability measure p j on K with Fourier transform exp(-l IP) whose q-th moments exist for 0 < q < p . By p; the product measure on K" and by pp the product measure on KOo will be denoted - as was done in the case

-

of the Rademacher and Gauss functions in section 8. The k-th projection K w 4 K will be called h k (to avoid the g k from section 8 which referred to Gaussian stochastic variables).

LEMMA3: Let a1,...,a, E K and

C;=l l a a l p = 1. Then the image measure of pp via

n

is the measilre p ; .

In the language of probability theory this means: The stochastic variables h k on the probability space ( K w,pp) (with values in K) are stochastically independent, identically distributed (their distribution is p;) and a k h k has the same distribution p i if I a k l p = 1. This is why the measure p i is called p-stable. For the present purposes it will be convenient to call the infinite product pp the p-stable Le'vy measufe on KOo.

c;=l

PROOF: Call v the image measure of pp via Z;=l a k h k ; then

319

24. Stable Measures

By the injectivity of the Fourier transform one has to show that S(Y)= S(pk). Take z E K (and note that R e ( z 1 3 2 ) is the scalar product of C = E'). Then

by the defining property of pk. 0 Everything is prepared for proving the main result of this section:

THEOREM (LQvy): For 0 < p < 2 let pp be the p-stable Lkvy measure on K". Then a k h k maps t?, into L q ( K o o pp) , for the operator I on t?, defined by I ( ( a b ) ) := all 0 < q < p and there are constants c ~ such , ~ that

xr=l

cP,qllalllp= 1 1 ~ ( 4 1 1 L , ( K - , / l p , =

(lKrn

I(%4 P P P ( W ) l'q

for all a E 1,. The constant

is (~~,~,1)'/q in the real and moment of vp".

(cp,q,2)1/f

in the complex case, where c~,~,,,,is the q-th

The mapping I will be called LGvy embedding.

PROOF:Take a = ( a 1 , .,.,an) E Kn with ((a([L, = 1; by lemma 3 the measure p i is the image measure of pp via a k h k . Therefore, lemma 2 gives

c:=,

This relation obviously implies the result. 0 24.5. The following consequence is obvious:

-

COROLLARY 1: For 0 < q < p < 2 or: p = 2 and 0 < q p and an isometry I : A!, L q ( p ) .

< 00 there are a finite measure

320

24. Stable Measures

For p = 2 this was already stated at the beginning of this section with the help of the Gauss measure. It is interesting to note that the map I cannot be positive, see Ex 24.2.. For the tensor norms wp one obtains the

COROLLARY 2: If 1 5 q < p < 2 or: p = 2 and 1 5 q < 00, then w,\ 5 wp PROOF: The isometry z E M @ N and factor

c-+

L, implies L y ( i d p ) = 1. &call that

Then w,\(z;M, N) = L?j(T,) 5 IlRll IlSllLfj(idt;) 5 llR11 llSll infimum over all such factorizations gives the result by 17.10.. 0

.Cp

w,\

, take

. Passing

to the

N

COROLLARY 3: If E i s an 2:,x-space and 1 5 q < p < 2 or: p = 2 and 1 s q < 00, then there are a Borel-Radon measure p and an operator I : E -+ L , ( p ) such that

PROOF:Since L y 5 Lp by corollary 2, it follows that, by theorem 23.2. on the characterization of C:-spaces, L?j(idE) 5 L p ( i d ~5) A. By the definition of and

Cp

the factorization theorem 18.6. for q-factorable operators there is a factorization

with IlIll

5A

and

IlS(l= 1. This gives the result.

0

In particular, for every measure v and 1 5 q < p < 2 (or p = 2 and 1 5 q < oo) there is a measure p such that Lp(v)is isometrically embedded into L , ( p ) . It is easy to derive this result directly from the L6vy theorem with the canonical embedding of a space into the ultraproduct of its finite dimensional subspaces (see 18.5.). The case p = 2 (Hilbert spaces) was already settled in Ex 18.7.(c). 24.6. Lkvy embeddings have interesting consequences for tensor norms and operator ideals. The following result is due to Saphar [242].

24. Stable Measures

PROPOSITION: For 1 I q < p < 2 or: p = 2 and 1 _< q

321

< 00

the relation

\. 5 C&P,,& holds on E 8 1, for all Banach spaces E .

PROOF:If I denotes the LBvy embedding lpc* L, := Lp(pp),then c;,:I is an isometry. This means that for t E E 8 1, r-, E 8 L, a\(z; El t p ) = cp',+\(z; El L,)

4%; El Lq) I cp,,a(r; E , t p ) for all tensor norms a.Since A, 5 9;) (Chevet-Persson-Saphar inequality 15.10.) and pp is a probability measure, it follows that A\(%;

E,tp)= c~,:x\(z;El L i ) 5 c~,:,:"(z; El Li) = cL,;A:(t; El Li)

I .;,,:A:(.;

<

El Lq) I c,;:g:'(t; El L,) 2 c ~ , ~ c p , , g ~ E,tp) '(z; .0

Note that the proof not only used embeddings tpc* L, but also that, for fixed p, the same embedding can be used for all q < p ! Since ?r = T\ on t 1 8 tp, it follows that

= 5 ~ ~ , : c ~ , ~ on g : el ~ 81p . Recall the "little Grothendieck theorem" 11.11.: .(3(i1,t2) = f&I2(t1,l2) = J2(Ll1.f2). The proposition implies the following generalization:

COROLLARY: If 2 < p < q

I 00

PROOF:Since 3, C f&I, and 3,

,

39(e1 ep)

or: p

-

= 2 and 1 < q 5 00, then

go, it follows that

= (el 8g;

fp))'

= (ti gW t p t ) ' = c(ti,t p )

.

The llP-local technique 23.1. clearly implies (for p, q as in the corollary) that

q E , F ) = Pq(E1 F ) = J,(E, F ) whenever E is an C .-,:

and F an Z;,+-apace, and in this case

P , P ) IIq(T) I ~ ; ~ l ~ P ' , n ' ~ P *l l ~ l l Note that 2(!1, &) = 'Q1(.f1,.&)by Grothendieck's theorem, but 3(t1,&) = n(t1,&) # . C ( l l , t 2 ) . See Ex 24.6. and Ex 25.7. for some consequences of this result.

322

24. Stable Measures

24.7. As it is the caae in the little Grothendieck theorem (i.e. &(Elf5) = !&(El e,) for E = el or loo) the last corollary also holds for tooinstead of C1. This will also be shown using LBvy embeddings; the range space will be even more general than ep, namely of cotype p . Recall from Ex 8.12. and Ex 11.20. the definition of the Banach ideal ('&+, Pr,a) of absolutely (r,s)-summing maps (1 5 s 5 r < m). In 8.9. it waa shown that for all Banach spaces F and elements z1, ...,znE F

Hence, the definition of cotype q gives the

REMARK: The identity map idF of every Banach space F of cotype q E [2,00[ is absolutely (q, 1)-summing, and Pq,l(idF) 5 C , ( F ) . The crucial step is to show that Vq,l(.foo, .) C Vq+c(&, , .).

LEMMA:For 1 < s 5 r < 00 and T E &(Pool F)

PROOF: By definition (see Ex 11.20.) and duality

Hence, the conclusion follows from

and the calculation of the projective norm in * @

. [7

Now cleverly chosen Lkvy embeddings (in the form of Ex 24.8.(b)) give the

PROPOSITION: Let 1 < s < r < p < 00. Then for every Banach space F

24. Stable Measums

323

As a consequence, this inclusion holds for every ZCB,-space E instead of -too.

PROOF:It is enough to show that Pp(T)5 CPr,#(T) for all T E C(Pm,F) - with a constant independent of n. Fix 0 < q < 2/s' and define O < q < po := qp' < ro := qr' < so := qst < 2 To apply the lemma take Y;,

(using Ex 24.8.(b) with q

,

...,dmE F'. Then

< po < so)

Applying Ex 24.8.(b) once more, this time with q line is

< p o < r,, it follows that the last

k=l

Since p o / q = p', the foregoing lemma implies

with a constant c, depending only on

( 8 ,r , p ) . 0

One can show (see Pietsch [214], 22.6.4., where the proof just presented was taken from) that -Cp,r := -4 cro,q 410

exists and obviously can be taken to be the constant c in the last inequality. Everything is prepared for the following result due to Maurey [186] - who in this way generalized a result of Kwapieri's [159]:

324

24. Stable Measures

THEOREM: If F as a Banach space of cotype q and E > 0, then

and for every Banach space E

PROOF: By Saphar's duality result in 20.19. it is enough to prove the first statement. Take E > 0 and choose 1 < s < q < r < q + E which satisfy 1 - l / q 5 l/s- l / r . Then, by Ex 11.21. and the proposition

now

idF

E

vq,l(by the remark) concludes the proof. 0

Note that for q E [2, m[ the space A!, has cotype q (see 8.6.), hence

for all E > 0. For q = 2 the little Grothendieck theorem allows even E = 0; that this is also true for arbitrary range spaces of cotype 2 will be shown in section 31. 24.8. We end this section with a remark concerning averaging. It is not too difficult to see that the average with respect to the LBvy measure dominates the Rademacher average: "stable" type implies Rademacher type (see Ex 24.5.). The converse is true up to an 6 :

PROPOSITION: Let 1 5 s < q < p 5 2 and E be a Banach space of type p. Then

.

for all t l , ...,zn E E and c := T P ( E ) ~ ; f ~ q , r ~ p , q

PROOF: Take p the Rademacher measure on D = (-1, 8.5.). Since the LBvy measure pi is symmetric, one has

and &k the projection (see

24. Stable Measures

325

Using the type p inequality and the LQvyembedding, one obtains

Changing the order of integration and applying again the LQvy theorem, the relation (u, ( h k ( w ) ~ ~ Z k = ~ ~( )w) , ( h k ( v ) l l z k l l ) ) shows that the last line is

by the definition of cp,q. 0

Exercises: Ex 24.1. A matrix A E t(P,d?’)is called positive definite if (AzJz)2 0 for all z E fJ”. US^ diagonalization of A to show that the Schur product (ai,jbi,j) . ._ ._. of two positive definite matrices ( a i j ) , ( b i , j ) E C ( P ,Q?’) is positive definite. Hint: ai,j =

CZ1At(eiIyt)(ej IVf).

-

Ex 24.2. If 1 5 g < p < 00, then there does not exist any positive isomorphism .fp L q ( p ) . Hint: 11.2. and P,(& 2 nl/q. -A$)

Ex 24.3. Show that for every m E 4v and 0 < g < p < 2 there is a probability measure p and an isometric embedding .fp(y) ~1L q ( p ) . Hint: Follow 24.4.. Ex 24.4. Formulate and prove proposition 24.8. for operators of type p.

326

24. Stable Measures

Ex 24.5. (a) Show for p, 8 E]O,2] that for the Rademacher measure p and the patable LBvy measure pp the averages satisfy

for all 21, ...,znE E. Hint: As in Ex 8.9.. (b) A Banach space has stable type q E [1,2] if it satisfies the inequalitiy of proposition 24.8. for some s E [I, 21. Show with Kahane’s inequality 8.6. for the Rademacher averages that stable type q implies type q. Ex 24.6. (a) If 2 < p < q < 00 or: p = 2 and 1 < q < 00, then every compact operator from an Z;-space into an Ei-space is q-nuclear. Hint: 24.6. and 22.4.. (b) Let 2 < p < q < 00. Then every compact operator from an EG-space into an 2;apace is q-nuclear. Hint: 24.7. and 22.4.. Ex 24.7. Let I : 1 2 L+ Lp(-y) be the embedding Iek := gk and denote for the complex case Ioek := gk. Take p , q E]l,oo[ with l / p + l / q 2 1. (a) Show that the mapping l a

(a)

dp,q

I

1:

L,,(T)

Lp(y)

:= IIgiII~~IIgiIIq/,KIIgiIIp/,~ satisfies Lp,q(idfAId p , q and

Qp,q

IdP,PP

l2

is just Ilgll[f,Kidta.

*

It can be shown that this constant is actually the best one (see Pietsch [214], 22.1.4. and 22.1.1.) and that dp,q= bp,bq/ (see 12.8.) in the real case. With the formulas given in Ex 11.24.(a) for the moments of the Gauss measure one can show, however, that in the complex case in general dp,q< bp/bq, (take p = 2 and q = 4/3 for an example). Ex 24.8. (a) Take p, q E]O,2[ and P < min{p, q } . Then for all a1, ...,a n E K

Hint: Express the first integrand in terms of p q , the second in terms of pp and use F‘ubini’s theorem. (b) For 0 < r < q < p < 2 and a = (a1,...,a,)E K” prove the inequalities

Hint:

Ilalltq= (ck(s[ a k h k [ r d ~ p ) q ’ r ) l ’ q C ~and , ~ the continuous triangle inequality.

Let q E [2,00[ and E > 0. Use theorem 24.7. to show that E ( l l , l q ) = ‘J3q+c(ll,t!q)(which is the statement of corollary 24.6.). Hint: The dual of the Lkvy embedding into L1 and lifting.

Ex

24.9.

327

25. Composition of Accessible Operator Ideals

25. Composition of Accessible Operator Ideals

The main topic in this section is an extremely useful result concerning the composition of operator ideals: If ?2lo ?B c C, then - under certain accessibility conditions - C* o Q C ?B* holds. Various applications of this result are given. It turns out that the theory of accessible operator ideals is quite smooth.

25.1. It will be crucial to exploit the accessibility of operator ideals, defined in section 21: A quasi-Banach operator ideal was called left-accessible if (in brief) each operator T E C ( E ,N) factors

i.e. a ( E ,N)

o s ( E , N), and right-accessible if

i.e. Q ( M ,F) f 5 o g ( M , F). The aim of the first part of this section is to give a characterization of accessibility in terms of the minimal kernel 3Lmi" of Q.

LEMMA:Let (a,A) be a p-Banach operator ideal. (1) For each Banach space E the following are equivalent: F ) f o r all Banach spaces F. (a) 50a ( E ,F) amin(E, (b) Q o z ( E ,N ) f %(E,N ) f o r all N E FIN. ( 2 ) For each Banach space F the following are equivalent: (a) Q o z ( E ,F ) f Qmin(E, F ) for all Banach spaces E . (b)

50O ( M ,F) A Q ( M ,F ) for

all M E FIN.

PROOF:Only (1) will be proven, (2) follows along the same lines. Assume (a); then Q 0 3 ( E ,N )

f 5 0 a 0 5 ( E ,N ) f

!2P"(E,N ) f

3 0 Q(E,N )

% ( E ,N )

25. Composition of Accessible Operator Ideals

328

- the first and last equalities being obvious consequences of N E FIN. Conversely, take T E 3 o %(ElF ) ; it is necessary to show that T E Qmi" with the same norm. There is a factorization

with S E 5. For S n E S(G,F) with IIS-Snll+ 0 define Tn := S n 0 R and fix n,m E A'. For N := (Sn - Sm)(G) E FIN(F), by (b), n m i n ( E ,N ) A a o Z ( E , N ) &. %(ElN ) and therefore,

Amin(Tn- Tm : E 1

--

=A((Sn - S m ) R : E

F ) 5 Amin((&- Sm)R : E N)5

llSn

-

N)=

- SmllA(R) .

It follows that (Tn) is an Amin-Cwchy sequence, which clearly converges in 1: to T and hence in Am'";moreover, since amin is q-normed for some q (see 9.10.), Amin(,) = lim Ami"(Tn)5 lim JJSnJJA(R) = IlSllA(R) 5 n+oo

n-w

I (1 + €1[II II0 A1 (TI which shows that Amin5

I

[[I [Io A].The converse inequality is trivial.

0

25.2. Corollary 16.9. about the factorization of finite rank operators in spaces with the metric approximation property gives immediately the

COROLLARY: Let (%,A)be a p-Banach ideal and El F Banach spaces. (1) If E' has the metric approximation property, then 21min(E,F ) = f o %(ElF )

isometrically.

( 2 ) If F has the metric approximation property, then

%min(E, F ) = !2l o $(El F )

isometrically.

In other words, in Qmin = 5 o % o T ( E ,F ) one 3 can be cancelled if E' or F has the metric approximation property. If % is accessible, the same is true - and even more:

PROPOSITION: Let (a,A) be a p-Banach operator ideal. (1) The following statements are equivalent: (a) (%,A)is left-accessible.

25. Composition of Accessible Opemtor Ideals

(b)

329

amin = 50IZL isometrically.

(c) 2lmin(E, N ) %(ElN ) for all (El N)E BAN x FIN. (2) For right-accessibility the following characten'xations hold: (a) (a,A) is righi-accessible. (b) amin = 2l o 5 isometrically. (c) Qmin(M,F )

a ( M ,F ) f o r all ( M , F ) E FIN x BAN.

This result was motivated by an idea of Lewis (see [169], lemma 1).

PROOF: Again, only (1) will be proved. Lemma 25.1. gives the equivalence of (b) and (c) - and (a) implies (c) by the very definition of a being ieft-accessible. Assume (c); in order to show that (a) holds, observe that

A

o $(El N). Denote, for the It follows that it is enough to show that 2l o z ( E ,N ) moment, by F and F the operator norm 11 11 on $ and 5. Clearly, A O F 5 A o F. Conversely, take T E E(E, N) with A o F(T)< 1, choose a factorization T = S o R -+ 0. Since with A(S)F(R) c 1 and R,,E 5 ( E ,G) such that llR -

it follows that ( S o &) is Cauchy in the complete (see Ex 9.18.) space % o $(El N) with limit T. Since Q o 5 is q-normed for some q (this has the same proof as 9.10.),

A o F(T)= n+m lim A o F(S o R,)5 lim A(S)F(&) = A(S)F(R) < 1 . 0 n-+m In particular, if

(a,A) is accessible, then

holds isometrically (see also Ex 25.1.). This is a very useful fact, see for example Ex 25.2. for an application to p-compact operators. - 1 -

25.3. Since 505 = 5 (shown in 22.1.), the relations

hold isometrically, and hence COROLLARY: Minimal p-Banach operator ideals are accessible.

25. Composition of Accessible Opemtor Ideals

330

This is a rather interesting result - since it will be shown in section 31 that there are non-accessible maximal normed ideals. It is easy to see (Ex 25.4.) that the ideal 92 of nuclear operators is not totally accessible. 25.4. The following result allows one to obtain various composition formulas from a given formula of the form rzlo % c C. The proof will reveal that the “cyclic composition theorem” is nothing but a condensation of trace duality: this is why it is natural that some accessibility assumptions have to be made. Recall from 17.9. that

Itr(S o T)( 5 A(S)A*(T)

whenever S E C ( M ,N) and T E 2 ( N ,M) for finite dimensional spaces.

CYCLIC COMPOSITION THEOREM: Let

(a,A), (’23, B) and

(C, C ) be quasi-Banach

operator ideals with

a023C C

and

C5c[AoB]

for some c 2 0 . (1) If D is right-accessible (or C* left-accessible), then C* 0 % c W

( 2 ) If

‘1)

and

B*5 c [ C * o A] .

is left-accessible (or C* right-accessible), then

23 o C* c %*

and

A* 5 c [ B o C * ] .

PROOF: To see (1) fix R E Q ( E , F ) and S E C*(F,G). To calculate B*(So R) take ( M L) , E FIN(E) x COFIN(G) and U E 2(G/L, M) with B(U) I 1 : E

RE 2I

’F

SEC*

’ G

a is right-accessible, take 2 := N E FIN(F) and R, E 2 ( M , 2 ) such that A(R,) 5 (1 &)A(R)and define So := Q o S o 1; . If C* is left-accessible, choose 2 := F/K with K E COFIN(F) and So E C ( Z , G / L )with C*(So) I (1 + & ) C * ( S )In . either case there are R, and So with If

+

25. Composition of Accessible Opemtor Ideals

331

and QSRI = S,R,. It follows that

Jtr(QSR1U)J = Itr(S,R,U)l C(R,U)C*(S,)I < cA(R,)B(U)C’(S,) 5 c ( l + c)~A(R)C*(S), which shows that B* 5 c[C* o A]. The proof of (2) is much the same. 0 Note that applying the cyclic argument ( 1 ) twice gives the statement (2) - but under stronger hypotheses than stated. We shall also use the term “rotating” for applying the cyclic argument. If % is right-accessible, then 2l o 1:c % implies

2 l * o 2 l ~ J and

I
- a formula which generalizes the one from 17.19. (proposition 1) to arbitrary quasiBanach ideals. Oertel [194] showed that right-accessibility is necessary:

PROPOSITION 1: Let (%,A) be a maximal normed operator ideal. Then (%,A) is right-accessible if (and only i f ) %*o%C

J

IIA*oA.

and

PROOF: It will be shown that the condition implies that 2 l o 3 = amin, which is enough by proposition 25.2.; for this fix T E %(E,F). Then for every Banach space G the map (FGatG‘)‘ = 2l’(F, C’) S has norm

-

J ( E ,G”) = (E&G’)’ SOT

5 A(T), and hence

which implies that

has norm 5 A(T) as well. But O(S) = ToS, which shows that the norm inequality holds; the other inclusion is obvious. 0

3103C amin and that

The cyclic composition theorem also gives results for the composition with minimal kernels; this is a consequence of the following easy

PROPOSITION 2: if (31, A), (?B, B ) and (C,C ) are quasi-Eanaeh operator ideals with Q o S

cC

and

C

5 c[AoB]

25. Composition of Accessible Opemlor Ideals

332

for some c 2 0 and ( C , C ) is accessible, then amin

a

0

'15

23min

cmin

and

c Pin

and

Cmin5 c[Amino B] CminI c[A o Bmin].

PROOF: If (C, C ) is left-accessible, then Pin = 3 0C by 25.2., and hence,

The other relation follows in the same way. 0 25.5. An immediate application is the multiplication table for absolutely p-summing

and pnuclear operators: In 11.5. it was shown that

-

if r,p,q E [ l , ~and ] 1/r = 1/p + l / q . Using the facts that PP gi) is totally accessible, ?; = gp# gp/ is accessible and Jp c PP and = 'YIP, one obtains by cyclic composition (and the proposition about minimal operator ideals) the PerssonPietsch multiplication table from [209]: For 1/r = l/p l / q

-

JFn

+

(read: 'YIP 0 Pq C

nr ... ).

25.6. There have already been examples given of operators T having the property that S E Q implies TS or ST E !€? for some ideals; Grothendieck's inequality (in operator form 23.10.) had this flavour: For rzL = 2 , (for some 1 _< p 5 2) and T = idt- the

operator S = ST was in Q = '&. For quasi-Banach operator ideals (a,A) and ('13, B) define for T E C

I I

A-' o B(T):= sup {B(ST) S E a, A(S) 5 1 and ST defined} E [0, m] B o A-'(T) := sup {B(TS) S E Q, A(S) 5 1 and T S defined} E [0,oo] (with the convention B(U)= 00 whenever U $! 23). The quotient ideals 2l-l o 23 and 2 3 o Q-' consist, by definition, of all operators T such that A-l o B(T) < 00 and B o A-'(T) < 00, respectively.

25. Composition of Accessible Operator Ideals

333

It is obvious that (24-'0B,A-'oB) and (%024-', BOA-') are quasinormed operator ideals - and the completeness is also not difficult to show. Note (Ex 25.9.) that T E 24-1 o % if (and only if) ST E 93 for all S E 24. In other words, 24-' o B is the largest ideal C with the property that 24 o C C B. If B is normed, then B o 24-1 and 24-' o B are normed as well. Since, obviously, 240(24-'0B)c B and (B024-')024c 2 3 , the cyclic composition theorem gives (ra-~oB)oB*

c24'

%*o(Bo%-')C2l*

if B* is right-accessible if B* is left-accessible,

and hence the

COROLLARY: Let 24 and B be quasi-Banach operator ideals, '23' accessible. Then 24-l

oB

c a* o (%*I-' and B o %-' c (B*)-' o 24' .

Saphar's result in 20.19. is a special case: for example,

applied to T = idE gives the equivalence of (a) and (c); the other one follows using the fact that ?pp(tw,.) = 7p(loo,.). 25.7. This already shows the power of these formulas concerning quotient ideals. Another quite useful result is the

QUOTIENT FORMULA: Let (24, A) and (%, B) be quasi-Banach operator ideals. (1) If 24 is accessible (or: 24 and 93 right-accessible), then (240B)*=%*024-'

isometrically.

(2) If 23 is accessible (or: 24 and 9 left-accessible), then

(ra 0 23)* = %-lo

24'

isometrically.

Recall that 24 is right-accessible if it is injective - and left-accessible if it is surjective.

PROOF:The formulas will be a consequence of various applications of the cyclic composition theorem. Assume first that 24 and B are accessible. Rotating 24 o '23 C 24 o % with the cyclic composition theorem gives (24oB)*or#c%* andBo(O#o23)*CIZL*,

25. Composition of Accessible Operator Ideals

334

which means (a o %)* c %* o Q-' and (a o %)* c 93-l o a* (the norm-inequalities will not be written). Conversely, it is obvious that (%* oar-')

oQ

a* .

C %* and 230 (W' oQ*) C

Rotating this yields 24 o %** c

(Wo %-')*

and 24"

o

B c (B-'

o

a*)*.

Since C c C**, this easily gives the other inclusions. Now the assumptions will be weakened, using the facts that Qmin is always accessi- the latter since C* is determined by the components ble and that C* = (Pin)* C ( M , N ) = C m i " ( M , N ). Note first that

- (rn 0 %)mi" = 5 0 Q 0 23 0 5 L 50 Q 0 50 23 0 5 = 50 Q 0 5 0 5 0 93 0 3 = - amin s m i n if !2l is left-accessible or 2 3 is right-accessible by proposition 25.2.. Therefore, (Q %I* = ((a

B)min)*

= (amin

%min)*

= %*

(amin)-'

=(~min)-l

by the first part of the proof. Clearly, %* o (amin)-' 3 %* o Q-' and

o Q*

3 23-l o Q*

To show equality take T E ? o (Omin)-' .I?* and S E a with A(S)

.

5 1. Consider

T

SEP

E - F - G

M(:

G/L

B(U)
WEB

Then Amin(S,) = A(S,)

5 1 if Q is right-accessible

and hence the inequality

Itr(UQTS1)l = Itr(UQTSo)l5 B(U)B*(QTS,) 5 B* o (Ami")-'(T) shows that %* o (Qmin)-'

c %* o Q-'. The same argument implies that (Wi")-'o

if %3 is left-accessible. 0 Note that the proof showed that

a* c 23-1 0 a*

Q*

335

25. Composition of Accessible Operator Ideals

whenever 2l is left-accessible or B right-accessible. Moreover, it can easily be seen from proposition 25.2. that (Brnin)-l

%*

(arnin)-l

A (%rnin)-l A (%*)rnin

if a* is left-accessible if %* is right-accessible.

(a*)rnin (arnin)-l

25.8. The quotient formula as well as the following interesting characterization of

(a o %)* in terms of the associated tensor norms appeared first in [41].

(a,A) and (%, B) be maximal normed operator ideals with associated tensor norms a and p, respectively. I f p is accessible, then the following statements are equivalent for each operator T E C ( E ,F ) :

THEOREM: Let

-

(a) T E (a 0 % ) * ( E F , ). (b) idc, 8 T : C, @a E C, @pF is continuous for some Johnson space C,. (c) idG @ T : G @a E G @pF as continuous for all Banach spaces G . In this case,

PROOF: Lemma 17.16.shows the equivalence of (b) and (c). The representation theorem for minimal operator ideals gives the following diagram

21rnin(G,E ) S

Go

(%*)m'"(G,F )

AAP+

ToS

I

which clearly commutes. Hence, if (c) holds, then

T E (%*)rnin o (amin)-' = (a o %)*

-

-

by the quotient formula (since % /3 and %* p' are accessible). Conversely, assume (a) and take G = C; which has the approximation property. Then the vertical maps are isometries, and hence (b) also follows from the quotient formula. Moreover, SUP llOcll= G

(B*)rnin o (Arn'")-'(T) = (A o B)*(T)

which gives the relations about the norm. 0

336

25. Composition of Accessible Opemtor Ideals

-

This result generalizes the two tensor product characterizations of Q given in section 17 since Q = (2 o a*)*= (24. o 2). and Z 6. Checking the proofs of the cyclic composition theorem, the quotient formula and this last characterization, reveals that accessibility was always used to put in an additional finite rank operator at an appropriate place. The results at the end of section 16 about the factorization of finite rank operators show that this can be also done if some metric approximation property is at hand. It becomes clear - for example - that for fixed Banach spaces E and F the assumption of accessibility in the last characterization of T E (24 o B)*(E,F ) can be replaced by the assumption that E has the metric approximation property. We omit the details. Some results of Kwapieri's concerning subspaces/quotients of Lp-spaces can be proved using what was developed so far. RRcaiI from 20.12. that for a maximal normed operator ideal Q with associated tensor norm a,the ideals associated with the projective associates a/ and \a were denoted by a/ and \%, respectively. They were characterized by factorization (into the bidual) by 25.9.

To formulate the following lemma in an adequate way recall from Ex 22.6.the regular hull (a,A)reg of a quasi-Banach operator ideal (a,A)

areg(E, F) := {T E C ( E ,F ) I IEF o T E % ( E ,F")} A " ~ ( T : E + F ) : = A ( K ~ OE T - F: " ) . It is straightforward to show that this is a quasi-Banach operator ideal, that (a,A) is Areg) = (a,A) and that Qreg(M,N) % ( M ,N) for finite dimensional regular if (are#, spaces. In particular, ((a,A)reg)* (a,A)*

A

A

for the adjoints. Since F / L = F"/Loo if L E COFIN(F), it is easy to see that maximal quasi-Banach operator ideals are regular (see also 17.8.).

LEMMA:Let (a,A) and (B,B) be maximal normed operator ideals such that the composition ideal (a o % , A o B) is maximal and normed as well. Then the following relations hold isometrically: (1) \(a 0 B) = (a 0 (\%))reg. (2) (ra 0 B)/ = ((a/)0 %)reg. (3) \(a 0 BY = ((a/)0 ( \ B N r e g .

25. Composition of Acceasible Operator Ideals

337

PROOF: For (1) it is enough to show that \(% o % ) ( E , F f t )= a o (\%)(E, F t f )- but this is immediate from the factorization. (2) and (3) follow in the same way. 0 To apply these formulas to Kwapieri's factorization theorem 9p,g = pPaIo ppfrom 19.3.,observe first that 20.14.implies that 3inj \vp

--

gp\ = 9;' \$I

- pp

= 3;'

= (gp'\)* = g;* = gp

- DP .

An immediate application of the quotient formula now gives a result on the injective and surjective hulls of the ideal 2p,g of ( p ,q)-factorable operators - a result which at first glance might terrify the reader.

PROPOSITION: The following relations hold isometrically: (1) 2,,,= (,$,I 0 VP')* = JP 0 (py)-'= p ;' 0 3 y .

(2) 2Ej = (pya'0 JP')* = pp0 (

p y) - I

= 3;'

0

(3) 2 ; ; ; = ( 3 y 0 pp')* = JP 0 (JqdY/)-' = p;1

0

(4)

22; aur = ( 3 y 0 gP')* = !pP 0 ( 3 p y = 3;'

3 y .

py. 0 pp1.

PROOF:All ideals involved are accessible by the results in section 21; therefore the quotient formula can be applied and all four formulas become a simple consequence of the lemma; for example, the last one:

As a special case, take the ideal 2, = 2p,p~ of p-factorable operators

(see section 18). Simple arguments show that (1) T E if and only if T factors through a subspace of some L p ( p ) . (2) T E 2rrif and only if ~ C oF T (or, equivalently, TI') factors through a quotient of some L p ( p ) .

22'

25.

338

Composition of Accessible Operator Ideals

tTrur

(3) T E if an d only if T factors through a subspace of a quotient (or, equivalently: a quotient of a subspace) of some Lp(p).

.Cp,2y,2rr2Prur

(See also Ex 20.8.). Thus, and are operator ideals 24 with the property that every T E % factors through some G E space(%) (or at least IEF o T or 57"). So one may concentrate on space(%) when investigating 24. The following result is due to Kwapieli [160]. Recall that space(.Cp) constists exactly of all e:-spaces by 23.2..

COROLLARY: Let E be a Banach space and 1 5 p 5 00. Then in each of the statements ( 1 ) - (4) the statements (a) - ( d ) are equivalent: ( 1 ) (a) El'(= E i f 1 < p < 00) is isomorphic to a complemented subspace of some LP ( P ) .

(b) % y O l ( * , E ) c 3p(', E ) . (c) '.Ppf(E,*) C J$"l(E, .). (d) Gad;, E 3 G @gF E is continuous for all Banach spaces G. (e) E E space(&).

(2) (a) E is isomorphic to a subspace of some L p ( p ) .

9pa1(*,

(b) E ) c (Pp(', E). (c) 3,#(E,*)C 3$'"(E, *). (d) G @df,E G @g;# E is continuous f o r all Banach spaces G.

-

(e) E E space(Cpj).

(3) (a) El'(= E if 1 < p < m) is isomorphic to a quotient of some L p ( p ) . (b) 3Ta1(*l E ) C 3p(er E ) .

-

(c) Yp@, .) c P;Yal(E,*). (d) Gad, E G agrE is continuous for all Banach spaces G. (e) E E space(CY'). (4) (a) E is isomorphic t o a subspace of a quotient of some L p ( p ) (or equivalently: t o

a quotient of a subspace).

(b) 3 y ( * ,E ) c Vp(*1 E ) . (c) 3,1(E, c P3,dY'(E, (d) G@d, E G @g;f E is continuous for all Banach spaces G. a)

-

a).

(e) E E space(Cy ,",).

a(.,E ) and idE E 24-l

PROOF:

i d E E 240 %-' is equivalent to %(., E ) C

alent to

Q(E,-) c % ( E ,.). Hence, the equivalences (a) - (c) follow immediately from

o % is equiv-

25. Composition of Accessible Operator Ideals

339

the proposition. The equivalence of (c) and (d) is, in each case, a simple consequence of the representation theorem for maximal operator ideals. 0 Clearly, the equivalence with (d) could also have been deduced from theorem 25.8.. 25.10. The Chevet-Persson-Saphar inequalities (15.10.) say that

- and there is even equality if E = Lp(v).The following result describes conditions on E which imply that equality holds in some of these inequalities; it is due to Kwapieh [160] and Gordon-Saphar [87].

PROPOSITION: Let E be a Banach space, 1 I p 5 00 and p, a measure such that Lp(po) is infinite dimensional. Then in each of the statements (1) - (3) all statements are equivalent:

(1) (a) E i s isomorphic to a subspace of some Lp(v).

(b) diI I g;l I cd;, on L p ( p , ) 8 E f o r some c 2 1. ( c ) di, 5 Ap 5 cd;llon L p ( p o )8 E f o r some c 2 1. E if 1 < p < 00) is isomorphic to a quotient of some Lp(v). (b) dp I gp 5 cdp on Lp(po)8 E f o r some c 2 1. (c) Ap 5 gp I cAp on L p ( p , ) 8 E f o r some c 2 1.

(2) (a)

Ell(=

(3) (a) E i s isomorphic t o a subspace of a quotient of some L P ( v ) (equivalently: quotient of a subspace).

(b) dp 5 g;, (c) Ap 5 g;l

5 cdp on Lp(p,) 8 E f o r some c 2 1. 5 cA, on Lp(po)8 E f o r some c 2 1.

(d) dp I A,

I cdp on Lp(po)8 E f o r some c 2 1.

PROOF: The statements (d) in the last corollary show that (a) always implies (b). The Chevet-Persson-Saphar inequalities show that (b) implies (c) (and (d) in case (3)). For the converse the fact that d, and g i I are the tensor norms closest to Ap will be used, see 15.11.. Assume, for example, (3)(d): then (by the version of 15.11.with fixed E) g;l cdp on G 8 E for all G - this is statement (4)(d) of the last corollary. The remaining implications follow in the same way. 0

5

A short look at the constants shows that for c = 1 one has to replace “isomorphic” by “isometric”. Operator forms of this result will be given in 29.9..

340

25. Composition of Accessible Operator Ideals

If E is isomorphic to a quotient of a subspace of some Lp(c(),then the right-tensor norm Ap on Lp(v)@ E is equivalent to a tensor norm (namely dp and g;, by (3)(a) of the proposition). It follows that

is continuous for all A E .C( Lp( q ),L, (v)).

COROLLARY: Let Lp(p,) be infinite dimensional and E a Banach space. ifthere is a tensor norm a such thaf

holds isometrically, then E is isometric to a subspace of a quotient of some L p ( v ) .

PROOF:By the Chevet-Persson-Saphar inequality and the remark after proposition 15.11. it follows that on Lp @ E , I ff I dp 5 Ap Ig;l hence, a = dp on Lp @ E. Now (3) of the last proposition gives the result. 0 $1

Obviously, one can replace "isometric" by "isomorphic". An operator form of this result will be given in 28.4..

25.11. The description of amin, as given at the beginning of this section for accessible %, was crucial for the results which were obtained. It is obvious that it might also be : advantageous for the investigation of the minimal kernels of 218". and

PROPOSITION: Let (Q, A) be a p-Banach operator ideal and E , F Banach spaces. (1) If F has the approximation property or: if E' has the approximation property and (Q, A ) is right-accessible, then (%8"r)min(E, F ) = (Q"'in)8ur(E, F)

isometrically.

(2) If E' has the approximation property or: if F has the approximation property and

(a,A ) is left-accessible, then (ginj)min

( E ,F ) = ( + p i n ) i n j(E, F)

isometrically.

The inclusions c are true without any additional assumption - but both are strict in = (3Tin)inj general: For 2 < p < 00 Reinov [228] constructed an operator To E which is not even approximable; dualizing the factorization of I o T , : E + F L+ &,(I?), (which is in 3 03po z ) , it is obvious that the dual operator Ti is in ((3~o')min)8ur -

fly

but Ti 4 5 =

pa', so T: is an example showing that (1) also does not hold in general.

25. Composition of Accessible Operator Ideals

341

PROOF: The details concerning the norms will be omitted - they will be obvious from the constructions. (1) Take T E (Qr"r)min(E, F). Then there is a factorization

and the lifting R, can be chosen to be compact=approximable by corollary 2(2) in 3.12.. It follows that

hence T E (2fmin)'"' . Conversely, take T E (2fmin)'"'(E,F) and factor To QE = S o Too R as follows:

Po E Q'"' and R,,S, are compact. If E' has the approximation property, then R, E 3 and T E Q""' o 5; if, additionally, Q is right-accessible, then the ideal 2fau'

Now

is totally accessible (see Ex 21.1,(e)) and hence Q " ' o 5 = by proposition 25.2.. If F has the approximation property, then So E 5 and therefore T E 5 0%'"'. Since a'"' is left-accessible, proposition 25.2. again gives T E (Q#r"r)min. (2) Any T E (Qinj)min(E, F) factors

G Clearly, I o S E IZL and hence I o T E % o ~ ( ( E , l , ( B p ) )= Qmin(E,Coo(B~~)) by corollary 25.2. (an argument like this could have been used in (1) as well). On the other hand, if T E (Qmin)inJ(E, F), then it can be factored aa follows:

342

25. Composition of Accessible Operator Ideals

Clearly, Po E gin’. Now the same type of arguments as in part (1) (using 25.2. and Ex 21.1.(d)) give T E (Qinj)min. 0

Exercises: Ex 25.1. (a) A p-Banach operator ideal IZL is accessible if and only if Q o 3 = 5 o Sf holds isometrically. (b) Let Q be a maximal normed operator ideal. Show that Q o 3 = 3 o Q if and only if%* o $ = T o Q * . Ex 25.2. Every compact operator with values in an 2;-space is pcompact (see 22.3.). Hint: 2, o 5. This result was already shown in Ex 23.9.. Ex 25.3. Each p-Banach operator ideal Q of the form Sf = 5 023 is right-accessible, each 23 o 3 is left-accessible. Ex 25.4. Let a be a finitely generated tensor norm and Q its associated maximal operator ideal. The canonical metric surjection E‘&F -+ Qmin(E, F ) is injective for all Banach spaces E and F if and only if Qmin is totally accessible. Hint: Recall M’ N A gmin(M,N). In particular, 5 is totally accessible, but 3 is not. Ex 25.5. Grothendieck’s inequality in operator form 23.10. says that 21 o 2, C 9 2 and 2 2 o 21 C TI. (a) Use the cyclic composition theorem to show that these two results are equivalent. , c 9 2 is false. Hint: Rotate. (b) Show that 22 o 2 Ex 25.6. Let 1 5 p 5 2. Use Grothendieck’s result 2, o 2 , C !J32 (see 23.10.) and the cyclic composition theorem to show that: (a) ! J ? 2 0 2 , C !J?l;in particular, every abolutely 2-summing operator on 4, is absolutely l-summing. (b) 2,o T2 c 9 p t . Ex 25.7. Let 2 5 p < q < 00. Show that: (a) The dual of an absolutely q’-summing operator T E C ( $ , F ) is absolutely 1summing. (b) Every absolutely q‘-summing operator T E C ( E ,el) is p’-dominated. Hint: Rotate Saphar’s theorem 24.6..

25. Composition of Accessible Operator Ideals

343

Ex 25.8. For 1 5 q < p 5 2 every absolutely q-summing operator E 4 l , is p dominated. Hint: Rotate Maurey's theorem 24.7., which implies that 2,)o C , c 3,) by 8.6.. Ex 25.9. Prove that 24-l o '13 consists of all operators T such that ST E % whenever S E 8, and the analoguous statement for '13 o 2l-l. Hint: l,(Gn). Ex 25.10. If 8 is accessible, then Q* = 3 o !2l-'. This is a sort of converse of 17.19.. Hint: Q = IZL o 2.Compare this proof with the one in Ex 17.15..

Ex

25.11. If 8 and

a* are accessible, then

Ex 25.12. Let 2l be a regular quasi-Banach ideal and El F Banach spaces. If Q is right-accessible and E' has the approximation property, then: T E 12Lmi"(E, F ) if and F"); moreover, A m i n ( ~ , o T=) Amin(T). Hint: amin(E, F)= only if KFOTE ami"(E, 8 o z ( E ,F) and im(R) C S-'(F) for a factorization K F O T= S o R E 203.For Banach operator ideals Q there were results of this type in 22.8.. Ex 25.13. Let 1 _< p 5 00 and E be a Banach space. Prove the following results (Takahashi-Ohazaki [265]): (a) E is isomorphic to a subspace of some L p ( p ) if and only if %,,(El C ffl,p)(E,-). a)

(b) E" is isomorphic to a quotient of some L p ( p ) if and only if ff~,~(.,E) C 'YIP(., E). Hint: Corollary 25.9.(3), (a) equivalent to (d).

Ex 25.14. Reprove Kwapieli's test 11.8. for an operator to be absolutely psumming by using the quotient formula. Hint: CP,o gP,. Ex

25.15. Let ( % , A )and ( B I B ) be p resp. q-normed operator ideals and G a Banach space. Use proposition 25.2. to show that: (a)

If

(a,A ) is accessible and !Bmin(E,G) %(ElG)for all Banach spaces El then ('13 o Q)min(E, G ) = 93 o Q(E,G)

holds isometrically for all Banach spaces E. (b) If ('3,B) is accessible and Qmin(G,F)

A %(G,F) for all Banach spaces F, then

('13 o 21)min(G,F ) = '13 o %(G,F )

holds isometrically for all Banach spaces F.

Ex

25.16. If IZL, %, C and 9 are accessible quasi-Banach operator ideals such that C c 9,then 9*o a o ' 1 3 c C*, C o g * 0 % c %* and '230 C O D * c %*. Hint: Ex 21.2.. Qo% o

344

16. More About

Lp and Hilbert Spaces

26. More About Lp and Hilbert Spaces

This section provides some additional information about the behaviour of ap,q, its associated operator ideals and Ap on Lp and Hilbert spaces. In particular, it will turn out that for Hilbert spaces the situation is quite simple.

26.1. Most of the results in this section will be stated for spaces E = .tponly. The Zplocal techniques easily show that these statements also .hold for arbitrary 2;-spaces E - in particular, for Lp(p)-spaces: in this latter case the constants are the same as for t,. Recall first from 12.8. that aPlq 5 bq8bpt w 2 (if p, g €11, m[),where br is the constant from the Khintchine inequality 8.5..

PROPOSITION: Let p,q E]l,oo[with l / p + l / g 2 1 and r, s E [1,2]. Then 6

<

5 KGbq#bp#& 5 * 5 KGbqfbpfaL,q

on fr @ t d on @ f8,.

(Yp,q

f,f

PROOF: Grothendieck’s inequality (in tensor form) gives, by duality, that w; 5 K G E on el @y, and hence on Ll(p)@L ~ ( Y )Since . w 2 5 wi holds (see Ex 19.5.), the tensor norms w2 and 6 are injective and t r LI L 1 ( p r ) by the Ldvy embedding 24.4., one gets

for a = w2 and a = E. It follows that w 2 5 KGEon 1, @ 4.Since ap,q5 12.8., the first result is proven. The second follows by dualization. 0

bqfbptw2 by

-

A more general result due to Pisier will be given in section 31: E ap,qon tensor products of cotype 2 spaces (p, q # 1). In terms of operator ideals this proposition is a result of Lindenstrauss-Pelczyriski [173].

COROLLARY: Lei 1 5 s 5 2 5 r 5 00 and p,q ~ ] 1 , 0 0 [with l/p+ l / q 2 1. Then

2(4-,!8) = .Cp(er,C) = Z p , q ( e r , L ) and

-

Lp,q(T :t r &) IK ~ b q # b p # ) J T.J l I n particular, every T E L!(lr, 18) factors through a Hilberi space. This is immediate by the representation theorem for maximal operator ideals. In Ex 26.2. an “adjoint” form of this result is given; see also Ex 26.1.. For factorization through Lp if 1 5 s 5 p 5 r 5 00 see 26.5..

26. More About

L, and Hilbert Spaced

345

26.2. Operators tr -+ e, do not automatically factor through a Hilbert space if the condition 1 5 s 5 2 5 r 5 00 is violated: There are three cases to consider. First, let r, s E [l,2[ (the case r, s €12, co]follows by duality); if C ( t r ,J?,)= &(&, t,) were to hold, then there would be a constant c such that

But this contradicts the following result - which is itself of interest (see Ex 26.4.):

which gives the reverse inequality. 0 To see that C(tr ,1,) # (see Ex 4.3.)

&(4. ,&) if 1 5 r < 2 < s 5 00, consider the Fourier matrices

The A, are unitary and hence "An : 4 --* 4 11 = 1; moreover, llAn : --* = n-ll2 obviously holds. The interpolation theorem of Riesz-Thorin (see Bergh-Lofstrom [ll]) gives 1 1 : ~ ~ ~ ~ 1 1n-'/WP' 1

c-

if 1 5 p 5 2, which easily implies

for 1 5 r < 2 < s I 00. Clearly, the adjoint matrix A: has the same properties as A n ; moreover, An o A: = id since An is unitary. Grothendieck's inequality (see 17.14.) gives P~(A:,:~-~;)~KGIIA:,:~",-~;II~KG.~

346

26.

More About Lp and Hilbert Spaces

and the little Grothendieck theorem (11.11.) implies that

Since !& o !& c 31 (see 19.4.),

-

and hence (Kc K ~ G ) - '5 L2(An : 4 there is no constant c such that for all n

e)5 L2(An :

-+

c).It follows that

and hence, by the closed graph theorem, C(trll#)# f!2(lrTt,) if 1 5 r < 2 < s 5 00

26.3. Let

.

E C ( L , ( p i ) , Lp(vi)); when is

continuous? If one of the operators is positive, then 112'1 @Tzll= 112'111 llT2ll by theorem 7.3.; Beckner's result (7.9. and 15.12.) gives the same norm estimate if the operators rrl. are arbitrary and 1 5 q 5 p 5 00. Maurey's factorization theorem treated the case 1 5 p 5 q 5 0 0 . Other results follow from the factorization 26.1. and the

PROPOSITION 1: For arbitrary p, q E [l,001 and operators TIE 2(Lq(ji1),Lp(vl)) and Ta E C(E, F)

-

Note that Lz(T2 : Lr(p2) L,(vz)) 5 KcllT2II if 1 5 s 5 2 < r 5 00 by corollary 26.l.(together with local techniques), hence llT1 @T' : ...I1 5 K&FTl(((IT211in this case. PROOF:Assume 11T111 = 1 and consider the maximal Banach operator ideal (!?Jl, M) of all T E C ( E ,F) such that

(see 17.4.). M is surjective and injective by 7.4.; since the ideal J of integral operators is contained in 9R (it is contained in every maximal normed ideal because of E 5 a) and M 5 I, it follows that M 5 (Isur)in35 KGL2

by the operator form of Grothendieck's inequality in 20.17. 0

26. More About Lp and Hilbert Spaces

347

(For other results in this direction see 28.4.,Ex 28.10.,29.12., 32.3., 32.6., 32.8. and 32.9..) It follows, in particular, that for Hilbert spaces H the operator

is always continuous and has norm 5 KcllTll -for p, q < 00 a result of MarcinkiewiczZygmund [182](clearly, with another norm estimate). For p = q this was already shown in Ex 23.14.. For p = 1 and q = 00 it gives Grothendieck's inequality (see Ex 28.12.), which indicates that the use of Grothendieck's inequality in the proof was natural. In the case q 5 p the constant can be taken to be 1 - but even a little bit more is true:

REMARK1: Take p, q, r E [l, m]. Then, for any T E 2 ( L q ( p ) ,L p ( v ) ) , the operator

has norm IlTll whenever q

5p

and r satisfies one of the following five conditions:

(1) r = P (2) r = 2 (3) q < r < 2

(4) r = Q (5) 2 < r < p . PROOF:The first three cases follow immediately from the version 7.9.of Beckner's result and an isometric embedding of .tr into some L , (see corollary 3 in 24.5.) by taking El = F = f, . The last two cases follow by dualization of (1) and (3). 0 This result has a nice application observed by Figiel-Iwaniec-Pelczyliski [65]: For any operator T :~ : ( p ) , real spaces

-L,R(v)

denote its natural complexification by

-

T" : ~ r ( p )

~ f ( v ), ~ " ( f ):= T ( Rf) ~

+ i T(Im f) .

Since t f ( p ) = L F ( p ) @A? Ri holds isometrically and T" = T @ i d q , it follows from theorem 7.3. that for positive operators T and all p, q E [l, m]

holds. Remark 1, case (2) gives:

348

26. More About L, and Hilkrt Spoces

Recall Ex 14.4.showing that this result is false for q = 00 and p = 1. For more results about the norm of the complexification see Ex 26.9. and Ex 28.14.. In the casea 15 p = 8 < q = r < 2 and 2 < p = 8 < q = r 5 00 the operator TI@Tzin the proposition need not be continuous. The following counterexample (as well as the last proposition for p = 8 and q = r ) is due to Bennett [lo], see also Rosenthal-Szarek [233].For 1 5 q < p < 2 let p, be the LBvy measure on KOo and I : t, 4 L p ( p q ) the Lkvy embedding 24.4.:

PROPOSITION 2: I €9 I : A?,C+A~tq+ L p ( p q )€ 9 Lp(pq) ~ ~ is not continuous (1 5 P < < 2).

Q

PROOF:As in section 24 the k-th projection K m + K will be denoted by h k and therefore I(c;=l akek) = a k h k . Take Zn := n - l / q e k €9 e k ; then Aq(zn;tq,tq) = 1. On the other hand,

cF=l

c;=,

holds. The functions I h k Iq are identically distributed, stochastically independent functions on the probability space ( K w , p q ) .Since they are not integrable (see 24.3.), the converse of Kolmogorov's law of large numbers (see e.g. Krickeberg [153],I11 3.7.) implies that

converges to 00 on a subset A c K m of positive measure. Fatou's lemma implies that the sequence (A,(I €9 I(zn);L,, L,)) is unbounded, which shows that I @ I is not continuous. 0 Simple duality (see 15.10.)shows that

is also not continuous (2 < q' < p' 5 00). For estimates of non-continuity of I@ I on see Rosenthal-Szarek [233]and Carl-D. [22].

@q

26. More About L, and Hilbert Spaces

349

26.4. To investigate the tensor norm gp = op,lit is advisable to first study the ideals of absolutely p-aumming operators. Let us recall some results which were already proven: In 23.10. it was shown that for all Banach spaces F

(1)

$p(ea,F)=~l(e,,F)andPIIKoPp for 1 5 p < 2 a n d l < s < 2

and in 24.7. (2)

pP(t,, F ) = g l ( l r l F ) and Pi 5 Fpi,rPp

f o r 1 Ip

< 2 and 2 I r < p’

(the constant can be easily deduced from the proof of theorem 24.7.).

PROPOSITION : (3) !Jlq(F,E ) = p z ( F , E ) and PZ5 Cz(E)b,P, f o r 2 5 q < 00 and E of cotype 2. Recall that Cz(4,) 5 a, if 1 5 s from the Khintchine inequality.

< 2 (see 8.6.) - where a,

and b, are the constants

PROOF:Take T E $ , ( F , E ) and 21, ...,z, E F. The definition of cotype 2 and the Grothendieck-Pietsch domination theorem give

-< C2(E)Pq(T)bpz(zk;F ) by Khintchine’s inequality 8.5.. 0 An obvious modification of the proof gives

The cyclic composition theorem 25.4. gives

350

26. More About

Lp and Hilbert Spaces

Since all tensor norms involved are accessible, the transfer argument 17.7. and the embedding lemma allow us to translate these results into inequalities about tensor norms.

COROLLARY 1: Let F be a Banach space and s , p E [1,2] and r,q E [2,00] , (1)

5 d, 5 KGd, 5 g& 5 KGg; d, 5 d, 5 Eq,rd, g; 5 g& 5 Z,,.tg,' 92 5 gp 5 arlbp1g2 d; 5 d$ 5 a6bpld; d,

gt

(2)

(3)

on t 6 8 F on t, 8 F. on t, 8 F if on t, 8 F if on er 8 F if on to8 F if

26.5. It was shown in 26.1. that every operator space if 1 5 s 5 2 5 r 5 00. More generally,

additionally additionally additionally additionally Lr +

r q'. p # 1 p # 1.

L, factors through a Hilbert

COROLLARY 2 (Kwapieli [160]): Let 1 5 s 5 p 5 r 5 00. Then

PROOF: One may assume s < p < r. Take p 5 2 (the case p > 2 follows by duality) and an operator T E E(lr,tg). According to proposition 25.9.(1) one has to show that TS E 3,(E,.f6) for each S E p?"'(E,&). The statements 26.4.(1) and (2) imply

Ex 17.7. yields TS E !P?"'(E,t,) C 3,(E,L,), which is exactly what is needed. 0 This time we find it worthwhile to explicitly note that the Cp-local techniques imply that for 1 5 s 5 p 5 r 5 00 every operator L,(p) + L o ( v )factors

(recall that &(v) is complemented in its bidual-

see Appendix B10.).

26.6. For Hilbert spaces many of the tensor norms/operator ideals that have been

considered coincide. First, recall from 11.6. that

26. More About L, and Hilbrt Spaces

351

the Hilbert-Schmidt operators, holds isometrically for Hilbert spaces H1 and Ha. Hence,

H1%; H2 C L v 2 ( H : , H 2 ) = A B ( H : , H 2 ) and therefore - for finite orthonormal systems ( e i ) and ( f j ) -

which implies that 9 2 = g$ d: = d2 on H I 8 HZ and gg is the Hilbert-Schmidt norm. These relations follow also from the Chevet-Persson-Saphar “equality” (15.10., corollary 2). For 12 8 12, and hence for all Hilbert spaces, 26.1. and corollary 1 now imply:

- and, by transposition, the same for d, and d;. So there are, up to equivalences, only three tensor norms among the ap,qand a;,qon Hilbert spaces: el n and the HilbertSchmidt norm g2. In terms of operators between Hilbert spaces this means that: p , q Ellloo[ : p , q €11, m[ : : p€[l,oo[ 4 €1114

cp,p = c, = c Pp,q = Pp= 3 = %

all operators nuclear operators

pp=ppduC’~=c1=cCm= = 3, = 3 y = A 6

Hilbert -Schmidt operators.

(That 3 = % follows from the Radon-Nikodfm property of Hilbert spaces). Pelczyriski’s result [198] for 1 I P < 00 *,(HI, H z ) = A B ( H 1 , H2) is a special case. For the associated minimal operator ideals see Ex 26.7.. 26.7. By what was said in section 12, the definition of a tensor norm on the class HILB of all Hilbert spaces is clear. Since every closed subspace of a Hilbert space is a Hilbert space and l-complemented, it is clear that tensor norms on HILB are projective, injective and totally accessible. The most prominent tensor norm on HILB is the Hilbert-Schmidt norm u defined by the embedding

HI 8 H2 CAG(H:, H2) .

26. Mom About L, and Halbert Spaces

352

Note that the dual H‘ of a Hilbert space is a Hilbert space with the scalar product (z‘ly’)~‘ := (R-’dlR-’z’)~

for z’,d E H’

,

where R : H --+ H’ is the “Riesz-mapping” - the (antilinear) mapping from the Riesz’ representation theorem. As has already been said

k=l f=l

k=l l=1

if (ek);!, and (ff)Elare orthonormal systems. If a is a tensor norm on HILB, then its dual a’ is defined as in section 15: first on the class HILB n FIN and then take the finite (= cofinite) hull. Clearly, H I @a Ha

always holds - hence, d., = g: = u’ .

d-( H i

Hi)’

I(%’,.)I 5 a(%;H I , Hz)a’(r‘;H i , H i ) . Recall that u = g2 = g; =

PROPOSITION: Let a be

a tensor norm on

for all z E Hi @ H2 (where Rk : Hk

-*

HILB. Then

f f i i s the Riesr-mapping).

See Ex. 2.10.(b) concerning the antilinear map R1 @ Rz. PROOF: Since (ek @ ff, Rlei = akfek @ ff that

c,,~

@ R2 fj)

= 6k,i .bt,j for orthonormal systems, it follows for

COROLLARY 1: u 5 (a. a’)lI2 holds for all tensor norms a on HILB in the real case - and in the complez case for all a which satisfy the Riesz-condition a(%;H i , H2) = a ( R i @R2(2); Hi,Hi)

.

It is easy to see that a’,a‘ and a* satisfy the Riesz-condition if a does. Since w p ( z k ; H ) = wp(&k;H’), it follows that all ap,g(and hence all a;,g)satisfy this condition. Actually, we do not know of any tensor norm which does not have this property.

26. More About

Lp

and Hilbert Spaces

353

COROLLARY 2: Let a be a tensor norm on HILB (which satisfies the Riest-condition in the complex case). If a = a', then a is the Hilbed-Schmidt norm u. PROOF:Corollary 1 shows that u that u = u' . 0

5

a,which implies that a = a' 5 u'. Now recall

26.8. The Hilbert-Schmidt norm u plays a central role in the theory of Hilbert spaces;

in particular, u = d = u* = d.Is there an extension a of u from HILB to the class NORM of all normed spaces which is symmetric (a= a') and self-adjoint (a= a*)? Since 92 = 91 # 9;' = d; = d2 on NORM (see 11.12. for this statement in terms of operators), none of the self-adjoint extensions 92 and d2 is symmetric. Does a selfadjoint and symmetric tensor norm a on NORM exist? Since a = (a$)*= a', such a tensor norm necessarily coincides with u on HILB by corollary 2. In terms of operators: Is there a (maximal) Banach operator ideal (a,A) which is self-adjoint (a = a*)and symmetric (a = ad""')? If there were a self-adjoint, symmetric tensor norm/operator ideal which were unique

- or at least with a natural description - this would undoubtedly enrich the theory of Banach spaces. From this point of view it is "unfortunate" that there are many self-adjoint, symmetric tensor norms, none of which seem to have a nice description. Self-adjoint and symmetric tensor norms on NORM were constructed in 1943 by Schatten [245].

THEOREM (Schatten): Let a be a finitely generated, Symmetric tensor norm on NORM which satisfies a 5 a*.Then them is a self-adjoint, symmetric and finitely generated tensor norm 7 on NORM with

7 is iotally accessible if a and a* are. If, in the complex case, a satisfies the Rieszcondition on HILB, then 7 does as well.

It follows from corollary 2 that all of these tensor norms 7 coincide on Hilbert spaces with the Hilbert-Schmidt norm u.The injective norm a = t satisfies the assumptions; the tensor norm a = w2 even gives a totally accessible tensor norm 7. For accessibility see Ex 26.8..

PROOF: For the convenience of the construction, assume a 2 a* instead of a

a*.

1. If a and p are tensor norms on FIN, then, obviously, $ ( a+ p) is a tensor norm.

26. More About

354

Moreover, using E 3 z-(z,

L, and Hilbert Spaces

z ) E (Ela/2)

(ElP/2) for E = M’ C3 N’, gives that

$1

Io5

5 2inf{max{~a’(z), ( 1 - A)P(~))

A

5 1) 5

by taking an appropriate A. It follows that

a+P

a + @ * a*+p+ ( 7 5 ) 7 .

and

on FIN and hence on NORM. Note that if a and /3 are finitely (or cofinitely) generated, then $(a /3) is also. 2. Now take a finitely generated tensor norm CY with a = a‘ 2 a* and define inductively

+

-1

It is easy to see that y is a tensor norm. Moreover,

+ 7 ( z ; E, F) = inf{T(z; M , N) I z E M 8 N , (M, N ) E FIN(E) x FIN(F)} = = inf inf an(%;M, N) = inf inf an(%;MIN) = inf an(%;El F) = M,N n

n M,N

n

= 7 ( z ; El F ) and hence 7 is finitely generated. A calculation of the same type, using 7 = sup a: , shows that 7 is cofinitely generated if a and a* (and hence all a:) are. Since a n = at, it follows that 7 = 7‘. Moreover,

1

MIN) = sup{l(u, %)I 7 y u ; M’, N’)< 1) = =supsup{((u,z)l aA(u;M‘,N’)< 1) =

y*(z;

n

I

= sup a;(%;MIN) = y ( r ; M , N ) , n

26. More About

L, and Hilbert Spaces

355

which shows that y is self-adjoint. Finally, if a satisfies the Riesz-condition, then a* does and so do all a, : this also shows that 7 does. 0 26.9. It is not a priori clear whether or not the Schatten construction may produce different y for different a. It waa PuhI I2271 who noticed that one actually obtains

infinitely many operator ideals extending the Hilbert-Schmidt ideal. In order to distinguish the various operator ideals he used limit orders. We shall give the main steps of his construction - in terms of tensor norms.

PROPOSITION: For every finitely generated tensor norm a on NORM there is a symmetric, finitely generated tensor norm /3 such that pg<(a.ff'.ff*.fft)1/4.

p

can be chosen t o satisfy the Riest-condition on

HILB if

a does.

PROOF: It is enough to show this on FIN. Define for M ,N E FIN f(u; M ,N ) := [a(u)a'(U)ab(u)at(U)]1/4

Then

E

foruEM8".

5 f 5 T , and P(z; M ,N) := SUP{I(% .)I

I f(u; M',"1

5 1)

is easily seen to be a symmetric tensor norm on FIN. Since f is positive homogeneous, it follows that

I(% 41 I f(u)P(.) and hence p' _< f. The fact that I(u,z)l implies that I(u,t>I

which shows that p(u)

P(.) = suP{I(uI .)I

5

I

a(u)a'(z) and

If ( M 4

5 a'(u)a(r) (and

so

on)

1

f(u); therefore,

I f(4 I 1) Isup{I(u, 41 I @(.) I 11 = B ( 4 = P * ( 4

*

The statement concerning the Riesz-condition follows from the fact that f satisfies it whenever a (and hence a',a* and at)does. I7

26.10. For u(") := C;=lek 8 ek E K" 8 K" and p, q E [I,001 the inequality

a ( u ( " ) ; q , q )< 7r(u'"';q,q) 5 n holds. The limit order X(a;p,q)of a tensor norm a on FIN is the infimum of a11 X E [0,1] such that there is a CA 2 0 with

ff(u(");q,q)I cxnx

for all n E I?.

356

26. More About

Lp

and Hilbert Spaces

The limit order A(%; p , q ) of a Banach operator ideal (24, A)is defined to be the infimum of all A E [0,1] with a cx 2 0 such that A(id :

C

-c)

5 cxnX

for all n E N.

This notion is due to Pietsch; see [214], sections 14 and 22, for the explicit calculation of various limit orders. Obviously,

if a and 24 are associated and clearly, X(a;p, q ) = X(a'; q , p ) . Since

It may happen that the sum is even 2 (see Konig [144]), but in many cases it is 1. For example (Pietsch [214], 22.4.12.),

if 2 5 p 5 00 and (r,s) E [l,21 x [l,001. If a is a tensor norm and Pff the one constructed in proposition 26.9.

then, clearly,

LEMMA: If A(a;p,q)+A(a;q',p') = 1, then

f o r all tensor norms 7 with Pa 5 7

PROOF: Inequality (**) gives

pff.

357

26. More About Lp and Hilbert Spaced

so the first terms in the two sums coincide. Again by (**) it follows that

This implies the statement. 0 Take a = g; for 2 5 p _<

00.

Then

1 = A(g;;2,l)+A(gp;O0,2)= q x 2 ) + A ( g p ; 2 , = ) = A(g;;2,4+A(gp;1,2) by the result (*) mentioned above. Moreover,

+

1 I A(g; ;O0,2) A(gp ;2 , l ) = W P p ' ;1,2)

+

291) L 0 + 1

by Grothendieck's theorem. It follows that

A(gi; 1,2)

+ A($;

2, 00) = 1

and

- the latter by Pietsch [214], 22.4.12 and Grothendieck's theorem. Denote by rp the self-adjoint, symmetric, finitely generated tensor norm which cornea from the Schatten-construction 26.8. when starting with rr = &; from Puhl's proposition 26.9.. Then the lemma implies that

for 2 5 p 5 00, which yields that the tensor norms rPare pairwise non-equivalent.

COROLLARY (Schatten-Puhl): There are infinitely many non-equivafent self-adjoint, symmetric, finitely generated tensor norms which coincide on Hilbert spaces with the Hilbert-Schmidt norm. 26.11. In Ex 12.9. it was shown that the unit vectors ek @el E 1&1,& form a basis for every tensor norm a,provided the ordering of (ek 8 et)k,c is the rectangular ordering: el @ e1,ea 8 e1,e2 8 e2, el @ e2, ... Since

358

26. Mom About Lp and Hilbert Spacer

it is obvious that this natural basis is even unconditional for the Hilbert-Schmidt norm, i.e. the series expansion along the basis converges unconditionally. Gelbaum-Gil de Lamadrid [80]observed that for a = B the basis is not unconditional. But much more is true; Kwapieh-Pelczyhski [163] showed that

THEOREM: I f a is a tensor norm on HILB such that (ek@et)L:,tENis an unconditional basis an .f2&&, then a is equivaleni to the Eilberi-Schmidt norm 6. The key to the proof is the following well-known

LEMMA: Let (xn) be an unconditional constant c > 0 such that

basis of a Banach space E. Then there is a

PROOF:e1&E is the space of unconditionally convergent series in E and

by the results of section 8. Since

m

m

n=l

n=l

is continuous (closed graph theorem), it follows that there is a c > 0 such that

which is the result. 0

PROOF (ofthe theorem): It is clear that it is enough to show that a and u are equivalent on 484 with constants independent of n . Define for z := aL:,tek8 e t f 484

ci,t=l

26. More About L, and Hilbert Spaces

359

(canonical embedding). Then, by the lemma, there is a constant c > 0 such that

Now fix z E t?!j QD l!j and choose two isometries Ul,lJ2 : l!j

+ l!j

such that

2 0 - this follows, e.g., from the polar decomposition UA of the operator t?!j (where U is an isometry and A positive) and the diagonalization A = DV, hence U-'T,V-' = D. with

Xk

T, :l!j

Isometries (onto) preserve tensor norms, therefore, n

.

u(z) = k=l

Now take the Fourier matrix (see Ex 4.3.)

and

Since all operators involved are isometries, it follows that a ( w ) = a(%).But

which implies a(z) = Cr(lw1) 5 c a ( w ) = c a ( 2 ) u(z) = a(lw1) 2 c-'a(w)

- and this was to be shown. 0

= c-la(t)

360

26. More About

Lp and Hilbert Spaces

Exercices: E x 26.1. Let p, q €11, m[ with l/p+ l / q 2 1 and 1 5 s 5 2 5 r 5 00.

(a) G OCr

c &,q

(b)

9p',p' 0

2, C 9,'

(c)

.CroSp',q'

c

*

.

Hint: 26.1., 23.1., cyclic composition theorem.

Ex 26.2. Let p,q E]1,00[ and

l/p+ l / q

5 1 and

1 58

5 2 5 r 5 00. Then

Hint: 26.1..

Ex 26.3. For p , q €11, cm[and l/p+l/q 2 1the projective associate is equivalent to wz and /a;,q\ is equivalent to w2. Hint: Grothendieck's inequality, 20.9. and 26.1.. EX 26.4. Show that L z ( i 4 ; ) = d(q:,G). Deduce from this and 26.2. that

for all p E [ l , 001.

Ex 26.5. Take g:, ...,gA E LP(pl) and g:,

...,g$

E Lp(p2).Then

if 1 5 q' 5 p 5 00 (with constant c = 1) or if 1 5 p 5 2 5 q' 5 00 (with constant = Ki). Hint: Use 11 Cea&n : --* Lpll = w , ( g t ; L p ) from 8.1. and 26.3..

c

q,

Ex 26.6. Use the foregoing exercise to show: If ( g : ) a E N are weakly q-summable sequences in Lp(pi) and 1 5 q' 5 p 5 00 or 1 5 p 5 2 5 q' 5 00, then (9: 8 gz),,tEN is weakly qaummable in Lp(pl 8 pa). In particular (see 8.3.), if p 2 and (9;) are unconditionally summable in L p ( p i ) , then (9: 8 g&,tEN is unconditionally summable in Lp(pi 8 p2). Using the counterexamples in 26.3., check that the result is false if 1 5 p < q' < 2 or 2 < p < q' 5 00. These results are due to Rosenthal-Szarek [233].

s

27. Grothendieck's Fourteen Notuml Tensor Norms

Ex

361

26.7. Deduce from 26.6. that on Hilbert spaces : %*,=A : A1 = A,

P,PEILoo[ p €11, oo[

-

= 9Ip = 36;

compact operators Hilbert-Schmidt operators.

1

Hint: Hi & H2 %(HI, H2) for maximal normed operator ideals. Ex 26.8. Show that the Schatten-construction in 26.8. gives an accessible tensor norm y whenever a is acceasible. Ex 26.9. (a) Take 1 q 5 p 5 00. If E C L F ( p ) and T E Z(E,L:(v)), then llT"ll = llTll. Hint: Use 7.9. as in 26.3.. (b) The best constants in the Khintchine inequality are the same in the real and complex case.

27. Grothendieck's Fourteen Natural Tensor Norms

The second chapter ends with a reference to Grothendieck's Rhumk: It will be shown that if one starts with the injective norm e and takes the adjoint, transposed, right or left injective or projective associate finitely many times, then one obtains, up to equivalence, only 14 different tensor norms. This Grothendieck result is, in a certain sense, a capsule summary of the results in the Rkumk.

27.1. Let (9 be the smallest set of (finitely generated) tensor norms which contains the injective norm E and is stable under the operations a-a', a f ,a\, and hence stable under a-a*, a/,\a,\a/,/a and /a\. A finitely generated tensor norm a is called natural if it is equivalent to a tensor norm in (9. Grothendieck's inequality states that /z\-

w2

and

\E/-

w;

,

hence the set Q 0 , consisting of the tensor norms Wl\ , \ E = % o , /w, , T\ = w&, , wL/, / T = W ; , \wi , wa/ [- d2 = 41 , \w2 [- ~2 = dl , w2 ,

E , e/=w1

1

T

(see 20.14. and the table in 20.15.) contains only natural norms.

362

27. Grothendieck's Fourteen Notuml Tensor Norms

THEOREM (Grothendieck): Every natural tensor norm is equivalent t o one of the fourfeen tensor nomas in @., Actually a more precise statement can be made, which (due to an idea of Grothendieck's) may be summarized in the following table of natural norms (and their associated maximal Banach operator ideals):

27.2.

J

I

I

1

c

I

J

1

1

J

I

c

1

1

I

I

J

1

1

J

I

I

1

I I

I

J

I

&

c

I

(See 20.12. for the factorization properties of the ideals VI/ and \'J33Sa'.) This table of the fourteen norms in 0, has the following properties, the first of which proves theorem 27.1. and the third of which shows (in particular) that all the fourteen norms in 0, are mutually non-equivalent :

(1) If a E a,, then a',at and a\ are equivalent t o a noma in 0,.

27.

Grothendieck's Fourteen Natural Tensor Norms

363

(2) A n arrow a -+ p means that a dominates p, i.e. there is a constant c with /3 5 ca. In other words, Q c 2?for ihe associated operaior ideals. ( 3) There are no other dominations in the table than those indicated. (4) If the locations of Q and p are symmetric with respeci to the point * in the middle, then a' and /3 are equivalent. (5) If the locations of Q and /3 are symmetric with respect t o the horizontal dotted line, then a' and p are equivalent. (6) If the locations of Q and /3 are symmetric with respect to the vertical dotted line, then at = p.

-

PROOF: The symmetries (4) - (6) are obvious - the only necessary work was already done in section 20 when showing that \w2 g2 = 92.. To see the stability property (1) - by the symmetries - it is only necessary to show that a\ is equivalent to a norm in 'Po for each a E 'Po : For a = \ ( / x ) , observe first that \w2 = (\w2)\ (by Ex 20.11.); now use the injectivity of w2, Ex 20.5.(b) and Grothendieck's inequality to see that \w2

= (\w2)\ 5 (\(/XI)\

5 \(/r\) 5 KG\w2

-

-- -

The remaining non-obvious casea are simpler: (/x)\ w2 by Grothendieck's inequality (wz/)\ ~2 by 20.15. (\E/)\ w;\ g2 by 20.15. ((x\)/)\ = T\ by EX 20.5. ( \ ~ 2 ) \ N \ ~ by 2 20.15. (\E)\ = (\di)\ = ( ( / d l ) / ) ' = d: = E by 20.14. (/(\&))\ = /woo\ = /gw\ = / E = E by 20.14.. The domination arrows on the right hand side are either trivial or can be deduced from the following (the others are a consequence of transposition; recall apz,ga 5 aPlrq1 for p1 5 p2 and q1 5 42 from 12.5.):

5 K L G ~=~KLc&/ / 5 K.cGSk/ by Ex 20.10. \w2 s; 5 \w2 Q; 5 w; w2 5 Q2 woo = goo 5 Q2 /woo = (g&/)* 5 KLGW;*= K L G W ~ by the first case. W;

--

It remains to show that there are no other dominations in the table than those indicated; the arguments will mainly use operator ideals: It will be shown that each ideal on the

364

27. Grothendieck's Fourteen Natuml Tensor Norms

right side of the diagram is only contained in those ideals indicated by the arrows. The rest then follows by transposing. , (see e.g. 23.3.), it is (by the definition of the surjective Since the space L1 is not in C hull) also not in Cr, hence 2 C r : Using the symmetry with respect to adjoints, this implies PI/ $ 3. By corollary 3 in 23.3. the ideals 21, C2, and 2 , are mutually uncomparable: It follows that 2; = ,pa' , 2 : = 9 2 , and 2: = g 1

c

are incomparable as well. In particular, y1 $ C 9 2 . By again using the symmetry , . Clearly, 2, $ !& $ !& - and hence with respect to adjoints, one gets 22' $ 2 all the vertical arrows on the right side are not reversable. As for the righthand side it remains to show that %31/

c %3y c 1

%31

21

1

%32

$21 1 2 , $tC y '1:2

$q

j

*

It is a result of Gordon-Lewis [84] that $t21, and hence V 2 $ 2 1 as well,. Moreover, the fact that t 1 4 space(C2') implies that 21 $ C g r ; this gives 2 , $ C;"j and also Cg' .?!A Using the symmetry with respect to the point *, one obtains !Q1/ from C , C? .

c~ ~ "

c

The deep result of Gordon and Lewis which was used solved one of the open problems in the Wum6. But note that $'2 $ 21 is much easier to show: If the non-compact embedding C[O,11 4 L 2 [ 0 , 1 ] actually factored through some L 1

then it would be in

o9 2

C

%32 o !fS2

C 3 by Grothendieck's theorem.

27.3. For the convenience of the reader - here is a list of Grothendieck's notation:

365

28. More Tensor Norms

Chapter III Special Topics In the first two chapters we presented the theory of tensor norms, the theory of operator ideals and their intimate relationship with each other. With this knowledge at hand, the final chapter touches some questions of a more specific character. Section 28 discusses examples of more tensor norms which are related to Laprestk’s 29 presents the powerful technique of traced tensor norms, which mimics for tensor norms the composition of operator ideals. Though at first glance a little bit formal, it also provides a very general setting for understanding the various tensor product characterizations of operator ideals given in Chapter 11. L Y ~ , , Section .

Sections 30 and 31 deal with the factorization of operators through Hilbert spaces in terms of type and cotype. In particular, this leads to the construction of a nonaccessible tensor norm/operator ideal due to Pisier. Maurey’s mixing operators are studied in some detail in section 32.

-

An important motivation for the sections 28 to 32 is to answer the question: Under what conditions are operators of the form S 8 T : L, @A, E L , 8~~F continuous? The results given include various characterizations and the study of two special operators S : the Fourier transform and the projection onto the Rademacher functions. Sections 33 and 34 treat two interesting topics from the theory of operator ideals: Lewis’ Radon-Nikodfm property for tensor norms, which serves to decide when 21min(E,F) = 2Pac(E,F) holds, and the tensor stability of operator ideals. The last section introduces the so-called tensor norm topologies on tensor products of locally convex spaces. It demonstrates that tensor norm techniques can be successfully applied to questions from the theory of locally convex spaces; in other words, there is also some type of a “local theory” (i.e. in terms of finite dimensional spaces) for locally convex spaces.

28. More Tensor Norms In this section some additional tensor norms will be studied. They are closely related to LaprestC’s tensor norms which were so important. The associated operator ideals have interesting descriptions in terms of factorization, inequalities and the continuity of operators A 8 T : e,,t 8 E -+ .!,8 F.

366

28. More Tensor Norms

28.1. For p, q E [l,w] with l/p

+ l/q

2 1 (equivalently: q1 2 p ) and

Pp,q(z; E , F) := inf Il(ak,t) : GI

rp,q(z; E , F ) := infll(ak,f) : 6p,q(z; E l F ) := infll(ak,t) :

G t

qt

+

+

+

z E E @ F define

$llwqt(xt; E)wpi(yk; F ) $)&i(Zt;

E)epi(yk;F )

$lleqi(Zt;

E)wpi(yk;F )

where the infima are taken over all finite representations n

ak,tzi@yk E E 8 . F ;

Z= k ,t=1

note that Aet := a k , t e k for the mapping A := ( a k , t ) : GI + $ . With exactly the same proof as for ( Y ~ (see , ~ 12.5. - or Ex 12.8.) one can show that these are finitely generated tensor norms satisfying Ppa,q3

IPP1,Ql

(analoguously for yp,qand lip,q) and PP,O

if

P1

L PZ and

41

5 q2

Pi,q = Pq,p(and this also for Y ~ , ~ Moreover, ).

Ia P , P and

PP4 I 6 P d

IYP,q .

Note that for p or q = 1 some of these norms coincide. Recall (Ex 20.2.) that rP,( is projective and hence rL,qis injective. It follows from 21.1. that yp,qis accessible and $,q totally accessible. It will be seen (28.6. and Ex 28.7.) that the tensor norms ,8p,q and are accessible as well. 28.2. For the study of the associated maximal (and minimal) operator ideals note that z = x k , t a k , ( t : @ y k E E1@F= Z ( E , F ) means that the associated operator Tzfactors

Recall from section 8 (in particular Ex 8.19.) that

L , ( x i ; El) = minIIS : E

-+

llDx : e", -+

CIl

where DAo S = x:Bek : E + is a factorization with a diagonal operator DA with non-negative entries. Since Dx is a positive operator, the s-integral norm satisfies

I,(Dx :

-

-

c c )= llDx :c c((

367

28. More Tensor Norms

by 17.17.. In particular, lpl(yk;

F) = minilS : F‘ = min IIR : q

= minIIR : q

--

5-

P , l llD, : P ,

F”ll /IDp: ~ “ 1 1 I,~Y’(D,:

411=

-

el(= C q ),

ci=l

with D, a positive diagonal operator such that Ro D, = ek@c&k) : * F” . It is clear now that the maximal operator ideals associated with pp,,,rPlq and bp,q are related with composition ideals of &,JP and J;fual.We shall treat Pp,, and 7p,qin order to demonstrate the methods and leave bP,, as an exercise.

28.3. First -yP,, : Take for finite dimensional spaces M, N n ak,LZ:

8 ?./k E M’

@yp,p

N

-

k,kl

Then for the associated operator T, : M

+N

and any factorization

TR the inequality (IFa’o I,I)(T,) 5 llSllI,~(Dx)llAII I$’a’(D,) IlRll holds. Passing to the infimum over all representations, it follows that (I$ o Iqi)(T, “ :M

-*

N ) 5 yP,,(z; M’, N )

-

by what was said before. To see that this is actually an equality, take for z E M’ @ N a factorization T, = R o 5’ through some Banach space G. Since J,i q , l and Jdual N O ~ , are ~ I accessible (section 21), the operators can be factored (see 18.1.) as P’ follows:

M-G

P ,

s

-q DA

R

G-N

368

28. More Tensor Norms

It follows that M' @7p,q N = (3$',' o 3 , ) ) ( M ,N) holds isometrically. This is in some sense the core of the following

PROPOSITION: Let l/p+ l/q 2 1. Then

and the associated maximal nonned operator ideal is \9ql,p,/

= (3;v

0 Jq,)reg

.

Recall from 25.9.that the regular hull %reg of a quasi-Banach operator ideal (a,A) is the class of all T E Z ( E , F ) such that t c o~ T E 2l, and Areg(T) := A(KFo T).The ideal \9q,,p,/ is, by 20.12.,the ideal of all operators T which factor

for some compact K and measure p.

vq,

PROOF: 9 q ~=, q$"" p~ o holds isometrically by Kwapieli's factorization theorem 19.3..Moreover, the formula \g; = gel from 20.14. implies that \QqJ

! @ a ' /

--

-

= gq1 3,) g;t/ = (\gi)' = g;,

\g;

-

5pv

by the relationship between tensor norms and operator ideals. 25.9. now gives that

\9q/,p// = \(!Q$O'

0

holds isometrically. Since remains to verify that

M'

!Qql)/

-

@7p*p

= ((W$O'/)

0

(\yqt))reg

= (3;Y'

0

Tq,)reg

is maximal, it and (3$'*' o DqOre9 = \9q,,p,/

N = 'a:$(

0

J q ' ) ( M ,N )

- and this was done before. 0 In particular, it follows that (3;Yoro 3 q ~ ) r e gis a n o m e d operator ideal - a fact which is not a priori clear. 28.4. The "adjoint" version of this result describes the operator ideal

[(3$0'

0 Sq/)+eg]*

-

= (3;Y 0 q)* y;,, = 7;,p = (\&J)r

= /aq,p\

-

c qi n , pj m r

28. Mom Tensor Norms

369

- and this implies a description of the operators in Cinj sur q,p

-

0

Jq4*

-

9

which for p = Q' is an operator form of a result of Kwapieri's [l60] and completes the characterization of subspaces of quotients of Lq-spaces given in 25.9. and 25.10..

THEOREM: Let l / p + l / q 2 1. For each operator T E $(El F ) the following statements are equivalent: (a) T E 'Ur(E,F). (b) For all A E C(tPl,1,) the operator

.CE/

A @ T :tpl

E

-

eq

@Aq

F

is continuous. ( c ) There is a constant c 2 0 such that for all natural numbers n E N ,all matrices (ak,f), all 21,...,zn E E and all yi, ...,y', E F' n

In this case,

PROOF: An obvious application of the closed graph theorem shows that (b) implies the existence of a constant c with llA @ TI1 _< cllAll. Since

and, for A = (ak,t),

it is easy to see that (b) implies (c). The converse is true by the density lemma 7.3.for p > 1 ; for p = 1 one has Am = E , and hence this implication follows by &,-local technique. By what was said before, the operator ideal ':2 l U ris associated with 7i,p.The representation theorem for maximal operator ideals shows that T'E:2 aur if and only if its associated bilinear form satisfies pT E (E @-,q,9 F')' . Therefore the definition of 74rP gives the equivalence of (a) and (c). 0

370

28. More Tensor Norms

In statement (b) the sequence spaces tp appear: they can be replaced by arbitrary infinite dimensional L p ( p ) ; this could be proved directly using ; : 2 Our = 3;' o (see 25.9.) - a fact which is related to the statement anyway. But there will be a general result in 29.12. showing this.

'J3Fa'

w2 5 ap,q5 a for p , q E [l,2], it follows that 3 = &, I C C 22. Moreover, 2 2 is surjective and injective and 3"'j IuP = 2 2 by Grothendieck's inequality in the operator form 20.17., hence

28.5. Since

(1) For p, q E [l,21 the relation ;:2

= 2 2 holds isomorphically.

Our

In particular, w2 and 7L,p(for 1 5 p, q 5 2) are equivalent tensor norms. For p E [l,m] the relation gil = gp\ (see 20.14.) implies that

and therefore,

(2) 2yj *P

=

'J3Fa'

IUc

and 22'

= Pychold isometrically (1 I p 5 m).

lUr

Clearly, the theorem is related with proposition 1in 26.3.: Since C(L,, L B )= C2(l+! L , ) if s 5 2 5 r (by 26.1.), it follows from (1) that every operator T : L, + L, is in ; : 2 IUr and hence, by (b) of the theorem (and 29.12.),

A @ T LpI(P1) @A,#

Lc(P2)

+

Lq(v1) @Aq L d ( y 2 )

-

is continuous. Note that in 26.3. this was shown for all p , q. Certainly, one of the most interesting cases is w2 = ~y;,~ 2 2 , the operators factoring through a Hilbert space.

COROLLARY: T E C ( E ,F ) factors through a Hilbed space if and only if there is a constant c 2 0 such that f o r all isometric n x n matrices A = (ak,t) and X I , ...,zn E E n

n

n

n

PROOF: By condition (b) or (c) of the theorem the condition is necessary. Conversely, fix t t ; then the convex function g(A) on the left side attains its maximum on the extreme points of the unit ball of C(4,G) - which are the isometries (see Halmos [99], p.263 ). It follows that the inequality holds for all matrices A = ( a k , L ) of norm 5 1 and this easily implies (b) or (c) of the theorem . 0 This result, due to Lindenstrauss-Pelczyliski [173], is in some sense central to the theory of operators factoring through a Hilbert space - e.g. in the proof of Kwapieri's

371

28. More Tensor Norms

important type/cotype theorem 30.5.. Note also that w; is projective, hence w; = 72,2 (by 28.3.) is an explicit description of the tensor norm associated with the adjoint ideal 2; = 9 2 of 2-dominated operators. See Ex 28.l.(d) for a 3ther description of 22. 28.6. Next, the tensor norm pp,q(z; E , F ) = inf{

ll(ak,L) :

*qllwqJ(tt)Wp’(vk)

and M’@ P , , ~N’ = (M @pi N)’, gives that for cp f (M @ N)’ OQ

lt’pll(M88;,qN)’

= inf Il4l IlRll IlSIl

7

where the infimum is taken over all factorizations ’p = 6 o (R @ S) with an operator R :M + an operator S : N -+ P,and 6 E @= $,)’ for some n E N.Following the ultraproduct-technique-proof oftheorem 18.5. word by word (with the exception that the forms 6 are positive there) gives the implication (a) r). (b) of the

c,,

THEOREM: Let l/p + l / q 2 1 and

(q,

’p

E ( E @ F ) * . Then the following statements are

equivalent:

(a) ’p E ( E @P;,q F)’ * (b) There are measures p and v, operators R E .C(E,L,,(p)),S E E ( F , L , ~ ( v ) and ) 6 E ( L * ’ ( p )@r LP’(V))‘ with

The measures can be taken to be Borel-Radon measures on locally compact spaces and

372

28. More Tensor Norms

PROOF:It remains to show that (b) implies (a): since ($# the 2p-local technique easily implies that

A ($)

@r $#)I

$#)I,

holds isometrically, which gives the result. Simple factorization arguments show the

COROLLARY 1: For l/p+ l/q 2 1 the maximal nomaed operator ideal associated with the tensor norm

Ppl9as isometrically

equal to the ideal

(XP0 f q r e 9

*

Again: (EPO C ~ ' )is~normed '~ - a fact which is not obvious. If % and '13 are accessible operator ideals, straightforward factorization and the strong principle of local reflexivity give that (% o %)reg is accessible as well. The ideals 2* wd are accessible, hence:

-

COROLLARY 2: The tensor n o m s pp,9are accessible, the tensor n o m s , i 3 ~ are , ~ even iot a1ly accessible.

The latter follows from the fact that 2 2 o Sq,= ( 2 2 o 1 3 9 ~ ) re9is an injective operator ideal, hence &,9 is accessible and right-injective, hence totally accessible by 21.1.(2). are maximal Note that .CPo22 = ( 2 , 0 C ~ ) ~ (by ~ g Ex 28.5.) and C202,t = normed operator ideals (1 5 p I 2).

28.7. This explicit description of the operator ideal associated with information about pp,9for special ( p , q ) :

pp,9gives further

PROOF: The description of \% in proposition 20.12. shows that

-

-

pp,l (2,0 C,)re9 = (\.Cp)'e9 = \2, \wp .

-

This gives (1) - and (3) follows from this using 20.14.and Grothendieck's inequality (tensor form 20.17.). (CPo C 'J32 g2 for 1 5 p 5 2 is corollary 1 in 23.10. (again Grothendieck's inequality), whereas 9 2

c .cz 0 c,

c (SP 2 o o y 0

373

28. More Tensor Norms

for 1 < p < 00 follows from the Grothendieck-Pietsch factorization theorem and the inclusion 2 2 c 2,. Statement (4) is a consequence of (2,o 2q’)reg C 22 for p 5 2 5 q’ (see 26.1.) and 22 c 2*o 2,’for 1 < p, q’ < 00. The last claim is obvious by corollary 1. 0 28.8. The adjoint operator ideal (2,o 2,,1)*= [(2,o 2q’)reg]* description in terms of inequalities:

-

p;,, has also a nice

COROLLARY 4: For l/p + l/q 2 1 and T E 2 ( E , F ) the following statements are equivalent: (a) T E (2,0 C , ’ ) + ( E F). , (b) For all A E 2(e,t,tq) the operator

A @ T : e,i

E

@lE--*

eq@ x F

is continuous.

(c) There is a constant c 2 0 such that for all n X n matrices and all Y;, ...,3/, E F’

I

(ak,t), at/ 2 1 ,

...,2, E E

n

a k , t ( T Z t , d ) l5 c ll(ak,t) : $’ - ~ I l w p ’ ( c t ) w q ‘ ( ? h )

*

k,t=l

In this case,

(L, o L,’)*(T) = sup IIA @ T ll4lI1

: @c

-

@xll

= inf c

.

1

PROOF:Since l, axF LI (e,! Bt F‘)’ and @en @ un;er,G) = Wr(Un;G) by 6.4. and 12.9.,it is clear that (b) and (c) are equivalent by the closed graph theorem and 2,-local techniques. The representation theorem for maximal operator ideals and

p’,sP = p*P 4

N

(2,0 &‘)*

1

show that T E (2,o Zqt)*(E,F ) + (E @pp,pF’)’ if and only if n

a k , t z f ‘8 k,k1

&)I 5

c Il(ak,t) :

-* ~

flwp’(zt)wq’(dk)

- and this is the equivalence of (a) and (c). 0 The relations in corollary 3 (and 17.12.)give:

(2, 0 2 , ) .

9y (2,0 Go)* = Y2 (210 Cm)*

(2,o

= 22

2,‘).= I D 2

for 1 I p s 00 for 1 < p s 2 for 1 < p , q I 2.

374

28. More Tensor Norms

One of the main objectives of this section was to establish conditions under which for a given T E 2 ( E l F ) the operator

28.9.

A @ T : tp#@ E

+t, @ F

is continuous (for certain norms) for all A E 2(tpt,l,). The results of 28.4., 28.8., Ex 28.8. and Ex 28.11. can be summarized in the following table:

r tensor norm

I

operator ideal

I

A@T:tp#@E-tq@F

Recall that ($Yo1 o 3,'). = 2ej * W . It will be shown in 29.12. that for these four characterizations A E 2(tpl,A!,) can be replaced by A E 2(Lpj(p)lL , ( v ) ) - with fixed measures p and v such that Lp'(c() and L,(v) are infinite dimensional. For related characterizations of this type for ; : 2 and XJ"' see Ex 29.4. and Ex 29.5.. PJ

Exercises:

Ex 28.1. Let 1 I p < 00. For z1,...,zmly 1, ...,y, m

m

k=1

k=l

E E write (zk)

for all X I E E . An operator T E S ( E , F ) is called (for the purpose of this exercise) p-strong if there is a c 2 0 such that (zk)

is pstrong with constant 1.

2p is pstrong with constant Ly(T). (c) If T is pstrong with constant c, then Ly 5 c. Hint: yk := C a k , t z ~and

(b) Every T E

28.4.(c). (d) T is 2-strong if and only if it factors through a Hilbert space. The converse of (b) is also true for arbitrary p : This will be shown in 33.7.. Ex 28.2. Let p be a measure and f lg E L 2 ( p 1E) such that

375

28. More Tensor Norms

for all 2' E E'. (a) If E is a Hilbert space, then I l f l l ~ ~ ( ~ _<) IlgllL,(E). Hint: f and g are separably valued, Parseval's identity. (b) If T E &(El F ) , then llT(f(i>)112dt5 L Z ( T ).f~Ilg(t>l12dtEx 28.3. Let 1 5 p 5 q' < s < 2 or: 1 5 p 5 q' < 00 and s = 2. (a) Use the Ldvy embedding to show that 2, C and ap,q\5 c#,q'cLiw,. (b) Take T E 2 ( E , F) and c 2 0 such that

s

2Ei

for d l

(Ok,f)

E 2(&,&)and sequences ( q ) in eb,(E).Verify that

for all (ak,t) E 2(tq~,4,) and sequences (zk)in t',,(E). Hint: (a) and 28.4.. EX 28.4. Show that E 5 /3p,q 5 APE on E 8 F if E is an 2:,A-space and F an 2$,r-space. EX 28.5. If (a,A) and (8,B) are quasi-Banach operator ideals, (a,A) is regular and ( 8 B) , injective, then (a o 8 ,A o B) is regular. Ex 28.6. Show that the finite dimensional Grothendieck constants K G ( ~from ) 20.17. satisfy:

-

Hint: 17.19. and (Pi"')' & , I . The following series of exercises will treat the tensor norm ap,q(t.;E , F )

:= inf{ll(at,t) : q,

-

I 5

~II~,'(zt)wP~(yk) z =

a k , m 8 yk}

.

k,C=l

The methods to be used are exactly those which were used for /3p,q and yP,, - in particular, the ultraproduct technique of section 18. The maximal normed operator ideal associated with 6p,q will be denoted by (q,ql Cp,q). Ex 28.7. (a) Show for T E 2 ( M , N) (finite dimensional spaces) that C p , , ( T )= inf llSll IlAll llDxll 11Rl1, where the infimum is taken over all n E N and factorizations

M

T

+ N

376

28. More Tensor Norms

Hint: 28.2.. Recall that I , # ( D x )= 11D~l1. (b) Show that for arbitrary T E C ( E ,F) the estimate C,,,(T) I Lp o 14#(T)holds. Hint: Use the maximality of and accessibility. (c) Use the ultraproduct decomposition of an operator given in Ex 18.8. and 17.17. to show that (2,o 3 , ~ ) ~ ~=gq,, Sp,, holds isometrically. (d) %,q and bP,, are accessible. Ex 28.8. Prove that for any T E E(E,F) the following statements are equivalent: (a) T E (Ep 0 &#)*(El F) . (b) For all A E E(l,~,l,)

G,,

-

A @ T :lpi BeE

-

l , 8~~F

is continuous. (c) There is a constant c 2 0 such that for all n x n matrices ( a k , f ) and all 21, ...,2, E E and d l...,s/, E F' n

In this case, (Lpo I,#)*(T)= supllAllllIIA @ T : + @A,II = inf c. E x 28.9. Verify the following relations with the aid of the associated operator ideals: (1) bp,l = \wp (2) 6 1 ,= ~ g,'/

---

(3) bP,l gz (4) SI,, w i (5) bP,, g2 (6) 6p,p#= gp

if if if if if if

1 5 p 5 00 . 1 I q 5 00 . 1

.

Hint Hint Hint Hint Hint

: 3, = E,, 20.12. or 28.7.(1). : 20.12. or 28.5.(2).

: 28.7.(2). : 28.5.(1). : Proposition 26.4. and

22

C Ep.

E x 28.10. (a) T E C(E,F) is absolutely 2-summing if and only if for some (for all) 15 q 5 2 5 8 < 00 and all A E C(4',,lq) the operator A @ T : l ,@c E @ A q F is continuous. Hint: Ex 28.9.(5), (3) and Ex 28.8..

-!,

-

(b) T E E(E, F) factors through a Hilbert space if and only if for some (for all) 1 5 q 5 2 and all A E C(l,, l,) the operator A @ T : l , @c E lq @ A ~F is continuous. Hint: Ex 28.9.(4). E x 28.11. Show that 6ilp (3F4'o and that T E (3:Y' o Eq#)*(E, F) if and F is continuous for all A E C(lpl,l,). only if A @ T : l p l @A,# E l, E x 28.12. In 26.3. it was shown (with the help of Grothendieck's inequality) that for every S E E(l, l l ) the operator 5' @ idt, :loo@c 42 41 @ x 1 2 is continuous. Show, viceversa, with 28.4. that this statement implies Ez c 2 : ; '"rl which by 20.17. is equivalent to Grothendieck's inequality.

--

-

377

28. More Tensor Norms

Ex 28.13. Use 28.4., 29.12. and the definition K G ( ~:= ) P i u c ( i d p )of the finite dimensional Grothendieck constants (20.17.) to show that

whenever L,(p) and L l ( v ) are infinite dimensional.

Ex 28.14. Complexificaiion of operators: For p , q E [1,00] denote by kq,p the best constant satisfying 11s" : L f ( p ) -+ L f ( v ) [ 5 [ k,,,IIS : L F ( p ) -, LF(v)[lfor all measures p and Y and operators S E .C(Lr(p),L ~ ( Y ) )Recall . from 26.3. that S" = S c?J idR; and that k,,p = 1 if 1 5 q 5 p 5 00. (a) Use 28.4. (as well as 29.12.) and 20.4.(2) to show that

and even equality whenever q = 2. Note that Ex 20.16. implies

(b) Conclude from the foregoing exercise that k,,1 = KE(2) and note that KE(2) 1r/2 by Ex 14.4.; actually k,,l = K g ( 2 ) = 4 by Krivine's result (see 20.17.). (c) Show that k,,p = kp1,,t and kr,* 5 k,,p if 1 5 p 5 s 5 r

q

5 00.

Ex 11.24. and

f!Eil o T?'"

(e) Deduce that the "complexification constants" k,,, satisfy for 1

5 p 5 q 5 00

(d) Use the embedding t!j L, LP(Sn-l,~ ~ (see 25.9.) to prove that for 1 5 p 5 q 5 00

- 1from )

(this result is taken from Pietsch [214], 22.1.5.).

c J!3,

5

29. The Calculus of Tmced Tensor Norms

378

29. The Calculus of Traced Tensor Norms

The basic philosophy behind our investigations is that finitely generated tensor norms and maximal normed operator ideals are in some sense the same - via the representation theorem for maximal operator ideals. In many cases the composition of operator ideals is a powerful tool for obtaining results about tensor norms; the reader is certainly aware of the important role of Kwapieh’s factorization theorem and the results of section 26. So it is natural to ask whether or not something like the “composition” of tensor norms exists. In fact, it is possible to define such compositions; they are the so-called “traced” tensor norms. They are quite simple and unify many of the results which have already been presented. However, their use may sometimes appear a bit formalistic (a sort of “calculus” is used with them) - and this is why we did not introduce them until now, although it might have shortened some proofs. The first systematic study of traced tensor norms appeared in [41].

29.1. For normed spaces E, F and G (where G vector valued trace)

CG : ( E 8 G’)8 ( G 8 F ) ( 2 8 u’) 8 (u 8 Y)

# (0))

-

the tensor contraction (or:

E 8F

(u’, U)G#,GZ 8 Y

is defined and surjective. If F is replaced by its dual, this mapping is just the composition of operators: (E@G’)@(G@F’) E@F‘

S(G,E ) 8 5(F, G )

T8S See also Ex 12.11.. If

since

CY

is a tensor norm, then

5(F, E ) ToS.

379

29. The Calculur of Tmced Tensor Norms

(this was Ex 12.11.(c)). Now let a be a right-tensor norm with respect to G (see 17.4., e.g. a a tensor norm or a = Ap if G = L p ( p ) )and p a left-tensor norm with respect to G' (i.e. p' is a right-tensor norm for G' @ ; example: p = A i l ) .Then

CG : ( E @ p G')@r (G 80 F )

-

-

( E @t G')

@ r (G @e

F ) -E@cF

is continuous, and hence ker CG is closed in the tensor product ( E @ p G') & (G& F). The projective norm A( ;E@p G', GC% F ) now defines a quotient norm 6( -;El F )on E @F via the tensor contraction CG.Straightforward arguments (e.g. by use of the criterion 12.2.) show that the assignment a @G

p : (ElF )

-

6( * ; E l F )

-

is a tensor norm on NORM which is called the traced tensor norm of a and p along 5 1 by a similar argument) that G. It is obvious from (*) (and 11 @p @ r @ c

aIa@PGIj

if a is a tensor norm if /3 is a tensor norm.

PIa@PGP

The reader might feel uncomfortable with the notation: the order of a and p in the four-fold tensor product and in a @ G ,B is different. This was done for purpose: it will turn out that the tensor norm a @c,,p is related to the operator ideal Q o %, if a and p '13 are associated. By definition of the projective tensor norm A ,

-

-

I

a @ . ~ p ( % ; E , F ) = i n f { ~ ( w ; E @ p GF ' ,)G @ CG(W) ~ =z}=

This implies the

REMARK:If a and p' are finitely generated from the right (see 17.4.), then the tensor norm a @G /3 is finitely generated. Recall that A, is finitely generated from the right (see Ex 7.16.). 29.2. The following lemma will be helpful for some calculations.

LEMMA: Let a be a right-tensor norm with respect to the dual G' of a normed space G and /3 u finitely generated tensor norm. Then a p on E C3 F i s the quotient norm of the canonical map &I

29. The Calculus of Tmced Tensor Norms

380

PROOF: Denote by 6 this quotient seminorm. Since E @ ~ p G is a subspace of E 8 0 G" (by the embedding lemma), it is obvious that (Y @G#p 5 6. Conversely, take z 6 E @ F and n w

=

8 V; E ( E 8~pG")@r (G'@af') 1 k=l

with C ~ ( W=) z and E;=,p(v:; E,G")a(vi;G',F) 5 (1 +&)a @G# p(z). Since p is finitely generated, there is an M E FIN(G") with p
+ E)P(v:; El G")

for all h = 1,...,n. Take representations m

j=1 m

j=1

and, by the weak principle of local reflexivity 6.5.,an operator R 6 2 ( M , G ) with llRll5 1 E and (Ru;,j, ui,i)G,G' = ( U t j , u:,i)G",G'

+

for all i ,j,k,.t.It follows that z is the image of n

C [ ( i d@ R)(v:)]@ v i E ( E @ p G ) @ m (GI @ a F ) k=l

and

n

n

n

29.3. Some examples:

PROPOSITION 1: Let 1 5 p 5 00 . (1) wp = E 8+ E and w l = e ace E . ( 2 ) dp = Ap @tpE (3) gp = E atp# A: and ?r = E @co ?r . PROOF: Statement (1) is just a reformulation of corollary 12.9. (with the help of lemma 29.2. for p = 1). Since d , = w l = E@L,

E

= A, 8~~E ,

29. The Calculus of Traced Tensor Norms

381

(2) holds for p = 00. For 1 5 p < 00 recall that dp(z;El F ) = inf{wpf(zr; E)tp(yk;F ) I =

c n

zt @ yk}

k=l

and, again from remark 12.9. and section 8, wp1(zk;E ) =k z ‘ ( &

Oer;E,tpl)

tp(yk;F) = A p ( C e r @ y t ; e , J )

This implies that Ap @ f p

E

5 dp. The converse follows from dp 5 Ap : dp

5 dp @tpE I Ap @tpE

*

(This was Ex 12.11.(b).) The last statement follows in the same way (for p = 1 use the previous lemma or Ex 29.1.). 0 29.4. The main result on traced tensor norms is the following

THEOREM: Let a and p be two finitely generated tensor norms, G a normed space and denote by (9, D ) the maximal n o m e d operator ideal associated with the traced tensor norm a @G p. Then f o r each T E C ( E ,F ) the following statements are equivalent: (a) T E D*(E,F ) . ( b ) idG @ T : G@.,E G@$ F is continuous.

-

I n this case, D*(T) = IlidG @ T :G @a E

-

G 8; FII.

If G = L p ( p )for some measure p , then the equivalence also holds in the following three cases: (1) a = Ap and p a finitely generated tensor norm, (2)p = Ail and a a finitely generated tensor norm, *--

(3) a = Ap and /3 = A i f ,where (Ail)* := Ap . PROOF: It was proved in 17.4. that the class of all T satisfying (b) forms a maximal operator ideal CG with the norm LG(T):= IlidG

@ T : G @a

E -G@F

Thus, it suffices to prove that D * ( T : M + N ) = L G ( T : M The definition of the traced tensor norm 6 := a @G ,f3 and

Fll.

-

N) for M , N E FIN.

29. The Calculus of Traced Tensor Norms

382

(by the duality theorem 15.5. for tensor norms and 15.10. for p = A;,) give the commutative diagram

and the fact that C& is a metric injection. This implies that the lower horizontal map is a metric injection, which is the statement. 0 This result unifies many of the tensor product characterizations of operator ideals which have been presented so far. Recall from 15.7. that for each Banach space G

G @ - F = G @ pF P’

if p’ is totally accessible or: if p is left-accessible and G has the metric approximation property or: p is right-accessible and F has the metric approximation property. wp1 = E @ f p E from 29.3. and IDp concerns pdominated operators:

29.5. Using

PROPOSITION: For 1 5 p 5

(4 T E BD,(E,F ) (b) idt, 8 T : ep @c E *

-

00

-

wpl * (see 17.12.)’ a first application

and T E 2 ( E , F ) the following are equivalent:

-

.tp@r F is continuous. In this case, D p ( T )= Ilidt, @ T :fp @c E t p @T Fll. For p = 2 this is a result of Grothendieck’s ([93], p.42). It is easy to see from this that a Banach space G which contains the c ’ s uniformly complemented has the property that T E C ( E ,F ) is p-dominated if

i d G @ T : G B CE - G @ . , F is continuous. This property of G implies (by the theorem for E, F E FIN only) that e @G E cwp1 which gives (see Ex 29.9.(b)) that G contains nearly the q ’ s uniformly -+ G and complemented: There exist a A 5 0 and, for every n, operators I k : Pk : G + q such that m

m

29. The Calculus of Traced Tensor Norms

383

(rn depending on n as well) - these spaces are called &-spaces (see also Ex 29.9.). It was shown in [42] that Tp-spaces for p = 1,2 and 00 contain the C ’ s uniformly complemented. 29.6. The following observations establish a sort of “calculus” for the traced tensor

norms:

REMARK1 : For appropriate right- and left-tensor norms (a@G

P ) 698 7 = a @ G (Pa8 y) =:

@G

Pa8 7 -

This follows immediately from the facts that the projective norm 7r is associative (Ex 4.4.) and projective, and from the transitivity of forming quotient norms.

REMARK2: Let a and P be tensor norms. Then (1) ( a @ P)/ ~ =O/@G P . (2) \(a@G P) = a @G \P (3) \(a @G P)/ = a/ @G \P

-

.

PROOF: Only ( 1 ) will be proved; the proof of the other two statements is the same. It suffices to check that for 6 := a @G P and y := a/@G P

-

E @a L i (BF) is a metric surjection: This follows from

F

@-,

1

( E @ p G’) @= (G @a l i ( B ~ ) )

E

?r

Li(BF)

1

.

since

@6

respects quotients. 0

Finally, some useful formulas:

REMARK3: For every tensor norm a the following holds: (l)CY@GA=7r@Ga=r.

( 2 ) a @G a* = a* @G a = ?r if a is finitely generated. PROOF: (1) follows from A 5 a @G A (see 29.1.). (2) follows from theorem 29.4.: If D* is the maximal normed operator ideal associated with (a@G a*)*, then

-

llTll 5 D*(T) = l l i d ~@ T : @a

5 (li& T :@a

@a11

@1I; 5

= IlTll

384

29. The Calculus of Traced Tensor Norms

--

This shows that (a@G a*)* C

E

and hence

since a @G a* is finitely generated (see remark 29.1.). 0 Though it may appear quite formal, the calculus of traced tensor norms will turn out to be very useful: it allows one, for example, to determine without any effort those T E C ( E ,F) such that

-

A @ T :Lp[O,1 ] @ ~ E,

G€L F

is continuous for all Banach spaces G and all A E C(L,[O, 11,G) - see Ex 29.7. (the answer is: T E o %*)*(El F) if a is right-accessible). It is for results like this that traced tensor norms along a “general” space G have to be introduced. 29.7. For two finitely generated tensor norms a and

/3 and a Johnson space

C2

(see

6.3.) define the tensor norm a @ p := a @C2

p;

it is called the tensor product of a and /3. By 29.l.it is finitely generated. The maximal normed operator ideal associated with a @ /3 is denoted by (a@ 23, A @ B ) , the tensor product o f ( a , A) a and (%, B) p. It follows from theorem 29.4. and lemma 17.16. that T E (a @ %)*(El F) if and only if for all Banach spaces G

-

-

idG @ T : G @a E

-

G 8; F

I

is continuous; in this case (A@ B ) * ( T )= sup{llidG @ T : ...I] G E BAN}. are independent of the particular choice In particular, the definitions of a@pand a@% of Johnson space C2. Recall that C2 is reflexive, has the metric approximation property and is isometric to its dual Ci. It follows that one may take Ci in the definition of ff

8 P.

The name “tensor product” for

a @23 comes from

for MIN E FIN. Since every L E FIN is (1 subspace of C2 and R E % ( M ,C2) factors

+ &)-isomorphic to some l-complemented

49. The Calculus of Tmced Tensor Norms

385

it follows that n

A @ B(T : M

-+

N ) = inf

A(Sk)B(Rk), k=l

where the infimum is taken over all factorizations T = Rk E C ( M ,L k ) and sk E g ( L k , N ) for some L k E FIN. 29.8. If /? is accessible, then

sk

o Rk with operators

C2 @ - F = C2 @ p a F and theorem 25.8. and 29.4. imply P'

that (B @ %)* = (ao %)* holds isometrically, the operators T E (a @ %)* being those for which for all G (or only some CZ)the map

is continuous; ( A @ B ) * ( T=) llidca @T : ...[I. This result also follows from the following proposition, which is independent of 25.8.:

-

-

PROPOSITION: Let ( a , A ) a and (%,B) /? be such that Q is left-accessible or % right-accessible. Then 8 % as the smallest maximal normed operator ideal containing the quasi-Banach ideal B o '13.

(1)

(2) ('21 @ %)* = (a o %)* holds isometrically.

PROOF:(1) If (C, C ) is a maximal normed ideal containing 9 l o 8,then there is a p 2 0 with C

5 pA o B. For T E 2 ( M , N ) with a representation

as at the end of 29.7. it follows that n

n

k=l

k=l

and hence C 5 pA @ B.Thus, it remains to prove that % o '13 c a @% : again it is enough to check this for finite dimensional spaces. Take T E Q o % ( M ,N) and a factorization

29. The Calculus of Traced Tensor Norms

M-N

T

/

\ R

A(S)B(R) 5 (1

+ E)A

0

B(T);

G then G can be taken to be finite dimensional as well since Q is left- or 23 rightaccessible. It follows that A @ B(T) 5 A o B(T) for arbitrary T E 2 ( E , F ) .This completes the proof of (1). To see (2), first observe that the last statement implies that

(AoB)'

5 (A@B)' .

Since (2l o %)** 3 Q o 23 and (A o B)** 5 A o B, the first part of the proof of (1) implies that (A o B)**5 A @ B, which gives

(A @ B)'

5 (A o B)*" = (A o B)' .

0

Clearly, this result (together with theorem 29.4.) gives another proof of 25.8..

COROLLARY: Let a and 2l be associated. The following statements hold isometrically: ( 1 ) % @ 2 = 2 @ Q = Qand a @ ~ = ~ @ a = a . (2)2l@3=3@%=3 and a @ r = r @ a = r . (3)%@2l'=%'@%=3

and a @ a * = a * @ a = r .

PROOF: The first statement follows from E

-

2 and

( 2 l @ ~ ) ' = ( Q 0 2=%' )* and the others were already shown in 29.6. since r Note that by 29.3. E @ E = E proposition implies that

# wp = E @+

E.

-

3 .0

If 2l o 23 is maximal normed, then the

2l023=%@23.

In general, 2l o 23 may be "much smaller" than Q @ 23 :

Using ultrastability one can show (see Ex 29.8.): If Q and 23 are two maximal normed operator ideals such that 2l is left-accessible or 23 right-accessible and g o 2 3 is normed, then (2l o Q @ 23. In particular, Q o 23 Q @ 23 if and only if 2o 23 is normed and regular.

A

A

29. The Calculus of Tmced Tensor Norms

387

by the way, also shows that 3 o 3 is not normed, since

The result about (% o

(3 0 3 y g c

%reg

c %dual C 3 = 3 @ 3 . #

The same type of argument shows that !& o 'J32 is not normed. 29.9. Kwapieli's factorization theorem 19.3. now gives

- and this is the first of the following formulas:

PROOF:Using the formulas from 29.3. and 29.6. and \g; = gpl (see 20,14.),one obtains = \(d: @ $) = d; @ \g; = df @ gp' = (di @ 6) @t, A;, = df

- d4+@ ( E @tpA;,)

@tp

(3) and (4) follow in the same way. 0 Via the main theorem 29.4. for traced tensor norms (and 29.7.) each of these formulas can be interpreted in terms of the adjoints of the associated operator ideals: cinj p,q

2 ; ; ; ;:2

--

= (\a;,q)*= (d; /ap,q = (a;,(/)* = ...

ap,q\

*ur

-

bP,P\

@ gpo*

= <\a;,(/)*= - * *

Take T E 2 ( E , F ) . Then, in (1) COROLLARY:

-

= (d;

at, A;,)*

*

(3), all the sfatentents (a) - (c)

(resp. (a) - (d)) are equivalent:

--

(1) (a) T E CE:(E, F ) . (b) idt, @ T : fp@d; E (c) idG @ T : G @d; E G = tP).

T E 2;;qYEl F ) . (b) idto, @ T : fql@+

(2) ( 4

E

,'k @A, F is continuous. G @g;, F i s continuous for all G E BAN (or only

-

k ' , ~agP F is continuous.

388

29. The Calculus of Traced Tensor Norms

--

(c) idG @ T : G a d q l E G =GI). (3) (a) T E i!Ej6U'(E, F). (b) idt, @ T :ep @dql E E (c) idt,, 8 T :!,I

(d) i d 0 @ T : G @dql E G = e,l or only G = tp).

G @g, F

i s continuous for all G E BAN (or only

ep @A, e,l

F is continuous. @ p l F is continuous.

G Bg;, F

is continuous for all G E BAN (or only

2p,

lu' , in particNote that for g = p' one obtains characterizations of 2r' and 2'"' P ular, Kwapieri's results in 25.10.. Moreover, it will shown in a moment (29.11.) that the sequence space .fa in the statements above can be replaced by any infinite dimensional L d ( p ) .We omitted the obvious results about the norms.

29.10. Another application of the "calculus" of traced tensor norms is the characterization of %'"jand ad"'in terms of tensor product formulas, which was already announced in 20.13.: The formulas give

and hence, for the maximal normed operator ideal

(2(

-

a:

The main theorem 29.4. shows:

PROPOSITION: Let a be afinitely generated tensor norm, (a,A ) its associated matimal nonned operaiot ideal and T E E(E, F ) . (1) T E gin' if and only if

is continuous. In this cuse, A'"J(T)= Ilidt, @ T : . .I] . (2) If a is right-accessible, then T E Qbu' if and only if

is continuous. In this case, A'"'(,)

= Ilidt,

@T :

...I]

29.11. In many of the formulas and results above the sequence space lpcan be replaced by any infinite dimensional L p ( p ) . The key to this is the following

389

29. The Calculus of Traced Tensor Norms

PROPOSITION: If a i s a finitely generated tensor n o m and 1 5 p 5

00,

then

Ap @tpa = Ap @ L , ( ~ a ) = dp @ a a atpA;, = @ t , ( p ) A;, = @ gp’ for every infinite dimensional L p ( p ) . One may replace tooby co.

PROOF:If (A,,) is a sequence of pairwise disjoint sets with p(A,,) mapping 1 8 : t 8 + L,(p) defined by

> 0, then the

00

IS(&) :=

C
b(~4n)l-’”XA~

fl=l

is an isometry and its range is l-complemented with the averaging operator

@a E c* L 8 ( p ) @a E is an isometry onto a l-complemented subspace - and this is also true for a = Ad (by Ex 7.1.). Since

as a projector. It follows that I, C3 i d E : t,

( I P ,IpY)L,,,L, = ( 2 , Y)fp’,fp by construction; it follows that

commutes and hence that Ap B L , ~a)5 Ap @ t , a. (Note that the projection does not commute with the tensor contractions!) To see the other inequality the calculus for traced tensor norms helps:

Ap atp = Ap @tp( E @a) = (Ap @tpE ) @ a = dp @ a I - ( d p @ ~ , ( p )E ) @ a = dp @ ~ , ( p )( E @ a)= d p @ L , ( P ) a I <

I AP @L,(,)

a

- which also gives the equation with dp 8 a. The second relations follow in the same way. That tcocan be replaced by c , follows from Ex 29.1.. 0 This shows again how “close” dp is to Ap : Recall that the proof of 15.11. concerning the closest tensor norms to Ap used the tensor contraction as well.

390

29. The Calculus of Traced Tensor Norms

-

29.12. This result has an interesting consequence for the continuity of all A @ T : Lp(p)@aq E

Lp(v)@a,,F

for a given T E 2 ( E , F ) . For this a little lemma concerning a @G E @H p will be presented - a lemma which is also useful in other situations (see Ex 29.4. - Ex 29.7.).

LEMMA:Let G and H be normed spaces and a and p finitely generated tensor norms (or a = A, if G = L,(p), or p = A;, if H = Lp(u)).If 9 is the mazimal normed operator ideal associated with a @G 6 @ H p, then T E 9 * ( E ,F ) if and only if there is a constant c 1 0 such that

for all A E S(G,H ) . In this case,

PROOF: Define 7 := a @G E . The main theorem says that T E 9 * ( E ,F) if and only if @T : H & E -P H @ ; F is continuous. By the definition of traced tensor norms

idH

this operator is continuo& if and only if the map (same norm):

ip

:= (idH @ T )o CG is continuous

F

Identifying H

G' and S(G, H ) this leads to an operator

which is A-A@T. In other words, idH@T is continuous if and only if 4 is continuous (and then with same norm). This is the claim. 0

COROLLARY: For p1q E [l,04 let a be a finitely generated tensor noma or A, and: p a finitely generated right-accessible tensor norm or Ap. Take measures p and u such that L,(p) and Lp(u)are infinite dimensional. Then f o r T E C ( E ,F ) the following statemenis are equivalent: (a) For all A E C(L,(p),Lp(v))the operator

391

29. The Calculus of Traced Tensor Norms

is continuous. (b) For all A E 2(t,,tp)the operator

i s continuous. One may replace In this case,

t , by c,.

PROOF:Since 1, is “nicely” complemented in L , ( q ) (as was pointed out in the proof of 29.11.), (a) always implies (b). If a and P are finitely generated tensor norms, then simple f!p-local technique gives the converse; note A, = ?r and Am = E . For the remaining cases, note first that L , ( p ) has the metric approximation property. with llRbll 5 1 converging pointwise to It follows that for a net Rb E 3(L,(p),Lq(p)) the identity, R,@idE also converges on L,(p)@E to the identity pointwise with respect to I and hence with respect to a. Therefore, AR, @ T ( r ) converges to A @ T ( z ) with respect to I and hence for p. Consequently,

To apply the previous lemma take a = A, and /3 = Ap and p , q €11 ,co[.Then proposition 29.11. implies that

which is independent of p and v: the lemma and (*) give the equivalence of (a) and (b) in this case. The ideal of operators satisfying (b) is associated with (a@, c@tpp)’ by the lemma. Moreover, a @tqE @tpA;, = 0 @tq( E @ ~ ~ ( A;)) v )

A,

P = (A, @ L q ( p ) E ) @fp P

@tqt: @tp

.

In the first case the lemma shows that the operators satisfying (b) are those for which

for A E g(t,,L,(v)). If a is a tensor norm, &,-local technique and (*) give (a). The same type of argument applies in the last case: a = A, and /? a right-accessible finitely generated tensor norm. 0

29. The Calculus of %xed

392

Tensor Norms

Note that the lemma together with

gives an alternative proof of the characterization theorem 28.4. of the operators in '"', the ideal associated with (\a;,(/)*. Recall from 28.3. that \a;,(/ = yq,p.The tensor norm pP,( and Sp,( will be calculated in Ex 29.6. as traced tensor norms.

22;

Exercises: Ex 29.1. Take a and /3 finitely generated tensor norms and G a normed space. Show that

-

Ex 29.2. Let 2l a be a maximal normed operator ideal and 1 5 p 5 00. For T E C ( E ,F) the following three statements are equivalent: (a) T E (a 0 3,)*(E, F ) . (b) idG @ T : G &, E t.G Bg; F is continuous for all G €BAN . ( c ) idt,, @ T : Z,t @u E -+ ZPt @+ F is continuous. Hint: 29.11., the main theorem and 29.7.. Ex 29.3. Deduce theorem 17.15. and corollary 17.16. from the main theorem. Ex 29.4. Show that T E 2 E i ( E , F ) if and only if for all Banach spaces G and all A E 2(G,Zp) the operator

is continuous. Hint: Reduce to A E S(C~,ep), prove (ap,(\)* = ... = d; €3E @ipAkl and use lemma 29.12.. Ex 29.5. Show that T E C;Yl(E, F) if and only if for all Banach spaces G and all A E E(Z,t ,G) the operator

A @ T : ~ q l € # AEq- lG @ g P

F

is continuous. Hint: As in Ex 29.4.. Note that corollary 29.12. allows one to replace Z, by infinite dimensional L , ( p ) in the previous exercises.

393

29. The Cnlculus of Traced Tensor Norms

Ex

29.6.

Use 28.8. to show that Pp,q

= E @tPIE @tqE

and Ex 28.8. for hp,q

1

= E 8 1 p l E @l*A,, .

Hint: Lemma 29.12..

Ex 29.7. Let a be an accessible finitely generated tensor norm. Describe the maximal normed ideal of all T such that for all Banach spaces G and A E 2(L,[O, 11, G) E

A 8 T : Lp[O,

---*

GBa F

is continuous. Hint: A, @ L ~E @ a* = ..., metric approximation property of L,. Ex 29.8. An operator ideal (%,A)is called ultrastable if for every ultrafilter LL on I and T,E 2l(E,, F,) for L E I with supl A(T,) < 00

(a) Use the canonical decomposition of an operator T E 2 ( E , F) into its finite parts to show that for every ultrastable quasinormed operator ideal (a,A) (%, A)maD= (Q, A)reg.

In particular, every ultrastable, regular quasinormed ideal is maximal. Kiirsten [156] (see also Heinrich [104]) showed the converse: pnormed maximal operator ideals are ultrastable (and regular). (b) The composition 2l o '13 of two ultrastable operator ideals Q and '13 (pnormed resp. q-normed) is ultrastable. (c) Deduce from Kiirsten's result, (a), (b) and proposition 29.8.(2) the following: If Q and '13 are maximal normed operator ideals such that % o '13 is normed and: % is left-accessible or '13 is right-accessible, then (2l o %?)reg % @ '13. This is due to 1411, 3.4.5.. Ex 29.9. (a) Let G be a Banach space which contains the 5 uniformly complemented. Then there is a A >_ 1 such that Wpl

5 Wpl @G

5 AWpl .

- -C s,

R,

Hint: Definition of up# and id :$ (b) If a Banach space G satisfies E @G 29.5.). Hint: Consider

(5@c

Wpl

E

G with IlRnll llSnII 5 A. 5 AwPi, then it is a Tp-space (in the sense of

G') 8 s (G@c

$1)

By the result from 1421 mentioned in 29.5. and (a) is true if p = 1,2 or 00.

+$ E @G E

@c

5t

< w,'

*

@G

wpl,

the converse of

30. The Vector Valued Fourier Transform

394

30. The Vector Valued Fourier Transform

The Fourier transform 5 : &(R) + L2(aZ) is a powerful tool in Analysis so it is worthwhile to determine the conditions under which the vector valued Fourier transform is continuous: Unfortunately, the Fourier transform for E-valued functions is continuous only if E is a Hilbert space. This is the main result of this section - a result which can also be formulated in terms of vector valued Fourier series (i.e. in terms of the Fourier transform L2[-7r, r] -+ & ( Z ) )For . the proof one first uses Gaussian averaging to get Kwapied's important type/cotype factorization theorem C2 o 2 2 C 2 2 : The composition of a type 2 with a cotype 2 operator factors through a Hilbert space. All the main results of this section are due to Kwapieri (1611.

30.1. The Fourier transform

5 :L2(R)

-+

L z ( R ) , defined by

[5(f)l(t) := (2x)-1/2 J

m

f(s) exp(-ist)ds

i f f E L 2 ( R ) vanishes outside of a compact set, is an isometry onto L 2 ( R ) . Its inverse is, up to a sign, of the same form:

where

3 is the isometry of &(R)defined by

The maximal normed operator ideal ( 2 3 , L3) of all operators T E 2 ( E , F) (between complet Banach spaces) such that

(see 17.4.) is injective and surjective: this is a direct consequence of proposition 7.4.. The operators in 2 3 are sometimes called Fourier operators. REMARK: (23,Lg)

= (28-1,Lg-1)

.

PROOF: This follows immediately from the fact that for f

E L 2 ( R ) C9 G

90. The Vector Valued Fourier Transform

for all Banach spaces G and formula for 5 8 T . 0

395

5-' 8 T = (8 8 T)o (3 8 i d E ) - and the analoguous

Since (8fIg)La

= (fIS-lg)La

for f , g E L2(R), it follows that 5' = 8 : L 2 ( R ) + L 2 ( R ) (watch the complex conjugate in the scalar product!). It is easy to see (use 15.10.) that gdual

3

= 23'

holds isometrically and hence:

COROLLARY: 2 3 = Z$'O'

holds isometrically, i.e. the ideal 23 of Fourier operators is

completely symmetric.

It was shown in 7.5. that the identity map on el is not a Fourier operator - and in Ex 7.8. that L 3 ( i d ~= ) 1for Hilbert spaces H. The ideal property implies for the ideal 2 2 of operators factoring through a Hilbert space that 2, c 2 3 and L3(T) 5 L2(T) for all T E 2 ( E , F ) .

30.2. Obviously,

holds for all h E &(R) 8 E . The following simple result is basic:

LEMMA:For a > 0,

a complex Banach space

E and x - ~ x-n+l, ,

..., xn E E

k=-n

PROOF: The first formula is obvious. For the second one, observe first that a a 2exp(-i-)sin(2) = i[exp(-ia) - 11 2

define

80. T h e Vector Valued Fourier Transform

396

and hence, for all k E Z and t E R,

nr+:

exp(-ist)ds = -i [exp(-i(k

+ 1)at) - exp(-ikat)]

=

t

a

2

at

at

= - exp(-ikat) [exp(-iat) - 11 = - exp(-ikat) exp(-i-) sin( T t t 2

With the substitution t

).

= 21ru/a this gives

But for all u E RZ \Z

c

mEZ

sin2(nu) a2(u+m)2 =

(consider Parseval's identity for the Fourier coefficients of the function exp(iu.)) and therefore.

Another obvious subsitution gives the result. 0 30.3. For T := [-r,r[ and f E L ~ ( T ) ~ A = , EL 2 ( T , E ) C L l ( T , E ) the Fourier coefficientsare defined by

f(n) := (2n)-'/2

/1

f(s)exp(-ins)ds E E

J-1

for n E Z; here E is again a complex Banach space. Recall that ((2n)-'I2 exp(in*))nEz is an orthonormal basis in L2(T) and that for f E L2(T) Parseval's identity

30. The Vector Valued Fourier Transform

397

holds. It is now quite interesting to see that T being a Fourier operator means that T satisfies certain Parseval/Beasel-like inequalities.

THEOREM: Lei E and F be complez Banach spaces, T E 2 ( E , F ) and c 2 0. Then ihe following jive siaiements are equivaleni: (a) T E C s ( E ,F ) and L3(T) 5 c . (b) For all n E N and x-n, ...,Xn E E

(b') For all f E &(T, E )

(c) For all n E N and x-n,

...,t n E E

(c') For all f E &(T, E )

PROOF: T is a Fourier operator if and only if A2(3 8 T ( h ) )5 c A z ( h ) for all h E &(R)8 E. Since the functions h of the form treated in lemma 30.2. are dense in L z ( R ) 8~~E, the equivalence of (a) and (c) is obvious. Moreover, for h as in the lemma

( 2 IITzkll2)'lz = Az((3-l 8 T) (8 8 i d E ) ( h ) ;~52(R), F) 0

h=-n

1

holds. Since 2 3 = 28-1,this shows that (a) implies (b) - and the converse is also ) dense in L 2 ( R ) 8~~E (recall true because the functions of the form ($8i d ~ ) ( hare that 5 is an onto isometry). The Fourier coefficients of the function n

(2~)-'/~exp(ik.)xtE L2(T, E)

f(.) = k=-n

398

SO. The Vector Valued Fourier livlnsform

are f(k) = xk for lkl 5 n and zero otherwise; moreover, these vector valued exponential polynomials are dense in Lp(T)&E and hence in L2(T)&3a2E : these two observations prove the equivalences (b) c, (b’) and (c) c+ (c‘). 0 30.4. For T = idE the theorem implia that if in Parseval’s identity

one inequality holds for all f E L2(T,E), then there is equality - and this is equivalent to L s ( i d ~ )= 1. A neat trick due to Kandil [141] (see Ex 30.2.) shows that then E satisfies the parallelogram identity and hence is (isometrically) a Hilbert space:

COROLLARY 1: A complex Banach space E satisfies L s ( i d ~ )= 1 if and only if it i s isometric to a Hilberl space:

The isomorphic version of this result is more complicated and will follow in 30.6. from Kwapieli’s type/cotype theorem 30.5.. Before entering this topic, an example:

COROLLARY 2: For 1 5 p 5 00 and n E N Ls(idi;) = n11/p-1/21 .

PROOF: Since By 26.2.

.C3

is completely symmetric, it is enough to check this for 1 5 p 5 2.

La(idp)

5 L z ( i d 4 = n1/p-1/2

holds. For the converse note first that

In particular, 1, 4 space(&) for 1 5 p 5 general result is true.

00

and p

#

2 - but as announced, a more

30.5. A rather general theorem about factorization of operators through Hilbert spaces will be proved next. It is based on Gauss averaging. The following lemma shows that sometimes Rademacher averages can be replaced by Gauss averages.

30. The Vector Valued Fourier Transform

399

LEMMA:Let E , F be Banach spaces (real or complex), T E %(El F ) and define dR := 1

and d c := &. (1) If T i s of type 2 , then for a l l z l , ...,zn E E

(2) If T is of cotype 2, then for all 2 1 , ...,z, E E

This result can be expressed as follows: Rademacher type 2 (resp. cotype 2) implies Gauss type 2 (resp. cotype 2). The additional constant 4 in the complex case is natural since the complex Gauss functions Q k have norm

a.

PROOF:The proof is - as in Ex 8.9. - a simple application of F’ubini’s theorem and the invariance of the Gauss measure ‘yn under isometries: Assume T to be of type 2.

for all w E K”. Integration with respect to 7n gives

where

Ilgkll2,K

= d~ by 8.7.. Statement (2) follows in the same way. 0

For T = idE Gauss cotype 2 implies Rademacher cotype 2, but for arbitrary operators this is false (see Pisier [225], p.36/37). Ex 8.9. implies that Gauss type 2 operators are of Rademacher type 2. For Gauss type/cotype p see Ex 30.3.. Note that with Gn := span{gk I 1 5 k 5 n} C Lz(m) and

Gauss type 2 means that

400

90.

The Vector Valued Fourier Tmnsform

and Gauss cotype 2

where, as in section 7, the norm A2 on Gn 8 E is the one induced from L2(yn) Q D A ~ E. The following type/cotype theorem is quite important.

THEOREM: The composition S o R of a type 2 operator R and a cotype 2 operator S factors through a Hilbert space, i.e. CZ o 3 2 c $2. Moreover,

In particular, a Banach space i s isomorphic to a Hilbert space if and only if it has type 2 and cotype 2 .

This characterization of Hilbert spaces is due to Kwapieri [161] - and it was observed by Maurey (see Kwapieli [162]) that the result also holds for operators.

PROOF:Let T = S o R E E(E, F) with R E %2(E,X) and S E C, ( X, F). Using the characterization 28.5. of 2-factorable operators, it suffices to check that for all n E N and all isometries A E .C(q,G)

Fix such an operator A and let A :L2(Kn,7") 4 L2(Kn,7") be the map fIt is obvious that A(Gn) c Gn and

f o A'.

A = I - ~ o A ~ , , ~ I : ~ - ~ . Moreover, A QD idx is an isometry on L2('yn) Q D A ~ X (by the invariance of 'yn under isometries). It follows that the diagram

commutes - and the remarks made before the theorem show that

which ends the proof. 0

30. The Vector Valued Fourier Transform

401

The natural quotient map Q : 11 + 1, (for 2 < p < 00) does not factor through a Hilbert space. Since 11 has cotype 2 and 1, type 2, it follows that in general neither the operators in 2 2 o C2 nor those in 2 2 n C2 factor through a Hilbert space. Note also that Kwapieli’s type/cotype theorem improves one of the results in 26.1. stating that every T : 1, -+ 1, factors through 1 2 if 1 5 s 5 2 5 r < 00. On the other hand, Pisier’s factorization theorem (to be proven in the next section) extends Kwapied’s result in certain situations since E’ has cotype 2 whenever E has type 2 (by Ex 8.14.).

30.6. Fourier operators are of type 2 and cotype 2 - this is the content of the next

PROPOSITION: max{Ts(T), Cz(T)} L 2 Lg(T) for all T E C ( E ,F). PROOF:In order to show that T E C3(E, F) has type 2 fix 30.3.(c) shows that for all w E 0,

z1,

...,t n E E. Theorem

Integrating with respect to p n gives

Kahane’s contraction principle (see Ex 8.16.) implies that

so that Tz(T) 5 2 Lz(T). The proof that the operator T has cotype 2 follows along the same lines. The constant 2 is actually superfluous: at the end of this section a more general result (with constant 1) will be proven - but we preferred for the sake of simplicity and elegance also to give the proof above. Now the main result of this section follows easily:

THEOREM (Kwapieri): A complex Banach space E is isomorphic t o and only if the E-valued Fourier transform

is continuous.

a

Hilbert space if

402

SO. The Vector Valued Fourier Transform

PROOF: If this is so, i d E E 23 and hence E has type 2 and cotype 2 by the previous propmition. Therefore, idE = i d E o i d E is 2-factorable by Kwapied's type/cotype theorem, which concludes the proof. 0 Recall that the isometric version of this result was already given in 30.4.: If L 3 ( i d ~ )= 1, then E is isometric to a Hilbert space. It is not known whether the operator version of this result holds: Does every Fourier operator factor through a Hilbert space? In other words, is 2 3 c CZ?A partial answer is the

REMARK: 2 2 = 23 o 23

.

PROOF: Clearly 2 2 C 2 2 o 2 2 C 23 o 2 3 and conversely 2 3 o 2 3 Kwapieri's type/cotype theorem. 0

c &2 o 2 2 c 2 2

by

30.7. It is worthwhile to have a look at the structure of the proof of the results above: If one says that an operator T E C(E,F ) has Fourier type 2 if it satisfies for some c 2 0 and all t k E E

and - only for the moment (since this notion is not the usual one) - Fourier coiype

then the following has been shown in this section: (a) An operator is a Fourier operator if and only if it is of Fourier type 2 and if and only if it is of Fourier cotype 2 (theorem 30.3.). (b) Fourier type 2 implies Rademacher type 2, the same for cotype (proof of proposition

30.6.). Rademacher type 2 implies Gauss type 2, the same for cotype (lemma 30.5.). (d) The composition S o R of a Gauss type 2 operator R and a Gauss cotype 2 operator S factors through a Hilbert space (proof of Kwapied's type/cotype theorem). (c)

30.8. In light of this structure the following type/cotype result with respect to an arbitrary orthonormal basis of an L2(p) is of some interest (the result is also due to Kwapieri [lSl]). For this, take an orthonormal basis ( f n ) n E ~of some separable La(R, p ) and consider the isometry

I : L2(p)

-

42

30. The Vector Valued Fourier Tran~form

403

defined by g-((glfn))nEN. For T E $(El F) continuity of I @ T : L ~ @ A -+~ E &@A~F means a sort of cotype 2 inequality:

- and the continuity of I-'

(g, T means a type 2 inequality (with respect to (fn)). It is obvious that every isometry I from L2(Q, p ) onto l 2 fixes (via the unit vector basis of 4 2 ) an orthonormal basis of L 2 ( n , p ) .

PROPOSITION: Let ( Q ,p ) be a non-atomic, separable measure space, I : L2(R, p ) -, 42 a surjective isometry and T E 2 ( E ,F ) . (1) If I @ T : Lz(Sl,p) ( g , E ~ -+ ~ C2 € 3 F~ i s ~continuous, then T is of (Rademacher) cotype 2 and C2(T) 5 111 (g, T : . .II. ( 2 ) If T : e 2 @aa E -+ L2(R, p ) € 3 F ~i s ~continuous, then T i s of (Rademacher) fype 2 and Tz(T)5 11I-l €3 T : . .)I.

PROOF:Clearly, one may assume that p is normalized. Then, by the Halmos-von Neumann theorem (see Brown-Pearcy [MI, p.181) the measure spaces ( Q , p ) and ([0, l ] , A ) (Lebesgue measure) are isomorphic, i.e. there is a measure preserving, monotone, additive and bijective map Qi from the classes of p-measurable sets onto the classes of A-measurable sets. Clearly, Qi induces a bijective isometry

between the classes of E-valued step-functions, hence a surjective isometry L z ( p , E) + L2(A, E ) . This means one may restrict one's attention to the Lebesgue measure on [0, 11. Thus, in case (1) there is an orthonormal basis (fn) of Lz[O,11 such that

for all 2 1 , ...,en E E. One has to show that the f k can be replaced by the Rademacher functions r k ; this will be done with a gliding hump approximation of the functions r k by certain fm. Fix x1 , ...,Zn E E. Since for the discrete Rademacher functions

holds for every strictly monotone 9 : N -+ N,the same is true for the sequence ( r k ) and 80 it is enough to find a subsequence of the (rk).

90. T h e Vector Valued Fourier Transform

404

Take c

> 0 and $( 1) E 4v such that for do) 9

:= E(rlI.fk).fk k=l

the function hl := Ilgll;'g there are

satisfies

11.1

-

5

2 - l ~ .Assume,

p(1) := 1 < ~ ( 2 < ) ... < P(n) and $(O) := 0 < $(1)

hk E S p a

{fm

1 $'(h - 1) < m Id'(k)I

such that llr9(k)- h k l l 2 5 2 - k ~: Since (rjlf,,,) p(n 1) > p(n) such that

+

lIhkIl2 4

< ... < $ ( n )

=1

0 for j

4 00,

+(n)

E

l(f$Y(n+1)1fk)I2

is small ;

k=l

+ 1) > +(n) and some normalized hn+l E span {fm I $(a) < m 5 4(n + 1))

this easily leads to some $(n

such that [lr9(n+~l - hn+l[12 5 2-("+')~. The construction showed that h n l h , whenever n

# m.

by induction, that

Now

- for all c > 0. The second claim follows in the same way. 0

there is an index

405

90. The Vector Valued Fourier Transform

The result does not hold for arbitrary orthogonal sequences: see Ex 30.7.. One application: If G is an infinite abelian compact group with normalized Haar measure, then the dual group G is a discrete topological group and the Fourier transform 5 : Lz(G) -+ &(G) is an isometry. If Lz(G) is separable, then C is countable.

COROLLARY: Let G be a compact, abelian infinite group such that Lz(G) is separable. If for a Banach space E the E-valued Fourier transform

5 8 idE : Lz(G)

E

6 9 ~ 2

as continuous, then

-

P Z ( ~8)A a E

E is isomorphic to a Hilbert space.

PROOF:Since G has no discrete points and each atom of a Borel-Radon measure contains a one-point atom, the Haar measure has no atoms in this case. Therefore the proposition shows that idE is of (Rademacher) type 2 and cotype 2, hence the claim follows from Kwapieli’s characterization 30.5. of Hilbert spaces. El The assumption that Lz(G) is separable is actually superfluous: this was shown by Rubio de Francia and Torrea [234].

30.9. Peetre defined a Banach space E to be of Fourier type p (for 1 Fourier transform satisfies the Hausdorff-Young inequality

p

5 2) if the

for all f E L , ( R ) 8 E. It is obvious that for p = 2 this means i d E E 28, which is the same as having Fourier type 2 in the sense of 30.7.;it follows that these are the spaces isomorphic to a Hilbert space. A deep result of Bourgain statys that each Banach space of type p > 1 has Fourier type r for r with r‘ = 18Tp(id~)P. See Konig [149] for references and the behaviour of Fourier coefficients of certain differentiable E-valued functions.

Exercises:

Ex 30.1. Recall that a norm (on a real or complex vector space E) comes from a scalar product if and only if for all t,y E E it satisfies the parallelogram identity IIt

+ Y1I2 + 1.

- Y1I2 = 2(11412 + llYll2)

-

(&)If II~+y11~+llz-y11~ <2(11~11~+llyll’)fora l l z , y E E ( o r >for a l l z , y E E), then the norm 11 11 satisfies the parallelogram identity.

406

90. The Vector Valued Fourier Transform

(b) If for a Banach space E the type 2 constant T z ( i d E ) or the cotype 2 constant C z ( i d ~ is ) 1, then E is isometric to a Hilbert space. Ex 30.2. Let E be a complex Banach space. (a) For z , y E E define f ( t ) := z for - A 5 t < 0 and f ( t ) = y for 0 5 t < A. Show that f E &(T, E) and

c Ili(.)l12(1. =

+ Y1I2 + 112 - Y1I2)A/2

*

nEZ

(b) Deduce from (a) and Ex 30.1.: If

for all f E L2(T,E) (or 2 for all such f), then E is isometric to a Hilbert space. This argument is due to Kandil [141]. Ex 30.3. Define Gauss type p and Gauss cotype q for operators. Use Kahane's inequality for Gauss averaging (8.7.) to show that for T E 2 ( E , F) and 1 5 p 5 2 5 q<m:

(a) T is of Gauss type p if and only if it is of Rademacher type p. (b) If T is of Rademacher cotype q, then it is of Gauss cotype q. Hint: 30.5. and Ex 8.9.. Ex 30.4. Use Kwapieri's type/cotype theorem to reprove the result from 26.1. that for 1 5 q 5 2 5 p < 00 every operator from an X;-space into an Xi-space factors through a Hilbert space. Ex 30.5. If 1 5 q 5 p 5 00, then every Fourier operator tp -+ t, factors through a Hilbert space. Hint: First q 2 and 30.5.. Ex 30.6. Show that for 1 5 q 5 2 5 p 5 00,every operator T E X ( t , , t , ) is a Fourier operator. Hint: 26.1.. Ex 30.7. If fn := 2 - " / 2 ~ p - 1 , 2 - n + 1 ~ , then (f,) is an orthonormal system in L2([0,11) and

holds for all Banach spaces E and 21, ...x, 6 E. This example (showing that proposition 30.8. does not hold for arbitrary orthonormal systems) is also due to Kwapieli.

Ex

30.8. Let

E be a complex Banach space.

(a) Show that 11f(n)11B 5 11f112 for all f E Lz(T,E) and n E N . (b) The E-valued exponential polynomials are norm-dense in L2(T, E). Hint: Use C(T, E) = C(T)&E. (c) Riemann-Lebesgue lemma: If f E L z ( T , E ) , then the sequences ( f ( n ) ) n E N and (f(-n))nEN converge (in E ) to zero.

31. Pisier’s Factorization Theorem

407

Ex 30.9. Show that for each Banach space the following statements are equivalent: (a) E is isomorphic to a Hilbert space. (b) If C= :o 11Zn112 < 00,then C2=oexp(in.)Zn E L2(T, E). (c) The sequence of Fourier coefficients of every f E &!(TIE)is in & ( E ) . Ex 30.10. If H is a Hilbert space and f E L2(T, H ) , then the Fourier series

converges in L 2 ( T , H ) to f.

3 1. Pisier’s Factorization Theorem

Kwapieli’s type/cotype theorem states that every operator T from a space of type 2 into a space of cotype 2 factors through a Hilbert space. Pisier’s factorization theorem - in some sense an “abstract” version of Grothendieck’s theorem - extends this to compactly approximable operators T E C(E,F) if E’ and F have cotype 2; note that E having type 2 is more than E’ having cotype 2. The proof of the theorem is taken from Pisier’s book [225]. Moreover, a non-accessible tensor norm, also due to Pisier, will be presented.

31.1. Let R : &(D,p ) + L2(D,p ) be the orthogonal projection onto the closed space Rad of the discrete Rademacher functions (see 8.5. and Ex 9.16.). As an orthogonal projection R is self-adjoint in the Hilbert space L2(D, p ) . Identifying L2(D, p ) with its dual through the usual duality bracket J fgdp, it can easily be seen that

- -

R’f= R‘f= Rf . Since the Rademacher functions en are real-valued, it is obvious that RT = R f and hence R’ = R. An operator T E C ( E , F) is called K-convez if

K(T) := llR @ T : L z ( D ,P ) @Aa E

-*

L2(Di P ) @Aa FII < 00 .

It follows (see 17.4. and 7.4.) that the class f f C of K-convex operators together with K forms a maximal, injective and surjective Banach operator ideal. Since

408

91. Pisier's Factorization Theorem

commutes, ffC C ffCd""' holds and hence, ffC = ffCd""' isometrically (17.8.): The operator ideal (ffC, K)of K-convex operators is completely symmetric. It is easy to see (using A:) that the identity operator on a Hilbert space and hence all 2-factorable are K-convex; moreover, K 5 L2. Recall from Ex 9.16. that idl, is not K-convex (proposition 7.8. stated that if there is a non-K-convex operator, then idc, is not K-convex) and Pisier's result that E is not K-convex (if and) only if E contains the 6 uniformly. By another result of Pisier's (mentioned in Ex 7.13.) it follows that E is K-convex if and only if it has proper type.

31.2. The dual of a type p operator has cotype p' (see Ex 8.14.) - but the converse is false: co does not have type 2 (Ex 7.14.), but el has cotype 2 (see 8.6.). It is clear from the definitions of type p and cotype q operators that this gap in the duality relation may be filled by K-convexity; more precisely,

PROPOSITION: For 1 < p 5 2 the following two relations hold: (1) RCo CTa'C 2 , and T , 5 K o C$"' . (2)

%, o ffC c %Par and TP"'5 C,) o K .

PROOF: Since ffCduo'= ffC, it is clear that L2

L~

R

* Rad J

(2) implies (1). For (2) use the operators JP'

-

+ l,# IP

Rad

e,

9

which satisfy (J,' o A)' = J o 1,. If S E ffC(E, F) and T E Cp#(F, G), the definitions show that the operator

has norm

5 C,#(T)K(S). The diagram

yields that T,((To S)') satisfies the same norm estimate - which is the claim. 0

409

91. Pisier's Factorization Theorem

In particular, if E is K-convex (i.e. - by the result of Pisier - does not contain the uniformly), then E has type p if (and only if) E' has cotype p'. 31.3. The following two lemmas prepare the way for the proof of Pisier's factorization theorem. Recall from Ex 8.15. that the Walsh functions E A (where A runs through the finite subsets of N)form an orthonormal basis of &(D,p ) .

LEMMA1: For every 6E]O, 11 there is a signed Bore1 measure D = { - l , l } N such that

I k & ~ d v 651 6 if A C N and IAl

(3)

V6

on the compact group

>1.

PROOF: For 0 E [-1,1] and n E N define n

It follows that

If"eld/l = fi(1-k k=l

and, for A

(kdpl)

=1

Di

c { 1, ...,n},

For each n E N put

1 Fa := 2 4

m).

Then the sequence F f d p in C(D)' is norm bounded (by Since C(D)is separable, it has a weak-*-convergent subsequence with limit V6 : It is obvious that this measure satisfies all the requirements.

K 5 Lz - but this estimate can be improved in a certain sense: This will be a crucial point in the proof of Pisier's factorization theorem. LEMMA2: Take T E &(E, F ) and 6 E]O,l]. Then

410

31. Pisier’s Factorization Theorem

PROOF: Fix

n f = x & A k @Xk k=1

ELz(DiP)@E

with distinct A1 ,...,An. The idea of the proof is to construct a decomposition

these estimates imply that

as desired.

Let

v6

be the measure constructed in lemma 1 and define n

.

since SD&,,,dv6 = 1. Note that the Rademacher functions E,,, (and hence all Walsh functions) are multiplicative on the group D - and p is actually the normalized Haar measure on D,i.e. invariant under multiplication. It follows that

31. Pisier's Factorization Theorem

411

m,

since llvall I and hence A 2 ( i d €3 T(f i ) ) 5 llTllJ17'8a~( f). The operator T is 2-factorable - so Ex 28.2. can be used to obtain the estimate for f 2 : Indeed, take 2' E E'; then

by the biorthonorrnality of the Walsh functions and lemma 1. It follows from Ex 28.2. that A 2 ( i d € 3 T ( f 2 ) L21 ; F ) I 6 L 2 ( T ) A 2 ( f L2, ; E) 1 which is the remaining estimate. 0

31.4. Now the main result of this section can be proven. It was obtained by Pisier [222] in 1980. PISIER'S FACTORIZATION THEOREM: Let E and F be Banach spaces such that E' and F have cotype 2. Then every operator T E & ( E l F ) (in particular, every finite rank operator) satisfies L2(T) I (2c2(E')c2(F))3/211Tll. In terms of tensor norms: E

5 ~2 5 ( 2 C 2 ( E ) C 2 ( F ) ) 3 / 2 ~

onE@F

whenever E and F have cotype 2.

PROOF:Since w2 and E are totally accessible, C z ( E ) = Cz(E") and 22 and w2 are associated, it is clear that the claim about the tensor norms follows from the one about the operators. Given T E & ( E l F ) Kwapiefi's type/cotype theorem 30.5. yields L2(T) I T 2 ( T ) C z ( F ).

Proposition 31.2. and lemma 2 imply that

+

1

T2(T)L CZ(E')K(T) I C 2 ( W [6Lz(T) zllTll] for every 6 E ] O , l ] ,and hence

412

91. Pisier’a Factorization Theorem

Choosing 6 := (2Cz(E‘)Cz(F))-’,gives the conclusion. 0

31.5. An operator T E C ( E ,F ) is called compactly approzimable whenever there is a net (T,) of finite rank operators converging to T uniformly on all compact subsets of E (see Ex 5.17. for a characterization of these operators). It is called A-compactly approzimable if the T, can be chosen of norm 5 X(lTI1.

COROLLARY 1: Let E’ and F have cotype 2 . ( 1 ) Then every A-compactly approzimable operator T E 2 ( E ,F ) i s 2-factorable and

( 2 ) In particular, if one of the spaces E or F has the bounded approzimation properly, then &(ElF ) = & ( E , F). Pisier [225] actually proved that this result holds for all compactly approximable operatom - and therefore ( 2 ) is also valid if E or F only has the approximation property.

PROOF: The fact that the unit ball of &(E, F) is closed in 2 ( E ,F) with respect to the weak operator topology (see 17.21.) together with the theorem shows immediately part (1) of the claim. 0 Using the fact that 2 ( E , E ) = C z ( E , E ) implies that E is isomorphic to a Hilbert space, gives the

COROLLARY 2: A Banach space E with the (bounded) approzimation property such that E and E’ have cotype 2 , is isomorphic t o a Hilbert space. As has aIready been mentioned at various times (see 4.8.) Pisier has constructed infinite dimensional Banach spaces P such that P P = P @c P; his construction even gave that P and P’ have cotype 2 . Obviously, such a space cannot be isomorphic to a Hilbert space and hence i d p shows that in corollary 1 (and 2 ) an approximability hypothesis is indispensable. 31.6. Pisier’s space P and his factorization theorem serve also to find a non-accessible tensor norm, or equivalently, a non-accessible maximal normed operator ideal. The following construction was given to us by Pisier. For M, N 6 FIN define

it is easy to see that this defines a normed operator ideal on FIN. Denote by ( % p , Ap) its maximal hull and by a p the associated (finitely generated) tensor norm.

91. Pisier's Factorization Theorem

THEOREM (Pisier):

( a p ,A p )

and

cup

413

are neither lef2- nor right-accessible.

PROOF:By 21.3. it is enough to treat the operator ideal. Assume that Q p is rightaccessible; it will be shown that this implies that P is isomorphic to a Hilbert space. By the maximality of the ideal of 2-factorable operators it is enough to check that La(15 : M c,P) is uniformly bounded for M E FIN(P). Obviously, Ap(Z&) = 1 and 80 accessibility gives M&P

It is straightforward from the defining inequality of cotype 2 that with P' also &(P') = (&(P))' has cotype 2. Since 1; o S has finite rank, Pisier's factorization theorem gives 0 S) I ~ ( ~ C ~ ( ~ ~ ( P ' ) ) C Z . (P))~'~ Lz(15) 5 JJRJJL2(Ig

: P -+

To see that % p is not left-accessible one considers in the same way Lz(Q! for L E COFIN(P).

P/L)

Qp

The injective hull of 2lp (which is associated with the tensor norm ap\) is rightaccessible (by 21.1,); however, the same argument as before, the metric approximation property of .&, and the fact that the ideal 2 2 of 2-factorable operators is injective imply the

COROLLARY: (a?, A?) and

cup\

are right- but not left-accessible.

These were non-accessible maximal ideals; see Ex 31.4. for non-accessible ideals consisting of approximable operators.

31.7. Since tk, and 4, are C:-spaces and hence have cotype 2 (see 8.6.), Pisier's factorization theorem (more precisely, its corollary 1) implies that

(*I

2(&J14) = .c2(&0,4)*

This is nothing other than Grothendieck's inequality, which can be seen as follows: The little Grothendieck theorem ( 2(!, ,.&) = !?32(.f,, &)) and Kwapieri's factorization theorem give

qt, ,4 ) = .c2(&0,e l ) = Y32du41 PZ(k3,e l ) = 9 2 ( & , 4 ) Since w i = wz

-

0

'

the representation theorem for maximal operator ideals yields is r 5 ew2 on t,@c, for some c :This is Grothendieck's inequality in tensor form. See Ex 31.3. for the constant c. 92,

( . f o o @ l r ~ o ) '= ( t , ~ w 2 c o ) ' which

414

91. Pisier’s Factorization Theorem

Thus, Pisier’s factorization theorem provides a new proof of Grothendieck’s inequality - and this justifies labelling this factorization theorem as an “abstract” form of Grothendieck’s inequality (in the form (*)). Note that toodoes not have type 2 - and hence Kwapiexi’s typelcotype theorem does not give (*). The use of the little Grothendieck theorem also implies part (1) of

COROLLARY 3 (Maurey): Let E have coiype 2. Then (1) .C(lm E ) = vz(tw E ) (2) !J3z(E,F) = v l ( E ,F ) for each Banach space F. 1

1

The second part follows from the first through Saphar’s result 20.19.. The content of the corollary was obtained in 1974 by Maurey [186]. Other proofs of Maurey’s result will be given in 32.9. and Ex 32.7.. 31.8. The last result and proposition 26.4. read:

-

!&,(ElF ) = PI(E,F ) i f p E [I,21 and E has cotype 2 P q ( E ,F ) = !&(El F ) if q E [2, co[and F has cotype 2

.

Since ipp gi, is totally accessible, the embedding lemma implies the following result concerning the equivalence of tensor norms:

COROLLARY 4: Lei p €]1,2] and q E [2, m] and El F Banach spaces. Then an E @ F g& gi i f E’ has cotype 2 dw d, i f E has cotype 2 g: g; i f F has cotype 2 d2 dp i f F’ has cotype 2 .

---

Exercises:

If E’ and F have cotype 2, then Z ( E ,F) C & ( E , F). Ex 31.2. (a) If E and F have cotype 2 and p , q €11, oo[satisfy l / p + l / q 2 1, then E F = E BeF holds isomorphically. (b) For the dual result let E‘ and F‘ have cotype 2: Then f and w; are equivalent on E @ F, but x and wz are in general not; a sufficient condition for x and w; being equivalent on E @ F is that E or F has the (bounded) approximation property. If E and F‘ have cotype 2, then it follows in the same way that x and wz are, in general, not equivalent on E’@ F, but Pisier ([225], p.47) showed that the nuclear norm N and Ex 31.1.

the 2-dominated norm Dz (associated with w:) are equivalent on %(El F).

32. Mizing Operatore

415

Ex 31.3. Show that the “abstract” proof 31.7. of Grothendieck’s inequality gives the estimate KG 5 8 K i G for the Grothendieck constant. Hint: 8.6.. Ex 31.4. (a) Show for the factorization ideal 2lp from 31.6. that %p o 3 is left- but not right-accessible. Hint: Ex 25.3. and proof of 31.6.. Recall that the minimal hull 2 p l = 5 0Qp o 3 is accessible by 25.3.. (b) Define the quasi-Banach operator ideal (Cp, C p ) by Cp(E, F) := {T E E(E, F) IT = Ti CP(T) := inf{(Ap

0

+ T2, Ti E ap 05,Tz E 5 0ap}

II ll)(Tl) + (I1 II 0 AP)(T2)1 -

Show with Ex 21.2. (b) that (Cp, C p ) is an ideal of approximable operators which is neither right- nor left-accessible.

32. Mixing Operators

The following situation has appeared already at various times: An operator T (or idE of a Banach space) has the property that S o T is absolutely p-summing whenever S is absolutely q-summing. This section is devoted to a systematic study of these

“mixing” operators/spaces. Mixing operators are characterized by interesting integral inequalities and by a certain splitting property (which explains the name “mixing”); moreover, they are almost equal to absolutely (r,s)-summing operators for appropriate r and 8 . Many of the results to be presented are at least implicitly due to Maurey [186]; the ideal of mixing operators was studied systematically by Pietsch in his monograph on operator ideals and by Puhl [226].

32.1. Let p,q E [ l , ~ An ] . operator T E E(E, F) is said to be (q,p)-mizing if for all Banach spaces G and S E V q ( F G) , the composition S o ‘ I is‘ absolutely p-summing. With Mq,p(T) := sup{Pp(SoT) Pq(S) I 11 it is clear that, in the terminology of quotient ideals introduced in section 25, the ideal 9JIq+, of ( q , p)-mixing operators is isometrically g;’ o gp:

I

(9JIq,p,Mq,p)= (V;l

O

Vp,

pi’ O PP) .

The quotient formula 25.7. implies that M,,,) is a maximal Banach operator ideal - and later on it will be easy to see that it is injective but not surjective (Ex 32.1.). The name “mixing”, due to Pietsch, will become clear in 32.7..

416

$2. Mixing Opemtors

Since VP is injective and 3y = ‘Q, it is also easy to see that mq,p

= 3;’

0

Vp

holds isometrically. The ideals Vp and DP are accessible and 31, = 3pmin is the ideal of pnuclear operators, hence the quotient formulas 25.7. give a result due to Puhl [226]:

PROPOSITION: The following relations hold isometrically:

mf,p:= ! p i l o

q.tp = 3;’

0

qp= p,t 0 3;’ = 3,, 0 371 = 31,)

0

3 ’1 ;

=

= pP’ 0 V,)*= pP’ 0 3J . Note that this result implies various composition formulas, as for example

32.2. Some observations - and translations of former results into the language of mixing operators deserve to be stated explicitly: (1) It is obvious that 9Jl,,p = C whenever q I p and M,,p = yp.So only the case 1 5 p < q < 00 gives something new. (2) Pietsch’s composition formula !Jlqo Pr C VPfrom 11.5. gives:

V r C M,,p and

M,,p 5 P,

if p , q , r E [1,00] with l / q

+ 1/r = l/p ;

this will be extended to absolutely (r, s)-summing operators at the end of this section. Since in Pietsch’s composition formula the index p is best possible (see Ex 11.23.), it follows that if 1 I P - 6 < P I q ; m,,p $t m,,p-a see also Ex 32.14.. (3) Saphar’s result 20.19. means that:

E E space(Mf,l)

if and only if C ( l , , E ) = ‘13q~(~wr E) ;

I

moreover, M , , l ( i d ~ = ) sup{P,t(T) 112’ : loo+ Ell 5 1). This also follows from the quotient formula % ,1 = I , 3,t o 3k1= !+oIf 2L1. ! F’urthermore, this formula also implies that 1F ) = V &J 1 F )

m,&a

holds isomorphically for all F. (4) The little Grothendieck theorem says that a(&,H) = V z ( l , , H) for every Hilbert space H , hence idH E space(9Jl2,l) by (3). Looking at the constants in Saphar’s result, one gets

M z , i ( i d ~5 ) KLG

417

$2. Mizing Operators

- even = KLG,see Ex 32.4.. Much more generally: E E space(9Jl2,l)

f o r all colype 2 spaces E

- this is just a reformulation of Maurey’s result in 31.7.. Conversely, every Banach lattice in space(m2,l) has cotype 2 (see Pisier [225], p.108) - but it seems to be unknown whether or not this holds for arbitrary Banach spaces. In 32.9. and Ex 32.7. alternative proofs of Maurey’s result are indicated. ( 5 ) For cotype q > 2 less is true. Theorem 24.7. gives:

E E space(%lq’--r,l)

if E i s of colype q > 2 .

Since L , is of cotype max{q, 21,it follows for the r-factorable operators that

2,.c mP,l

if 2 < r

< p’ < 00

and

2,

c M2,l

if 1

Ir 5 2 .

(All this was already stated in 26.4.(1)and (2)- including the constants involved.) (6) In particular, the sequence space el is in space(9Jl2,~).One can show, however, that el is not in space(mq,l)for q > 2;this will follow - as will some other counterexamples -from considerations about the ideal of absolutely ( r ,s)-summing operators: m,,~c !J?,I,l by 32.10. and t1 4 space(Vq,,l) by proposition 34.11.. Since C[O,l] c, L,(A) (Lebesgue measure) is in Vqbut not in VP for p < q (this was mentioned in 11.3.),

C(P, 11) 4 space(M*,p) and (by local techniques) also 1, !,I

if 1 I P


,

4 space(9Jl,,p). Later on (32.8.) it will be shown that

4 space();m,,p)

if 1

5p
- this reveals that the result in ( 5 ) (about cotype q > 2) cannot be improved. 32.3. The calculus of traced tensor norms and tensor products of operator ideals developed in section 29 (and the quotient formula 25.7.) shows that (see 29.8.) m,,p

= VF1 0 Pp = (V;0 Vq)*= (3p’ 0 Vq)*= (3pj @ Vq)’ 0 qp= (q;0 &)* = (JP‘0 3J = (DP’ QD 3J ,

mq,#= 3;l

and hence the associated tensor norm is, by definition of A @ B (see 29.7.):

Moreover, by 29.11. Qp’ 8 s;l

= Qp’ QD 9, = Qp’ @L&)

A:

418

92. Mizing Operators

for every infinite dimensional L,t(p). The tensor product characterization 29.4./25.8. of operators in (a@%)* and the accessibility of gq and g;, give the

PROPOSITION: Let p , q E [l,oo] and /3 sfdements are equiualenf:

(4 T E %P(E,

= gq o r g;,. For T E C(E,F ) the following

-

F) * E G @ pF is continuous for all Banach spaces G. (b) idc @ T : G (c) For one infinife dimensional Lqt(p) (and then for all) idLq,(p)@ T: Lq,(p) @gpn

E

-

Lq'(p) @Aq#F

is continuous. I n this case, Mq,P(T)= J J ~ ~ L , @ , (T ~I1 ) = Ilidc, 8 TI1 2 I l i d ~8 Tll.

This result has a nice application to the continuity of tensor product operators between spaces of Bochner integrable functions - in some sense an extension of Beckner's result 7.9. (see also 26.3.), which is due to Carl-D. [21] :

PROOF: Using the Chevet-Persson-Saphar inequalities 15.10., a look at the diagram

gives the result. 0 Various special cases of this result are important enough to be stated explicitly:

(Take 8 = p.) Certainly the most important case is s = 2. Since cotype 2 spaces are in ~ ) all p (by 32.2.(4) and Ex 32.2.(a)), this gives space(3J22,l)C ~ p a c e ( M 2 , for

92. Mizing Operators

419

(2) S 8 T i s continuous in each of the following three cases: - F of cotype 2 and p 2 2 . - E’ of cotype 2 and q 5 2 . - E’ and F of cotype 2 .

Note the special case: E = L, and F = L, for s 5 2 5 r. This gives a variant of the result in 26.3.. The fact that +-r E space(m&1) c space(rnq,p) whenever q

5 2 (see 32.2.(5)) has another nice consequence:

(3) Let 1 5 p 5 q 5 2 and e T E .C(Lq(pz),F) the operator

> 0. For all operators S E 2(L,+,(p1),Lp(v)) and

i s continuous.

PROOF: Take s = q . 0 32.4. There are two characterizations of mixing operators - one in terms of integral inequalities and one in terms of a splitting property. These descriptions of the quotient = pi1o ppof (q,p)-mixing operators make them important in many ideal 9Jtq,p situations. First the integral description, which has the flavour of the GrothendieckPietsch domination theorem:

PROPOSTION: Let 1 < p

<

q < 00 and T E C ( E ,F ) . The following statements are equivalent: (a) T is (q,p)-miting. (b) There is a constant c 2 0 such that for each probability measure p on B F l there is a probability measure v on BE’ such that

for all z E E. (c) There is a constant c 2 0 such that

420

92. Mixing Opemtors

f o r a l l q , ...,z m € E a n d 4 ,..., 5 / n € F ’ . In this case, M q , p ( T = ) minc, where the minimum is taken over all c satisfying (b) or all safisfying (c).

PROOF: If T E Mq,p(E, F ) = !Qglo !Qp(E, F) and p is a probability measure on B p , then

I :F Y

--

Lq(p)

(0)

is absolutely q-summing and hence I o T E !Qp(E,L q ( p ) ) : The Grothendieck-Pietsch domination theorem gives a probability measure Y on BE’ such that for all t E E

Since P p ( l o T) 5 M q , P ( T ) P q ( I=) Mq,p(T),statement (b) follows. In order to show that (b) implies (c) assume .tq(dk) = 1. Then

is a probability measure on B p , hence there is a v as in (b). Therefore,

o !Qp : By what was just Finally, it remains to prove that (c) implies that T E !Qil said, condition (c) means that all discrete probability measures p on B p satisfy

for all 21, ...,zm E E. Since the set of all discrete probability measures on Bp is u ( M ( B F I )C(Bp))-dense , in all probability measures p , it follows that this inequality holds for all such p. Hence, if S E !Qq(F,G), then another appeal to the domination theorem gives P,(S o T)5 cPq(S).0 Note that for q = co condition (b) can be interpreted as the domination theorem for !Qp

=~Ccl,p.

32. Miztng Operators

421

32.5. Using this characterization, one can “improve” the index p for E in space(%lq,p): this is the content of the following result which one might call the Maurey-Pisier eztrapolation theorem (see Pisier [225], p.60 and p.69).

THEOREM: For 1 5 p < q < 00 let E be a Banach space such that

%#,

F ) c Vp(E9 F )

for all Banach spaces F. Then

Qq(E,F ) c Vl(El F ) for all Banach spaces F. In other words, space(9??q,p)= space(mq,l) . Recall from 32.2.(2) that Mq,p@ %lq,l whenever p > 1, hence the result does not hold for operators. Moreover only the case q < 2 is interesting by 32.10..

PROOF: Fix a probability measure p on B E ’ . By the previous proposition one has to find a probability measure Y such that for all z E E

- with a constant c independent of p. Let vo := p. By assumption and induction there are probability measures un such that

where co := Mq,p(idE). Define v := Then Holder’s inequality gives 00.

Since p = vo,this implies that

c,”==, 2 - ” ~ n and 0 ~ ] 0 , 1by [ l/p = Q+(l-O)/q. M

.

422

32. Mizing Operators

which gives M,,l(idE)

5 (2Mq,,(id~))'ie .

For an application to spaces which satisfy Grothendieck's theorem (the G.T. spaces from 20.18.) see Ex 32.9.. 32.6. Maurey's factorization theorem 18.9. gives a very useful characterization of operators in me', which - for spaces - is also due to Maurey [186], th6orhme 23.

< <

+

THEOREM: Let 1 p q 5 00 and l/p = 1/r l / q . For T E C ( E ,F ) the following five statements are equivalent: (a) For every S E C(L,(v),G ) and U E C(F,L p ( p ) ) (for some measures p and v) the operator s '8 U T : &(v) @A, E G @A; & ( p )

-

is continuous.

(b) For every U E C(F,t p )the operator idt4 8 UT : e,

@A4 E

-

A?, @A;

ep

i s continuous.

(c) For every U E C(F,L,(p)) there are a g E L , ( p ) and R E C ( E , L , ( p ) ) such that U o T = MBo R. (d) For every U E C(F,A?,) there are a A E A?, and R E C ( E ,A?,) such that U o T = DxoR. (e) TI is ( q , p)-mizing. In this case,

I

Mq,P(T')= sup{llidt, '8 UTll IlU : F

= sup{min{llAll,llRII

+

tpll 5 1 ) =

I U o T = DAo R } I JIU : F

--+

5 1) .

PROOF: Maurey's factorization theorem shows the equivalences of (a) and (c), as well as of (b) and (d). Since obviously (a) f i (b) and (c) f i (d) the implications (b) f i (e) and (e) f i (a) remain to be shown. Assume (b): A closed graph argument garanties the existence of a constant c 2 0 such that for all m, n E hr Ilidt; @ U T : el; @b4E for all U E & ( F , q ) . For U = ~ 1 . . ., , c n E E

cj"=l $Be,

-

@A;

TJJ 5 ~Ilull

one has IlUll = w p ( $ ) , and hence for all

52. Mizing Operators

Since

BE is weak-*-dense in BE##, this inequality also holds for x i

EE

- and proposition 32.4. gives that T‘ is (q,p)-mixing.

423

E E” instead of

Now assume TI E 9J14,P(E,F)and take U and S as in (a). It follows from the tensor product characterization 32.3.(c) that

has norm

5 Mq,JT’). Since gi, is accessible, this implies that

hence the diagram

32.7. A consequence of this is Maurey’s splitting theorem:

+

COROLLARY 1: Let 1 5 p 5 q 5 00 and 1/r l / q = l/p. A n operator T E C ( E ,F) i s (q,p)-mixing if and only if for each weakly p-summable sequence (xn) in E there is a sequence (An) E t, and a weakly q-summable sequence (yn) in F such that

for all n E IV. In ihis case,

This splitting property of mixing operators is sometimes taken as a definition of (p, q)mixing operators. Pietsch calls weakly p-summable sequences (z,,), which split into Zn = Anyn as above, (q, p)-mixed sequences - and this explains why he gave operators in M,,p a related name: (q,p)-mixing.

PROOF:Let T be (q,p)-mixing and wp(xn;E) 5 1. Since T” is (q,p)-mixing, (d) of the preceding theorem gives that for V : -+ E defined by Ven := xn there is a #, ! .

factorization (read c, instead of looif p = 1)

424

32. Mixing Operators

F'-

4

T'

El-

V'

Lp

/

IlRll

ll4lr I Mq,p(T)

1

since llV'll = wP(2,) 5 1. It follows that X,Rle, = Tx, E F, which gives the splitting because of w q ( R ' e n ) 5 llRllw,(e,;t,') = 11Rl1. Moreover, c := supinf{ ...} 5 M 4 , P ( T ) . Conversely, take S E V q( F ,G ) and wp(z, ;E ) 5 1. Then

Lp(STxn) = e p ( L S ~ n )I tr(h)tq(Syn) I G ( L ) P q ( S ) W q ( Y n ) 9 which shows that S o T is absoluteiy p-summing, and hence T E 9Y?q,p. The calculation also gives M q , p ( T ) c . 0

32.8. The theorem has another quite interesting consequence:

COROLLARY 2: Lei E be a Banach space and 1 I p 5 q 5 2. Then El E s p a c e ( M , , p ) if and only if for all L,(vl) and Lp(v2) (or only L, and l p )and all L p ( p ) (or only $) S @ T : L q ( V 1 ) @ A ~E

Lp(v2)

@Ap L p ( p )

is continuous whenever S E 2(Lq(vl), Lp(vz)) and T E f!(E, L p ( p ) ) .

Note that for p < q, the Maurey-Pisier extrapolation theorem says that space(9Rq,p)= sPace(~q,l).

PROOF: By (a) of the theorem the condition is clearly necessary. To see the converse for p < q (for p = q there is nothing to prove since %Ip,, = 2)observe first that the condition (with 1, and tp)implies the continuity of all A @ T T Q q @ A q E--* Lp(V2) @A,

ep

= Lp(vZ)

@A; ep

by corollary 29.12.. Taking for A a Lkvy embedding

I

:

e, c ' ,L p ( p )

(see 24.5.) gives that

is continuous. Again, an appeal to part (b) of the theorem gives the claim. 0 The corollary has also a negative flavour: Since for 1 5 p (Lkvy embedding) the operator

I @ "4,

: Lq @ A ~Lq

-+

< p < 2 and I : t,

L p ( p q )@A,

eq

c*

Lp(pq)

425

32. Mizing Operators

is not continuous by proposition 2 of 26.3., one obtains that

32.9. The Banach spaces in space(9Jl2,l) have an interesting characterization in terms of the coincidence of tensor norms.

COROLLARY 3: A Banach space E i s in space(2R2,l) if and only if for one (and then for all) p E [l,2[

E@cep=E@w2ep holds isomorphically.

In Ex 32.8. this result will be interpreted in terms of operator ideals.

PROOF:E E space(%?z,l) implies, by Saphar's characterization (20.19. or 32.2.(3)), that C(eo0 E ) = 'P~(eo09 E ) = &(&a I E ) . Since

22

-

wz is accessible, this yields that

-

holds isomorphically. It follows that this is true for every L1(p) @ E; using a LBvy embedding tpr-, Ll(p) (see 24.5.) and the injectivity of E and w2, gives that E w2 on 1, 8 E for all 1 5 p 5 2. lpfor one p E [l,2[, then Conversely: If E BCtp= E

EI

< = EI

@wL

e;

4

( E @c

ep)'

=(E

ep)'

and hence

2 ( E ' , t p )= 2 2 ( E ' , t p ) c 3'"J d"'(E'ltp)c 22' '"'(E',tP)

-

by Grothendieck's inequality, operator form. Theorem 28.4. implies that

s@

: e-2 @ A ~E

'

ep @A,, ep

is continuous for all S and T : But now corollary 2 implies that E" E and hence E E ~ p a c e ( M 2 , ~Since ). p < 2, the Maurey-Pisier extrapolation theorem gives the result.

-

Since 1, has cotype 2 for 1 5 p 5 2, Pisier's factorization theorem 31.4.gives that e w2 on tp@ E if E has cotype 2 - and hence, by the corollary, E is in space(9Jl2,l); this is Maurey's result 31.7. (see also 32.2.(4)).

426

32. Mixing Operators

This corollary can be used to give, as in corollary 2, a characterization of those E such that for all S and T s @T Lq @A+ E - L p @ A p L p

is continuous - this time for p

< 2 I q.

COROLLARY 4: Let 1 5 p < 2 I q 5

00. Then the dual E' of a Banach space is in space(m2,l) if and only if for all S E C ( L q ( v l ) Lp(v2)) , and T E C ( E ,Lp(p))the operator

s 8 T : Lq(v1) @Aq E

i s continuous.

-

L p ( v 2 ) @Ap L p ( p )

PROOF: By 28.4. and 28.5. the condition is equivalent to the fact that C ( E , t p )C 2E:,8ur(E,tp) = C z ( E , t p ). Simple transfer arguments show that this is equivalent to corollary 3 gives the result. 0

E

-

w2

on E' 8 tp,hence

32.10. Recall from Ex 8.12. and Ex 11.20. - Ex 11.22. the ideal of absolutely (r,s)summing operators (1 5 s I r 5 00) - in particular, the inequality tr(TZk; F )

5 pr,,(T)w8(2k; E ) .

The splitting theorem immediately implies that:

REMARK 1: ?Blq,p C 'QJr,p and

P , p

5 Mq,pif

l/q

+ 1/r = l / p

Note the special case M 2 , l c - and recall that E E space(!J?Z,l) means that E has the Orlicz property (see 8.9.).

REMARK2: For 15 r < 00 and l f p - l / r < 112 the space ideal for q

> max(2,p)

of!&,p (and hence the space ideal of ?Blq,p) consists of finite dimensional spaces only.

PROOF: If E E is infinite dimensional, it contains all uniformly by Dvoretzky's spherical sections theorem (see [Sl]). Since 'Qr,pis injective, this implies that Pr.p(i4;)

for some c independent of n. But

Ic

92. Mizing Operators

427

which is a contradiction.

32.11. The inclusion Mq,pc !$.?r,p (if l/q+l/r = l/p) is in general strict: For 1 < q < 2 it is known from 32.8. that L,, fZ space(Mq,l),but L,’ E space(?13qi,l)according to Ex 11.22. (or remark 24.7.). The inclusion %Iq,,c ?J3,p,1 is even strict for all 1 < q < 00; see 32.12.. Talagrand’s Banach lattice (mentioned in 8.9.) which has the Orlicz property (hence is in space(!&,l)) but does not have cotype 2, shows that space(M2,l) # space(v2,l) since a Banach lattice is of cotype 2 if and only if it is in space(M2,l) (see 32.2.(4)). Nevertheless, the ideals Mq,pand ?J3r,p almost coincide: This result is contained implicitly in Maurey’s thesis [186] and was stated and proved explicitly by Puhl [226] :

THEOREM: %,p C Onq-c,p zf l / q

+ 1/r = l/p

and

E

> 0.

PROOF:Recall first from 32.1. that

and from proposition 24.7. Pq‘J

0

2 , c Pq’+q

+

(since ! J J q t , 1 c ?J3r,,,, if 1 < s1 < q’ < r1 < q‘ q with appropriate r1,s1 - see Ex 11.21.). Take T E ‘J3r,P(E,F) and S E JPz(G,E). By the factorization theorem 18.7. for p’integral operators and the fact that U E % if and only if U” E 2l for maximal normed ideals one may assume that G = C ( K ) ,which is an XL-space. It follows that

and hence by Ex 32.2.(c)

For appropriately chosen q this gives the result. 0

32.12. The “border case” E = 0 in the last statement was investigated by Carl-D. [21] : If T E C ( E ,F) has rank n, then

-

where l / q + l / r = l/p. For p = 1 and 1 < q < 00 this “growth” is sharp in the following sense: There exist a constant cq > 0 and operators T, : P, P, with cq(l

+ log n)’/q 5 Mq,~(Tn) and Pq,,l(Tn)= 1 .

428

32. Mizing Operators

(For (q,p) = ( 2 , l ) these results were first proven by Jameson [120]). In particular, it follows that

MgJ c ! P g # , 1 #

for 1 < q < 00. For q < 2 this result was already obtained in 32.11. - but with totally different methods (32.8. and 26.3.).

Exercises: Ex 32.1. Show that the operator ideal Hint: Splitting theorem, L1. Ex 32.2. Show that

is injective but in general not surjective.

(4 Me,,,, c M,,,P,

if Pl I P2 and 92 I Q1 * (b) m4,r 0 M‘,P c M9,P if P, q, s E [I, 4. (c) !Pp,6 0 m6,r c fPg,r if r, s,p,q E [I, m] such that 1/s l / q = I/p l/r. Hint: Maurey’s splitting theorem for s > r and Ex 11.21, for s 5 r. In all three statements there are no additional constants in the norm estimate. Ex 32.3. If E is an 2&-pace, then fPg#(E, F) = Mg,l(E,F) for every Banach space F. Hint: 3, = 2,. Ex 32.4. Show that

+

+

if 1 I p I 2. Hint: 32.2.(3), 11.11. and 23.10.. Ex 32.5. Show for 1 I p 5 q I 00 :

Hint: Quotient formula, Ex 29.1., 32.1., 32.6. and 18.10.. Ex 32.6. Show that T E 2 ( E , F’) is (q,p)-mixing (for 1 there is a c 2 0 such that

In this case, Mg,P(T)= rninc.

I p I q < co) if and onIy if

429

92. Mizing Opemtors

Ex 32.7. Use 26.4.(3') and the Maurey-Pisier extrapolation theorem to prove Maurey's result from 31.7.: E E space(m2,l) if E has cotype 2. Note that this again shows that t1 E space(m2,l) which, together with the little Grothendieck theorem, implies c(t1, t2) = 'JIl(t1, t2). This is another proof of Grothendieck's inequality. Ex 32.8. Show that the following statements are equivalent for a Banach space E : (a) E E space(m2,i) . (b) For one (and then for all) q E 12, m] 32(Edq) = 3(E,tq) (c) For one (and then for all) p E [l,2[: Every compact operator E' Hint: 32.9. and representation theorems for ideals. Ex 32.9. (a) If E is a Banach space such that

-+

t, is 2-factorable.

for some 1 5 p < 2, then L(E,&) = 'JIl(E,t,), i.e.: E is a G.T.space (Grothendieck theorem space) in the sense of 20.18.. Hint: E E space(9l?z,,). (b) G.T. spaces are in space(m2,1), in particular, they have the Orlicz property. Ex 32.10. Let 1 5 p < q 5 00 and q > 2. If a Banach space E is such that for all T E Z ( E , t p ) the operator

T 8 i4,: E BA; eq

-

tp@A,, eq

is continuous, then it is finite dimensional. Hint: 32.6. and 32.10.. Ex 32.11. space(m,,,) = FIN if and only if q > max(2,p). Ex 32.12. Show that Mq,,(tl, F) = mq,1(t1,F) if p 5 2. Ex 32.13. (a) Show that T E 2 ( E , F) is (q,p)-mixing if and only if @T : pq(F,&o)

-

@T(S) := S o y

'JIpfE'ec~)

is defined (and hence continuous). In this case,

ll@Tll= MQ,P(T) *

(b) Show that in (a) the space t , can be replaced by L,[O, 11. Ex 32.14. Use Ex 11.23. to show that

whenever 1 < r 5 q

< 00.

430

99. The Radon-Nikodh Property for Tensor Norms and Refleziuity

33. The Radon-Nikodfm Property for

Tensor Norms and Reflexivity

The Radon-Nikodfm property led to an understanding of the full duality of the projective and injective tensor norms, namely, describing the conditions under which E’&,,F‘ = (E & F)’ holds. Noticing the central role of the sequence space t!, for this question, Lewis [169] succeeded in obtaining many results of the form E‘&,,F‘ = (E&, F)’ or, in other words, results about 21min(E,F’) = % ( E lF’) if 2l is the maximal normed operator ideal associated with a. This section presents Lewis’ ideas including applications to the reflexivity of tensor products and operator ideals. The main results are due to Lewis, but the proofs are sometimes taken from Meile [189].

33.1. Let a be a (finitely generated) tensor norm and (a,A) its associated maximal normed operator ideal. The objective of this section is to find conditions under which the natural map

is a metric surjection (the reader may already be accustomed to recognize the use of the representation theorem for minimal and maximal ideals). Note that J is injective if E’ or F‘ has the approximation property (see 17.20.) or if F’ has the a-approximation property (see section 21) or if a is totally accessible. For a = T this question was investigated in section 16: It turned out that in order for

t o be a metric surjection a sufficient condition is that F’ (or E‘) has the RadonNikodfm property (16.5.). An alternative proof to the one given in section 16 (or Appendix D7.) uses two ingredients: (a) The Lewis-Stegall theorem: I f t h e Banach space G has the Radon-Nikody’m property, then every T E 2 ( L 1 ( p ) ,G) factors through e l ( & ) (or through el zfp is finite), and IlTll = inf llRll IlSll, where the infimum is taken over all these factorizations ( A p pendix D2., proposition 2; one may take R the natural quotient map e l ( & ) + G). (b) 3(E,L1) = ( E & co)’ = (cO(E))’= e l ( E t )= e,&,,E‘ = %(E,e1) isometrically. Now take T E 3 ( E , F’) with F‘ having the Radon-Nikodjmproperty. The factorization theorem 10.5. for integral operators and (a) give

99. The R a d o n - N i k o d h Property for Tensor Norms and Reflexivity

431

m

and it follows from (b) that T factors through

A %(E,tl) , which implies that T E %(ElF') and N(T) II(T)(l + E ) . To = S1IR E 3 ( E , t l )

33.2. This simply structured proof leads to the following definition, due to Lewis:

DEFINITION: A finitely generated t e n s o r n o r m a has the Radon-Nikodim property if el

=(E

c,)'

holds isometrically for all Banach spaces E .

It was just shown that a has the Radon-Nikodfm property - but follows from 1 E'&tl A & ( E , t l ) C(E,ti) = (E@X ~ 0 ) ' .

E

does not: This

The fact that l 1 has the metric approximation property implies that the mapping J : E'&,tl --+ ( E c,)' is a metric injection whenever a is left-accessible. Since the usual tensor norms are accessible, the important part of the definition is the surjectivity of J - in other words, it will be important to know when

21mi"(E,t1)= Q ( E , L l ) holds (where 2l is the associated operator ideal). Since

E'

el =1 E'

@m !I

and

E'

@af\ c ,

1

= E'

@af c,

(see 20.8.), it follows that

PROPOSITION: a has the Radon-Nikody'm property if and only if a/ has i t , does not have the Radon-Nikodfm property, but In particular, E / = d , = wl = it will soon be seen that all the other ap,gdo!

33.3. The following is simple:

LEMMA: If(%,A) i s a quasinormed operator ideal and E a Banach space such that Qmin(E,!l)= %(El4 ) holds isometrically, then

amin(~,e,(r)) = a(~,e,(r))

432

33. The Radon-Nikodb Property for Tensor Norma and Reflexivity

holds isometrically for a11 sets I'. Since 6 is 1-complemented in e l , the finite case is obvious. If r is infinite PROOF: and T E 2l(E,f!l(l?)), then for each bounded sequence (zn)in E the sequence (Tz")is contained in some 1-complemented subspace el(I',) with countable r0 C .'I It follows from the assumption that (Tz")has a convergent subsequence, hence T is compact. Therefore, T has separable range, which - again - is in some 1-complemented l1(r0), as before. This implies that T E 21mi"(E,!1(r)) (with the same norm). 0 This observation and the fact that operators in 2l/ factor through some L1(p) (see 20.12.) give a result which can be seen as a justification of Lewis' definition:

THEOREM (Lewis): Let a be a tensor norm with the Radon-Nikody'm property and (%, A ) its associated maximal normed operator ideal. If F' has ihe Radon-Nikody'm property, then E'&,/F' + ( E F)' is a metric surjection and (21/)mi"(E,F') = % / ( E ,F') holds isometrically for all Banach spaces E.

PROOF: The definition of T E 2l/(E,F') and the Lewis-Stegall factorization theorem give E

T

F'

.1 7 f. Q Since clearly, A/(R) = A(R) and since a/ has the Radon-Nikodim property (by proposition 33.2.), it follows from the lemma that

and hence T E (21/)m'"(E,F') and (A/)mi"(T)

-

5 A/(So o R)llQll 5 A/(T)(l

+E). 0

This result does not hold when a/ is replaced by a : It will be seen in a moment that a := w2 2 2 has the Radon-Nikodim property, but

33. The R a d o n - N i k o d ~Property for Tensor Norms and Reflexivity

433

33.4. It is interesting to note that whenever E'6pF' -W (E @pi F)' holds for some tensor norm p, then this holds also for p replaced by some injective or projective associates of p. This will be an easy consequence of the

LEMMA:Let P be a finitely generated tensor norm, E and F Banach spaces. (1) I f L w ( E ~ j ) ' 6 p F '--w ( ~ ~ ( B@pi E IF)' ) is a metric surjection, then

~ 'F' 6( E ~ ~F)' -W

is a metric surjection as well. (2) I f l W ( B ~ ) & p F-W' BE) @pi F)' is a nietric surjection and F' has the approximation property or: p i s right-accessible and E' has the approximation property, then

E

~

--o

(~ E qSlp)l ~ F)' ~

F

~

is a metric surjection.

PROOF:With I : E + L C O ( B ~part i ) , (1) follows from the diagram

I1&dFi

To see (2) take B ately that

-

3

1

-

p and observe that 23""' /p. The assumption implies immedi(Bmin)sur(E, F')

A BsUr(E,F') . 1

Proposition 25.11.(1) implies that (%mi")8ur(E, F') = (Bdur)min(E, F') under the conditions stated, which means that

~ ' 6f, (~~ a u~r ) m~i n ( F') t~ , A B ~ EF'), = ( E B

( ~F)'~ .) ~

Taking F := co, and recalling that \a and /a are always left-accessible, this shows the

COROLLARY 1: If a has the Radon-Nikodgm property, then \a and /a also have it. Together with proposition 33.2. this implies that along with a a/, \a,/a and \a/ 1 /(a/) (a/)/ 9

(and so on) also have the Radon-Nikodim property. But note that a\ need not have it: This follows from e = w,\ (see 20.14.) and from 33.5. below.

434

33. The Radon-Nikoddm Property for Tensor Norma and Rejlezivity

Combining this with the lemma and theorem 33.3., leads to another result of Lewis':

COROLLARY 2: If the tensor norm a has the Radon-Nihody'm property, then so do a/,\a/,/(a/) and (/a)/.If E and F are Banach spaces (E' or F' with the approximation property i f /3 = /(a/))such that F' has the Radon-Nikoddm property, then E'&p F' --Y ( E @pi F)'

p =

a metric surjection and Brnin(E, F') = % ( E lF') holds isometrically for the maximal operator ideal (%, B ) associated with /?.

is

33.5. LaprestB's tensor norms and their adjoints (this means: the ideals CP,,of (p, q)factorable and !Dp,q of (p, q)-dominated operators) nearly all have the Radon-Nikodjm property. The result is due to Lewis:

= E and 0 1 , ~= w l = E / do not have the Radon-Nikodfm The tensor norms = g& has it or not; since property. Unfortunately, we do not know whether 9; !J31 = Jinj, this is the same as asking whether or not

-

(by proposition 25.11.(2)) holds: Is every absolutely summing operator E nuclear (see Ex 9.13.)?

-

tl

quasi-

PROOF: Since all these tensor norms (and their associated ideals 2f) are accessible, it remains only to show that 5 0 Q(E,ti) = Q ( E ,ti) (see the initial remarks in 33.2. and 25.2.). (1) Each operator T E X,,,(E,tl) factors as follows

Ts

33. The Radon-Nikodh Property for Tensor Norms and Refleziuity

435

Thus, by the very properties of I,, it is enough to show that S is weakly compact (=compact=approximable): For 1 < p < 00 this follows from the reflexivity of L,, for p = 00 this is a consequence of the fact that every operator L , -,I1 factors through a Hilbert space (e.g. by Grothendieck’s inequality, see 17.14.). For p = 1 this is not true - a more complicated argument is necessary.

So assume that p = 1 and 1 < q < 00 (the case a 1 , 1 = 7r is already clear). It is enough to show that +(p)

C

L L(P)

.Crp.

s

I1

( p a finite measure) is in Since I, has the Radon-Nikodim property, the operator S has a Ftiesz-density g = (gn) with gn E L1(R,p) and

for all w E R. The sequence A = (An) defined by

is in Iq.It suffices to prove that the operator S o l factors through the diagonal operator Dx : 1,) + It and that m

The latter follows from

For the factorization define U : I, makes sense since for (a,) E Iq

-, L , ( p )

by Uen := A;’gn

- and

this definition

It is easy to verify that S o I = Dx o U‘ - and this ends the proof of (1). (2) For 1 < p < co take T E Zg/(E,Ll) and factor

33. The Radon-Nikodh Property for Tensor Norms and Reflexivity

436

Then it is obvious that R, E ZEj and So E Z((M,tl) since M is reflexive. It follows that T = So o R, E 3 0X:i(E,ei) = (2Ei)min(E,.t~) by the accessibility of ;:2

-

-

.

CY~,~\

(3) First recall that 3 p ~=, q ?pal ~ o q p t by Kwapieh's factorization theorem 19.3..Sinced;/ = d,, (by 20.14.), it followsfrompart (1) of the proof that for 1 < q 5 00 the tensor norm d;/ and hence di has the Radon-Nikodim property (33.2.). Since d; p$"', this means that

-

YfYo'(E,4'1)

A (!.@''')min(E,lI)

Ex 25.15.(a) (about (B o 21)min(E, G ) = B o B ( E , G)) gives the result. In the remaining case q = 1 one has has the = g;l = gpt\, and therefore - and

Radon-Nikodim property by (2) whenever 1 < p < 00

.0

33.6. The fact that the (right-projective) tensor norm dp has the Radon-Nikodim property (for 1 5 p < ca) implies that

-

is a metric surjection if E' has the Radon-Nikodim property (theorem 33.3.). Since $ = g p gP (the ideal of p-integral operators) and V, = grin (the ideal of pnuclear operators), it follows that

COROLLARY 1: Let 1 5 p property, t h e n

< 00

and E , F Banach spaces. If E' has the Radon-Nikodgm

J p ( E ,F') = %,(ElF ' ) holds isometrically.

For p = 1 this was shown in Appendix D. Recall from 21.10. the Pietsch pintegral operators, which factor as follows (1 5 p < m) :

33. The Radon-Nikodh Property for Tensor Norms and Reflezivity

437

T

E-F

Since Lp is 1-complemented in L : for 1 5 p

< 00, it follows from corollary 1 that

P&(Io R ) = Ip(Io R ) = Np(Io R ) whenever E' has the Radon-Nikodjrm property, and hence,

COROLLARY 2: For 1 5 p < 00 and E' with the Radon-Nikodgm property W p ( E ,F ) = %(El F ) holds isometrically.

33.7. One application of the coincidence of p-nuclear and p-integral operators: In Ex 28.1. an operator T E C ( E , F ) was called p-strong (for 1 5 p < co> if there is a constant c 2 0 such that

whenever

21,

m

m

k=l

k=l

..., zm,y1, ...,y, E E are such that m

m

k=l

k=l

for all x' E E'. It was shown that every operator T E p-strong operators are in C y I"'. Even more is true:

THEOREM: A n operator is in

Cy is p-strong - and that all

Cy if and only if it i s p-strong

(1 5 p < 00).

For T = idE this is a result of Lindenstrauss-Pelczyriski [173] (see Ex 33.5.) - and for arbitrary T the following proof was given to us by Tomczak-Jaegermann.

Cy

PROOF:That each T E is p-strong with constant c = L y ( T ) was shown in Ex 28.1.. For the converse take a p-strong T E &(ElF ) with constant c . It will be shown that L y ( T ) 5 c. Recall from 25.9. that = ppo (pFa')-l.Therefore it is sufficient to show that for S E !#Fa'(G, E ) with PpduO'(S)= 1 the composition TS satisfies P (TS) 5 c. By the definition of T being p-strong and the fact that Ci"j are maximal operator ideals, one may assume that E and G are finite P and $IFaP dimensional. Since 3y = QPholds isometrically, it follows that

Cy

IoS' :El

s'

I

G' k!,(Bc)

438

33. The Radon-Nikodh Property for Tensor Norms and Reflezivity

is p-integral and

4 ( Z o S’) = Pp(S’)= 1, hence

with Np(ZoS’)= 1 by corollary 1 before (or by the representation theorems 22.5. since E is finite dimensional and gp accessible). Therefore it is a consequence of the definition of gp (see e.g. 17.10.) that I o S‘ factors as follows

- or

dually

Now take z 1 , ...,zm E G with and define xk

:= S r k

Wp(%k;G)

5 1, assume without loss of generality

E E and y k := R i D A e k = A k R ’ , e k

These two m-tuples of elements in E satisfy for each xF E E’ m

m

k=l m

m

k=l

k=l

Since T is p-strong, it follows that m

m

m

k=l

k=l

k=1

which shows that P p ( T S )5 c . 0

EE

.

rn = n

$3. The Radon-Nikodh Property for Tensor Norms and Reflezivity

439

33.8. At the end of this section conditions under which the tensor product

and/or the component

% ( E ,F ) of the associated operator ideal is reflexive will be investigated. If E and F are # (0) (which will be assumed from now on) it is a necessary condition that the complemented subspaces E (or E') and F of E 6 a F (or %(ElF)) are reflexive. Clearly, this is not sufficient since, for example,

(see section 16) is not reflexive. Note that

(see 26.1. and Grothendieck's inequality, operator form) - so the spaces Cp,q(E, F) and qyr((E, F) are also not necessarily reflexive if E and F are. Recall that a reflexive space E has the approximation property if and only if its dual has - and this implies that E and El both have even the metric approximation property.

THEOREM: Let a be a finitely generated accessible tensor n o r m , E and F Banach spaces one of which has the approximation property. T h e t e n s o r product

hold isometrically.

PROOF: Clearly, the condition is sufficient. Assume now that E g a F is reflexive. Then E and F are reflexive and, e.g., E and El have the metric approximation property. It follows that the natural injections

are isometries. The commutativity of the diagram

440

33. The Radon-Nikalghn Property for Tensor Norms and Rejlezivity

shows that

11

is surjective and 1; injective: This is the claim. 0

33.9. If 24 is the associated maximal operator ideal, then (a')' = a* N 24. and the representation theorems for maximal and minimal operator ideals give

COROLLARY 1: Let (a,A) be an accessible, maximal normed operator ideal and E , F Banach spaces one of which has the approximation property. Then % ( E ,F ) is reflexive i j and only i j E and F are reflexive and 2lmi"(E,F ) = % ( E ,F ) (2i*)min(F, E ) = a * ( F ,E ) hold isometrically. Using again E @a F = F @ a t E , another consequence is immediate as well:

-

COROLLARY 2: Let a (a,A) be accessible and E , F Banach spaces one of which has the approximation property. (1) 1f E 6 a F is reflexive, then E'&IF' , F&tE

and

F'&-E'

are reflexive as well. If % ( E ,F ) is reflexive, then

(2)

Q*(F,E) and !21dua'(F', E') are reflexive. (3) %(E,F ) is reflexive if and only if E'G,,F is reflexive. 33.10. Theorem 33.8. shows that Lewis' theorem 33.3. and its corollary 2 in 33.4. provide conditions for reflexivity. Recall again that reflexive spaces have the RadonNikodjm property.

COROLLARY 3 (Lewis): Let a be a tensor norm such that a and a* have the RadonNikodgm property. For p := /(a/)or p := (/a)/and E , F reflexive Banach spaces, one with the approximation property, the spaces E & F , E ~ @ , FE,& F ,

E&=F

33. The Radon-Nikodbirn Property for Tensor Norms and Refleziuity

441

are reflezive.

The analogous statements for operator ideals are obvious.

PROOF:By the foregoing corollary and (/(a/))* = (/(a*))/ it is enough to prove that E&pF is reflexive for ,8 = (/a)/. The tensor norms /3 = (/a)/and p' = /(a*/)are tensor norms as in 33.4., corollary 2, hence (using the approximation property),

Since p' is (totally) accessible (by proposition 21.1.(3)), and therefore /3 accessible, theorem 33.8. gives the result.

33.11. Some examples: It has already been seen that tP,P(W2)

is not reflexive whenever p , q E]l,co[ and hence reflexive as well. What about gp and dp = gi?

L2&3a,,,12,

by corollary 2(3), is not

PROPOSITION 1: Let E and F be reflexive Banach spaces, one of which having the approximat ion property. (1) E@g,F is reflexive f o r 1 < p 5 00, but not i n general f o r p = 1 . (2) 3 , ( E , F ) , 3FaI(E,F ) are reflexive f o r 1 < p 5 00, but not i n general f o r p = 1 . (3) !&tp(E, F),!-@'"(E, F ) are reflexive f o r 1 5 p

< 00, but not in general f o r p = 00 .

For p = 1 the statement (3) is due to Gordon-Lewis-Retherford [85], the other results seem to be due to Lewis [169].

-

-

PROOF:Since gp DP and gpl pplit is enough (by corollary 2) to deal with gp or gi = dp = ( / d p ) / (see 20.14.). For 1 < p < 00 the tensor norms dp and d; have the Radon-Nikodfm property by 33.5., therefore the result follows from corollary 3. The space &&& = &$$jslf?2 is certainly not reflexive, so it remains to prove that E&d, F is reflexive: It follows from Ex 33.6. that

-

hold isometrically - this is one condition of the reflexivity theorem 33.8.. For the other one it remains to check (recall d, = w l 21)that

442

33. The Radon-Nikodim Property for Tensor Norms and Reflexivity

is surjective ( d , = w1 is accessible, one of the spaces has the metric approximation property). Take T E 21(E', F ) , use the Radon-Nikodjm property of F and the LewisStegall factorization theorem (33.1.) to see that T factors as follows:

E-F

T

Ll(P)

-

el(&)

so

where Q E 21 and So o R is weakly compact=approximable since E is reflexive. It follows that T E 21 o 3 = 2yinby 25.2.. 0 More results in this direction follow by considering the tensor norm 33.11.).

(see also Ex

~ t ~ , ~ /

+

PROPOSITION 2: Let p , q E [l,oo] such that l/p l / q 2 1 and El F reflexive Banach spaces with E having the approximation property. (1) For q > 1 the spaces

E 6 a p ~Fq /and E&a;,,\ F 2 P > 9 / W F ) and \ 2 9 , P W E ) !D$$,(E, F ) and !D;&(F, E)

are reflexive. (2) For q = 1 each of these statements fails for E = F = &.

PROOF: It is clear from 33.9., corollary 2, that it is enough to show the reflexivity of E 6 a p , q / F. The case al,oo/= d,/ = d, was settled in proposition 1. For 1 < q < 00 the tensor norm ap,qhas the Radon-Nikodjm property (33.5.), hence Lewis' theorem 33.3. gives

E 6 a v , , / F = (E'

F')' .

Applying theorem 33.8. again, it remains to show that

(!D$,$)min(E, F') = E'ga;,p\ F' (note that

A (E

@I(~;,~\)S

F)' = !D:,i,(E, F') ;

is totally accessible by 21.1.). Since

(!D7$,)min(E, F') = (!D7:$)inJ(E, F')

99. The Radon-Nikodbim Property for Tensor Norms and Reflexivity

443

by 25.11.(2), it is enough to show that

Since 'Pqj(E,G) = !PYin(E,G ) for all G (see Ex 33.6., here the approximation property of E - and not of E or F - is needed), this follows from Kwapieli's factorization theorem

Dqt,p,= y3pdF 0 PqI and Ex 25.15.(b) (concerning (23 o %)mi" = % o a). For q = 1 it is a consequence of Grothendieck's theorem (this was already mentioned in 33.8.) that

is not reflexive. 0

Exercises:

Ex 33.1. An operator T E 2 ( E ,F) is called a

Radon-Nikody'm operator if for every

S E C(L1[0,1], E ) the operator T o S is representable. Denote by r2) the ideal of all 11 11) is an Radon-Nikodfm operators and give it the operator norm JJ 11. Show that (9, injective Banach operator ideal which is not surjective and not maximal. Hint: Identity of k'l and L1[0,I]. Ex 33.2. If p is a finite measure, S E ~ ( L I ( , u )E) , and T E r2)(E,F), then T o S is representable. Hint: Proof of theorem Dl.. Ex 33.3. The ideal of operators T E C ( E ,F) which factor

with the norm inftlRll llSll is not maximal. through some Ex 33.4. Use corollary 2 from 33.4. in order to give conditions under which

Give an example for which %(ElF) = Qmin 22.9. for the counterexample.

reg (ElF

) # %""n(E, F). Hint: 31 = 3 and

444

33. The Radon-Nikod&n Property for Tensor Norms and Reflezivity

EX 33.5. For 1 5 p < 00 a Banach space E is isomorphic (resp. isometric) to a subspace of some L p ( p ) if and only if there is a c 2 1 (resp. c = 1) such that for all m E N and 25, gk E E the condition m

m

for all x’ E E’ k=1

k=l

Ex 33.6. If E’ has the Radon-Nikodh property and: E’ or F has the approximation p < 00 property, then for all 1 I

holds isometrically. Hint: 33.6. and 25.11.(2). Ex 33.7. Let a be a finitely generated left-projective tensor norm with associated maximal operator ideal Q. If at and E‘ have the Radon-Nikodjm property, then

Q(E,F’) = Qmin(E,F’) holds isometrically for all Banach spaces F. E x 33.8. Let Q be an accessible maximal normed operator ideal. Show that Q ( E ,F ) is reflexive for all reflexive E , F such that E has the approximation property if and only if Q*(E,F ) is reflexive for all reflexive E , F such that F has the approximation property. Ex 33.9. Prove again the result from 16.7. that for reflexive Banach spaces E and F (one with the approximation property) X(E, F ) is reflexive if and only if X(E,F ) = ff(EF , ) . Hint: 33.9., corollary 1. E x 33.10. Let E , F be separable Banach spaces, one of which has the approximation property, and Q an accessible maximal normed operator ideal. If Q ( E ,F ) is reflexive, then it is separable. Ex 33.11. Use the fact that the tensor norm a p , 2 is left-injective (for 1 5 p < 00) to show that is reflexive if E and F are reflexive Banach spaces and E or F has the approximation property. Hint: (/a)/. Note that proposition 2 in 33.11. did not cover the case that F has the approximation property. E x 33.12. Use corollary 26.1. to show that for p, q E ]1,00[ the spaces

are not reflexive.

94. Tensorstable Operator Ideals

445

34. Tensorstable Operator Ideals

This section is devoted t o the following natural questions: Given an operator ideal ?2l and a tensor norm p; is S 6 p T E % for all S , T E a? Is E 6 p F E space(%) if E and F are? Although this happens rarely for general % and p, there are some positive situations which have interesting applications. On the other hand, the number of results which handle these questions in the special case where 1,3 = E or p = ir is too large to cover all of them here. We only present a systematic approach for the more general setting. At the end of this section a method will be presented which improves inequalities with the help of tensor stability.

34.1. Let p be a tensor norm. A quasi-Banach operator ideal (%,A)is called ptensorstable if for all Ti E %(Ei,Fi) the continuous tensor product operator

is in the ideal %. It can easily be seen using an ll-sum argument that in this case there is a constant c 2 1 satisfying

say: % is P-tensorstable with constant c. For c = 1 the term metrically ptensorstable will be used. See Ex 34.2.(c) for an example showing that not every P-tensorstable ideal is metrically p-tensorstable. . 11) is metrically pThe metric mapping property of tensor norms says that (2," tensorstable for all tensor norms. The relation - we

5 of approximable operators is metrically p-tensorstable for all p. If Ti E C(Ei,Fi) is compact, then BE,) is relatively compact and hence

also shows that the ideal

is also relatively compact, which shows that the ideal ff of compact operators is metrically n-tensorstable. The ideal (%, N)of nuclear operators is easily seen to be metrically p-tensorstable for all p : just use expansions. Since e & 4 and l 2 & & are not reflexive, the ideal !ID of weakly compact operators is neither ir- nor E-tensorstable.

446

34- Tensorstable Operator Ideals

34.2. The metric mapping property implies that even llT16pT~ll= 11T111 llT211. Under what conditions does a /3-tensorstable operator ideal (a,A ) satisfy an estimate from below: A(Tl)A(Tz) 5 cA(Tl&pT2)? Using an argument of Pelczyriski ([203], lecture 14), the following can be shown for the adjoint operator ideal %* :

PROPOSITION: Let (Q, A ) be a quasi-Banach operator ideal which is P-tensorstable with constant c 2 1. Then

-

PROOF: Using the definition 17.9. of A*, take N, E FIN(Ei) and L; E COFIN(Fi) as well as Ri : Fi/Li Nj with A(R;) 5 1. Then A(R18pRz) 5 C, and hence

-

since tr(S1 8 S 2 ) = trS1 trS2 (check with rank-one operators). The definition of A* now yields the claim. 0 Since Q** = 2l for maximal normed ideals, this result (applied to %*) implies the

COROLLARY: Let (2l,A) be a maximal Banach ideal. If ( % , A ) and (%*,A*) are metrically P-tensorstable, then

for all TI, T2 E %. Operator ideals satisfying this relation will be called strongly P-tensorstable.

REMARK:The ideal (3inJaur,pnj aur) is g2-tensorstable, but not strongly gz-tensorstable.

PROOF: By Grothendieck's inequality (in the operator form 20.17.) 3'"J bur = 2 2 holds with equivalent norms. Since H&H = H&H (Hilbert-Schmidt norm; 26.7.) is a Hilbert space, simple factorization shows that Cz and hence 3'"j is g2-tensorstable. Assume now that I,, := 1'"j satisfies

447

34. Tensorstable Operator Ideals

Since Io(idt2)= KG,the Grothendieck constant, this would imply

which is not possible because KG > 1. 0 We do not know whether every P-tensorstable ideal Q necessarily admits a lower estimate A(Tl)A(T,) 5 cA(T16pT2).

34.3. More examples: Since llTl6pT211 = 11T1;11llT211, the ideals 2 and P-tensorstable and ff is strongly n-tensorstable. Using

5 are strongly

and the tensor stability of 5, it follows that the ideal ff of compact operators is strongly P-tensorstable whenever P is injective, e.g. /3 = E or w2. A similar argument with L1(BEl)6p11(EEa)shows that ff is strongly P-tensorstable if P is projective, for example, P = w i . It seems to be unknown whether or not ff is tensorstable with respect to arbitrary tensor norms.

LEMMA: For p = 1,2 the mapping e ,

-

e , @ e , extends t o a n isometry

el c ' . ep&.*ep. PROOF:T(C:=~ Anen 8 e,;eP,ep) 5 C,"==, IA,l gives one inequality. The other one follows from e163*e1= el(N x nV) and, for p = 2, from (see Ex 3.28.) m

If I : P1

L,

8

La, then obviously

commutes and this implies the

-

COROLLARY 1: I f t h e operator ideal Q i s injective and L1 !$ space(%), then the operator 16% I : el&.*el e26.*e2i s not in Q.

34. Tensorstable Operator Ideals

448

I n particular, for 1 n-tensorstable.

5 p < 00

the ideal

!Jlp of

absolutely p-summing operators is not

The latter follows from I E !J31 C Pp(see Ex 11.5.). Since the ideal Q% of quasinuclear operators is the injective hull ninjof the nuclear operators (Ex 9.13.),the same argument which was used for ff = gives that Q% is metrically P-tensorstable whenever P is an injective tensor norm. But Holub [113] observed that:

?J

COROLLARY 2: The ideal Q% of quasinuclear operators is not ?r-tensorstable. PROOF: Take

(An)

E co and T := C,”==l Anengen : el

-

P2,

Then the operator

is nuclear by Ex 11.5. and Appendix C8., corollary (recall that 3‘“j = PI),and therefore T is quasinuclear. Now look a t the diagram

To

lJ fl

,

‘J

Toen := Aien

.

fl

I f T&T were quasinuclear, then To would be quasinuclear and hence m

n=l

nuclear (T: E Q9Z2c J!J: in f4 (Ex 3.28.). 0

c !J3; c %). But this is only the case if the sequence (An)

is

For a positive result in this direction see Ex 34.3.. 34.4. Many of the arguments already used are valid in a more general setting. The following statements (taken von Carl-D.-Ramanujan [23]) allow a more systematic treatment of tensor stability.

PROPOSITION: Let (a,A) and (23, B) be Banach operator ideals and P a tensor norm. (1) If(%, A ) and (%, B) are P-tensorstable (with constants a and b, respectively), then (Sa o %, A o B) is P-tensorstable (with constant a . b).

449

94. Tensorstable Operator Ideals

(2) If (a,A) as P-tensorstable, then its minimal kernel (amin, Amin)is P-tensorstable (with the same constant). ( 3 ) If ,O is injective, (21max,Amax)right-accessible and A(Ti @p T2) I c .A(Ti)* A(T2) for operators Ti and T2 between finite dimensional Banach spaces, then the maximal hull ( ~ m a zAmax) , of is P-tensorstable (with constant c). (4) Let (a,A) be P-tensorstable (with constant c). If P is injective, then (IzLinj,Ainj)is /3-tensorstable; if ,8 is projective, then (Zl'"', A*"') is P-tensorstable (same constants). (5) If /3 is projective, (a,A) is P-tensorstable (with constant c) and (Qmar, Am,=) is

left-accessible, then

Amaz

( a m a x dual 1

dual

1

is P'-tensorstable (with constant c).

PROOF:(1) and (2) are obvious since 2 P n = 5 o a o 5 and 5 is P-tensorstable. Statement (4) follows as in the case of ff = $n' = 5""'at the beginning of 34.3.. TOprove (3) take Zl E a m a r ( E i , F i )and put E := El @p E2. Then, by the density lemma for maximal operator ideals (Ex 17.17.), it is enough to show that A ((T16pT2) o I i )

5 c . Amax(Tl). Ama2(T2)

for cofinally many M E FIN(E). Since for each such M there are Mi E FIN(&) with M C Mi @ M2 and P is injective, it follows that

M

Mi @p Mz

El @p E2

this implies that M of the form MI @p M2 suffice. Bmazis supposed to be right-accessible, hence there are Nj E FIN(Fi) and Sj E L!(Mj, N j ) with

- and

. I;: o I s i = 12 o Si and A(Si) I (1+ &)Ama"(Z) Now the desired inequality is a consequence of A ((Ti6pT2)o I & ~ ~ M =A ~ ((Ti ) 0 Izl)6p(Tz 0 12)) =

=A

((12 sl)6p(~zs2))= A (1NlqN2 0

F,&&

0

0

(slB~s2))I

< c * A(S1) - A(S2) 5 ~ (+lE)~A'"''(T~)* Amax(T2). To see ( 5 ) first observe that for operators TI and T2 on finite dimensional spaces, one has (Ti @ p t Tz)' = Ti @p Ti,and therefore,

-

Adua'(Tiapt Tz)= A(T{ Bp Ti)5 c A(T:) A(Ti) 5 c * Adual(Ti). Adua1(T2) .

450

34. Tensorstable Operator Ideals

Now (3) applied to !21dua' and

p' gives the result. 0

Note that ( l ) , (2) and (4) also hold for quasi-Banach operator ideals. 34.5. The most important positive examples come from factorization techniques (apparently first used by Holub [113] in this context). Factorization of an operator in 2 p , q ( EF , ) usually goes into F" (rather than F ) , and therefore the following simple observation is useful: If /3 is finitely generated, then any of the two maps 0, from the extension lemma 13.2. make the diagram

commutative; recall that we do not know whether or not the

0j

are isometries.

If Ti E 3(Ei,Fi), then the factorization 10.5. for integral operators yields for each finitely generated tensor norm @

1 C(K1 x Kz)= C ( K l ) & C ( K 2 ) - and

-

/

Ll(Pl)@rLl(PZ)= Ll(P1 63 P 2 )

the observation above implies the

REMARK:T h e ideal (3,I) of integral operators is metrically @-tensorstable for all finitely generated tensor n o m s p. Exploring this factorization idea for T E 2 p , q ( EF , ) , the factorization theorem 18.11. for (p, q)-factorable operators leads to the diagram

1 and hence the

/

34. Tensorstable Operator Ideals

451

PROPOSITION: Let p and 7 be finitely generated tensor norms such that

A,) 5 7 on L,t @ L,) P 5 A, on Lp @ L p Then f o r

E

2p,q(Ejl

Fi) the operator

-

is defined, ( p ,q)-factorable and LP,,(T16Tz)5 Lp,q(T1)Lp,q(Tz). It follows from 29.11. and 29.4. that in the hypotheses L,I and L, can be replaced by and ,! respectively. The result is interesting because of its special cases 3, = f!, and J?"' = 2 1 , , : the fact that dil = dp = A, = g;, = g p on L, 8 L, (see 15.10.) implies the !,I

COROLLARY 1: (1) For 1 5 p 5 00 the ideal 3, ofp-integral operators is metrically P-tensorstable f o r all finitely generated tensor norms satisfying P 5 A, on L, @ L,. Examples: E , d;, ,d,, g;l ,g,, wp,wpj and all smaller tensor norms. (2) For 1 5 q 5 00 the ideal 3F"' is metrically P-tensorstable for all finitely generated tensor norms satisfying P 2 A,, on L,I @ LqI. Examples: T , d ; , dqtlg; , 9,' , w;, ,w; and all larger tensor norms. It follows that for 1 5 p 5 2 the ideal 7, is metrically w,-tensorstable for all r E [l,001. Proposition 34.4. allows one to verify tensor stability for many other operator ideals (see Ex 34.2.). Restricting to E and T , it gives the related to 3, and 3":'

COROLLARY 2: Take 1 5 p 5 00. (1) The ideals 3, of p-integral, $' 3, of absolutely p-summing, and

'JlY

of quasi-pnuclear operators are strongly E-tensorstable, the ideal %, of p-nuclear operators i s metrically e-tensorstable. (2) The ideals 3Fa',q?"' and A:; are strongly ?r-tensorstable, the ideal Al,, is metrically x -1 ens o rsl a ble. Because of its application to the theory of eigenvalue distribution (see 34.10.) perhaps the most important statement here is that q, is E-tensorstable; this result is due to Holub [113].

PROOF:Corollary 1 and proposition 34.4. give that all these ideals are metrically tensorstable with respect t o E or A. Since 3; = $'3,' and V; = J , I , corollary 34.2. gives that the maximal ideals in (1) are even strongly E-tensorstable. Since g p = gp\ = g;l

452

94. Tensorstable Operator Ideals

on E' 60 looand g;, is totally accessible, the representation theorems for maximal and minimal ideals imply that

for all E , F - and this shows that with relations

!&, also T$'j

is strongly c-tensorstable. The

(ppal)*= 'Jpdyal give that the maximal ideals in (2) are strongly 7-tensorstable and

ffiY;(E, F ) that this is also true for

pra'(E, F)

ffb,y.0

34.6. Two applications of these results. First, recall from 23.6. that the projection constant A(E) of a finite dimensional Banach space is L , ( i d ~ )= I ,(i d~). Hence the relation I p ( m 3 c T 2 ) = I p ( W p ( ~ 2 )7

which was just proved, implies that

for all finite dimensional Banach spaces E and F. This is a well-known result. However, there is no such relation for infinite dimensional spaces! Even more:

PROPOSITION: Let E and F be infinite dimensional spaces. Then the Banach space E& F is not injective - i n other words: A(E& F ) = 00. This result was given to us by Pelczyhski.

PROOF: First take E = F = l, : By a result of Kalton [140] the space t , @ ~ ~=t , ff(41,l,) is not complemented in C ( t ,,t,) - and is therefore not an injective Banach space. Now, take arbitrary E and F and assume EI& F to be injective. Then, as complemented subspaces, E and F are injective as well. A result of Rosenthal's states (see [177], I, p.105) that t, is isomorphic to subspaces E, c E and F,, c F ; they are complemented (since toois injective), and hence t,@,,t, is isomorphic to the complemented subspace Eo&F,, of the injective space E&F. It follows that t,&t, is injective as well which is a contradiction. 0 Another application for bases (see Ex 12.9.) is also due to Holub [113] :

34. Tensorstable Operator Ideals

REMARK:The s n i t vector basis (en 8 em) in . t l & t ,

453

is not unconditional.

-

PROOF:The embedding I : l 1 L, l 2 is absolutely l-summing. By corollary 2 the l2&& is also in and hence maps unconditionally operator I&I : l l & l l convergent series to absolutely convergent series. If (en @ em) were an unconditional basis of 1, &el, then every expansion

would converge absolutely in r?2&4 which means that C lAn,ml c 00 would hold. In other words, every operator in f f ( c o ,e l ) = .f1&11 would be nuclear; but this is not true (see e.g. Ex 4.3.). 0 34.7. What about tensor stability of space(%)? Negative results will follow from the

LEMMA: Let spaces. Then

p

be an injective tensor norm and E l , E2 infinite dimensional Banach contains the uniformly.

El&@E2

PROOF:By Dvoretzky's spherical section theorem, which was already mentioned in 32.10., the spaces Ei contain subspaces Mi" with d ( M r , e ) 5 2. Since /3 is injective, all it4r @p M," are isometric subspaces of E163pE2; therefore, it is enough to show that Pa is isomorphic to a subspace of Ej @ p 6 with control of the norm. For /3 = E

is obviously an isometry. But t = 202 on 6 @ 4, hence Grothendieck's inequality (and again the injectivity of p) give

- this ends the proof. 0 Note that the lemma is false if /3 is not injective: Take the Hilbert space 12&)ga.t2 as an example. The following theorem and its corollaries are due to Carl-D.-Ramanujan 1231 :

THEOREM: Let (%,A) be a Banach operator ideal and /3 a tensor norm such that (a) /? i s injective and supA(idt&) = 00, or: (b) /? is projective and sup A(idt;) = 00. Then E1@3E2

4 space(%) whenever El

and E2 are infinite dimensional Banach spaces.

34. Tensorstable Operator Ideals

454

PROOF:Assume E163gEz E space(%) and the first condition (a): The lemma implies that there is a c 2 1 such that

which is not possible. Under condition (b) the dual follows that

P' of /3 is injective, and therefore totally accessible. It

6( E ~ ~ D E E~ space(0max )' dual 1

E I ~ ~ E ;

(recall 17.8., corollary 4) and hence,

Since sup Arnax

dual i n j

( i d f k ) = SUP A(idf;) = 00

,

the tensor norm P' and the Banach ideal Smar injsatisfy condition (a), therefore E;&E; 4 space (amax 'nj ) - a contradiction. 0 Since sup A(&) yields the

<

00

implies that

XP c

% whenever % is maximal, the theorem

COROLLARY 1: Let % be a maximal Banach operator ideal such that space(%) contains an infinite dimensional space.

P, then 2, c 0 . norm p, then C1 C a.

(1) If% is P-tensorstable for some injective tensor norm (2) If % is P-tensorstable for some projective tensor This has various consequences:

The ideal tPof pfactorable operators (1 < p < m) is not E-tensorstable and not n-tensorstable (since neither 21 nor 2, are in 2,). The ideal Cp of cotype q operators (2 5 q < 00) is not E-tensorstable (since t, has no proper cotype, Ex 7.14.). It is not known whether or not Cz is n-tensorstable: Tomczak-Jaegermann [271] showed that t 2 & t 2 has cotype 2, but it has not yet been decided whether t26,,t26,,t2has cotype 2 or not. The ideal Tp of type p operators (1 < p 5 2) is neither E- nor n-tensorstable (since t~ and toodo not have proper type, Ex 7.13. and Ex 7.14.). The ideal M2,1 of (2,l)-rnixingoperators is not c-tensorstable (since too4 space(9R2,~) by 32.2.(6)). See also Ex 34.4., Ex 34.8. and Ex 34.9.. More generally, one has the

$4.

455

Tensorstable Operator Ideals

COROLLARY 2: There is n o m a t i m a l Banach operator ideal ?2l # I: such that space(%) contains an infinite dimensional Banach space and which satisfies (1) or (2): (1) Q is injective and P-tensorstable f o r some injective tensor norm. ( 2 ) Q is surjective and P-tensorstable for some projective tensor norm.

PROOF: Corollary 1 would imply

and

E = 2;"' c %'"' = % in case (2)

.0

For another interesting application of the lemma see Ex 34.6.. 34.8. Since L&J?2 and L2&& are not reflexive, they are not contained in any L, (for 1 < p < m). It will be shown below that this is also true for p = 1. The proof of this result requires some preparation.

DOUBLEKHINTCHINE INEQUALITY:

f o r all a i , j E ( I and . ! 1 Ip inequality.

-

< 00, the a, and b, being the constants f r o m the Khintchine

PROOF:The left inequality is obvious for p 2 2, the right one for p 5 2. Denote Rad, C L p ( p ) the identification of Lz with the closed subspace Rad, by S, : e2 of L p ( p ) spanned by the Rademacher functions; the Khintchine inequality says that llSpll 5 bp and IISilll 5 a,. The variant of Beckner's result given in 7.9. shows that for p 2 2

456

34. Tensorstable Opemtor Ideals

and for p 5 2

- and

these are the missing inequalities. 0

The next result is due to McCarthy [188].

PROPOSITION: Let (a,v) be a measure space, 1 Ip < 00 and E a subspace of LP(Q, v). For given n-tuples (PI, ...,Pn) and (&I,...,Qn) of pairwise orthogonal projections in E which commute (PjQj = Qj Pi) define

Then

where ap and bp are the constants from the Khintehine inequality.

-

PROOF:Note first that R := Cti=,PjQj is a projection of norm Ia b and that the family (PjQj)rj=l of projections is pairwise orthogonal. It follows that it is enough to show for all f E E and fi,j := PiQj f E PiQjE that

xyj=,

fi,,. In order to use Rademacher averaging take w,w‘ E D = since Rf = {-ll 1IN. Then for gi,j E PiQjE

94. Tensorstable Operator Ideals

Integrating the p t h power of this inequality for Khintchine inequality, gives

$i,j=

457

e i j fi,j and using the double

again by inequality (*). This ends the proof. 0 Note that the constant can be taken to be aib;a2b2 if

Everything is ready for proving the announced result:

THEOREM: If /.3 is a tensor norm on HILB such that l&,pl2 is isomorphic t o a subspace of some L, (for 1 5 p < cu), then /.3 and the Hilbert-Schmidt norm u are equivalent on l2@ &. The result is a special case of a result of Lewis [168] (for p = 1) and of Pisier [218]. The argument for the present proof was given to us by Pelczyhski.

PROOF:By the Kwapieli-Pelczyiski result 26.11.it is enough to show that the unit vector basis (ei @ e j ) (rectangular ordering, see Ex 12.9.) is unconditional in the tensor product E = l2&& whenever this space can be embedded into some L,. A basis (2,) is unconditional if there is a constant c 2 1 such that

458

94. Tensorstable Operator Ideals

for all &k E (-1, l}, all n E N and all A, E K; this follows from 8.3., Ex 8.1. and a closed graph argument. The basis ( e , ) of f?z satisfies this inequality with c = 1. Denote by Uk := e k e k the k-th projection 4 4.Then

-

Pj := Ui@3pidi, and

Qj

:= idt,63pUj

satisfy the assumptions of McCarthy's result with a = b = 1, and hence

Since, obviously, U j 6 0 U j is the ( i ,j)-th projection of the basis (ei @ e j ) , this shows its unconditionality. 0

COROLLARY: Let /3 be a tensor norm which is not equivalent to the Hilbert-Schmidt norm on & 8 4 (for example: E , x ) and 1 5 p < 00. Then the ideals 2 7 and CC""'are not P-tensorstable.

-

PROOF:Since t?z is a subspace of Lp, the first case follows immediately from the theorem. Dually, k'z is a quotient of Lpt;assume that E = &63p&E space(2;Y'). Then E' E space(27' = s p a c e ( 2 y ) and therefore

&6&

t ' . (tz 8 p

!,)I

Lp

I

hence P' is equivalent to the Hilbert-Schmidt norm Q on 2'l @ & by the theorem. Since u' = Q (see 26.7.) and PI' = P on HILB, this is not possible. 0 34.9. The tensor stability with respect to e and x of the idea1 2p,q of ( p , q)-factorable :2 is comoperators, its injective and surjective hull and its minimal kernel ffp,q:= pletely known:

THEOREM: The following two tables show which of the ideals in the first column are or r-tensorstable:

E-

For example, among the ideals Cp,qonly the X P , 1 = gP are E-tensorstable, all others are not. The result is taken from Carl-D.-Ramanujan [23].

459

34. Tensorstable Operator Ideals

PROOF: The positive results have already been established in 34.5., corollary 2. First the negative results for the maximal ideals will be settled. Since, by proposition 34.4.(5), the ideal Qduo' is E-tensorstable if Q is a-tensorstable, it is easy to see that it must only be checked that the ideals not mentioned in the first column are not Etensorstable. Moreover, by 34.4.(4), the negative statements about ;:2 imply those about 2p,q. . . So take ;:2 with q > 1 (since 22' = is the positive case). For the ideal i$"i = E-tensorstability would imply (see 34.7., corollary 1) that Loo E space(Cil"f) but this is not true since, for example, Loo does not have cotype 2. If p, q €11, w[, then 2E; c !?JJ, the weakly compact operators. Now L 2 E space(ilp,q)C space(CE4) by 17.13., but L ~ & I ~isLnot ~ reflexive.

vP

2y

It remains to check the case ::2 quotient mapping e l --o e 2 is in

c !?JJ for 1 < q <

= 3Tr

00.

Since the natural

by the little Grothendieck theorem and 26.4.(3"), it follows that e, 6 space(3:') and hence t 2 E space(3yr also. The same argument as before gives that ::2 is not c-tensorstable. ; ; ; will be treated for ( p , q ) # ( 1 , ~ )It. will be shown that for all these ( p , q ) Now 2 the ideal 2;:; jnj is not E-tensorstable (which is enough by 34.4.(4)). Grothendieck's inequality shows that

.c2

Juur inj

caw inj

PA

- and the latter ideal is contained in ID if p or q E]l,oo[; the space E-tensorstability. The same applies to

L2&&

destroys

ID.

So it remains to consider Cz:Tl = 2gr- but this is not etensorstable as was shown in 34.8., corollary. The negative results for the minimal ideals follow from the negative results for the maximal ones: Those in the &-column using 34.4.(3), those in the r-column using 34.4.(5). 0 34.10. Tensor stability has interesting consequences for the distribution of eigenvalues of power compact operators (more generally: of Riesz operators, see e.g. Pietsch [216]). These operators T : E + E (in a complex Banach space) have at most countably many eigenvalues A # 0, each of which has finite algebraic multiplicity n(T, A) defined to be the dimension of 00 N w ( A i d ~- 2') :=

u

ker()ridE - T)" .

460

94. Tensorstable Operutor Ideals

With each such operator T E 2 ( E , E) one can associate a (not necessarily unique) sequence (An(T))nENof eigenvalues such that (1) Each eigenvalue A # 0 appears in the sequence exactly n(T,A)-times and consecutively. (2) IAl(T)I L IAz(T)I L * * . * (3) If T has exactly k eigenvalues A # 0 (counted according to their multiplicity), then An(T)= 0 for all n > k. Using Weyl's inequality in Hilbert spaces, one can show that Hilbert-Schmidt operators have eigenvalues (An(T))in Lz. Since every absolutely 2-summing operator T : E -+ E admits a factorization T = R o S through a Hilbert space such that S o R is HilbertSchmidt, it follows from the principle of related operators (due to Pietsch) that T has eigenvalues in 12 as well. In particular, nuclear operators have eigenvalues in 1 2 ; in Hilbert spaces they have eigenvalues in and it can be shown that a Banach space E is isomorphic to a Hilbert space if all T E %(E,E) have summable eigenvalues (Johnson-Konig-Maurey-Retherford [1311). In order to prove eigenvalue estimates the following result, due to Pietsch [215], is sometimes helpful:

PROPOSITION: Let (a, A ) be a quasi-Banach operator ideal of Riesz operators such that for all Banach spaces E and all T E B(E, E ) the eigenvalue sequence (A,(T)) is in Lp (for 0 < p < w). If there is a tensor norm p such that (24, A ) is P-tensorstable with constant c, then

n=l

for all T E % ( E ,E ) .

PROOF:First observe that there is a c1 2 1 such that the inequality (*) holds with c1 : If not there would be Tm E %(Em, Em)with A(Tm)5 (2d)-" (where d is the constant 2 m. Considering EmTm in in the quasi-triangle inequality for A ) and II(An(Tm))llp &(Em), gives a contradiction. Now take c1 t o be the best constant in the inequality (*). Since for eigenvalues A and p of T E % ( E ,E) one has N m ( A i d ~- T ) @ N , ( p i d ~ (8ee Pietsch

- T ) c i V , ( A p i d ~ @ -~ T 8 T )

[216], 3.4.7., for a proof), it follows that

94. Tensorstable Operator Ideals and hence that c1 5 (c1c)1/2 since follows that c1 5 c. 0

c1

461

was chosen to be the best possible constant. It

Note that for this “trick” only T&T E !2l was used and that the constant c does not depend on p : This will be important in a moment. By the way, the trick was already used in remark 34.2.. Using Konig’s Weyl inequality for Banach spaces (see [145]), one can show that in the case 2 5 p < 00 there are constants cp with SUP

IAn(T)ln”P 5 ~

p p p ( ~ )

for all T E yp(E,E ) (see [145], [131]). In particular, this “weak inequality” implies that (An(T)) E .t‘p+rfor all E > 0. Since !Jlpis metrically E-tensorstable (see 34.5.), it follows from the proposition that

n=l

for all E > 0 and T E V p ( E E) , and hence,

THEOREM: If T E 2 ( E ,E ) is absolutely p-summing for 2 5 p < 00, then

n=l

This important theorem is due to Johnson-Konig-Maurey-Retherford [131], the idea of proving it with the “tensor product trick” is due to Pietsch [215]. For more details, see the books of Konig [147] and Pietsch [216]. The proposition about the tensor product trick also has a negative flavour. For example, the inequality sup IXn(T)ln’IP < 00 n

actually holds for all T E P p , 2 ( E , E )with 2 5 p < 00, see [131] - but for 2 < p the eigenvalues are not in (see Konig-Retherford-Tomczak-Jaegermann [1501). The above reasoning implies the

!,

REMARK: For 2 < p < 00 the ideal Vp,2of absolutely

( p , 2)-summing operators i s not

tensorstable for any tensor norm.

See also Ex 34.8. and Ex 34.9. for the tensor stability of the ideal pv,*. 34.11. The ideals VP,land 9 X p j , ~ nearly coincide (see 32.10.-32.12.). A variant of the tensor product trick will give a nice improvement of the following theorem, due to Kwapieri [157].

462

34. Tensorstable Operator Ideals

THEOREM: For p E [l,001

1/r = 1 - Il/p

and

- 1/21

f o r T E 2(t,,t,) and 8 := 11 - 2/pl .

Note, the special cases p = 1 : 1 1 has the Orlicz property (see 8.9.). p = 2 : Grothendieck's theorem/inequality. p = 4/3 : an extension of Littlewood's result mentioned in 10.8. - this will be clarified in a moment.

PROOF (taken from Tomczak-Jaegermann [272]) : Using I!,-local to find a constant cp such that

technique, it sufices

pr,l(T) I CplITIl

for all T E Wl(zi;q)

2(e,$); equivalently:

= 1. Fix such

(21,

satisfies IPP

Since

:W

l

$

e

.tr(Tzi;$) I cpllTll for all 2 1 , ...,zm E with Then one has to show that CP,(T) := ( T z i )

...,zm).

-

t

T($)lI3 c p .

)

= P,(q), this means

I!(e,q) =P ,

-

It@, :e",(qJ)

-

Y(qJ>II 1 CP .

0, as a map KnX" Kmx"does not depend on p, so this is a typical situation for using interpolation. By Grothendieck's inequality and P z , l ( i d t , ) 5 C z ( P 1 ) 5 a 1 = f i (Orlicz's theorem 8.9., 8.6.) one gets

-

11%

:e",(e;)

11%

: c(q)

w3IlI q ( q ) l l sK G We).,)Il I Jz

II'P.0 : G(e).,)

*

First take 1 5 p 5 2 and 8 defined by

-1 = -1 - 0 P 2

+ -01.

Then 1/r = (1 - Q ) / l + 8 / 2 . Complex interpolation of P,(.) (n-tuples!) and of vector valued L, (see Bergh-Lofstrom [111, p. 107) gives

lppllI 2012KA-Q . For 2 5 p 5 00 start with l/p = (1 - 0 ) / 2

+ 0/00- and do the same. 0

463

34. Tensorstable Operator Ideals

.Before proving that the index r in this result is best possible the connection with Littlewood's result will be clarified: For p E %i[(co,co) = 2(co,t1) = c ( t 1 ) one has llL,+,ll = llpll = wl(L,e,;tl) (see 8.2.(1)) and hence Littlewood's inequality implies

E %il(c,,co), which means exactly that the embedding Z : tl 413 is in So for p = r = 4/3 the theorem generalizes Littlewood's result from Z to all operators T E f!(tl,t4/3). The theorem gives c 5 2'/4KAJ2 for the best constant in Littlewood's inequality. for all

'p

Q

?4/3,1.

PROPOSITION: L e t p , r E [1,00].lf2(k'l,tp) = Pr,1(t1,tp), then 1/r 5 1 - Il/p- 1/21. For p 5 2 this conclusion holds if only P , 1 ( Z : t, ~ -t,) r < 00. In particular, this means that the index r in the theorem cannot be improved (a result also due to Kwapieli [157]) and that in the Littlewood inequality the exponent p = 4/3 is best possible as well.

PROOF: Walsh matrices will be used. They are defined by induction:

-

A1 :=

2-"I2A, and A,

: t:"

o A,

(i

-:>

(2 -2,)

----5") and An+l :=

.

t:" is orthogonal and self-adjoint, hence llAn : tg" k':"ll = 2"12 = 2"id. Moreover, llAn : t:" too 2" 11 = 1 and the factorization

give llAn : 1% c 2 1 satisfying

-

,,

A, : t& 1:"ll

id

t;"

A n

e;"

id

t:"

5 (2n)3/2. To prove the result, observe first that there is P J ( T : t;-

a

5 CllTll t;" if 1 5 p 5 2 (and all n E N ) . Take

for all T E 2(t:", $") or only T = I, : t:" t!:" one has the latter case first: For A, : tz,"

k=l

hence, ((2n)r/p2n)1/r 5 ~ ( 2 " ) ~which / ~ , is only possible if the claim about r is satisfied. If 2 5 p 5 00, consider

464

- and

34. Tensorstable Operator Ideals

again r satisfies the desired inequality. 0

34.12. Next the announced improvement (due to Carl-D. [21])of Kwapieri's theorem 34.11. will be given. Recall that !Dlq,l C PSp,,1.

THEOREM: Let p , q E [l, m] with l/q = 11/2 - l/pl. Then

In particular, this space coincides with Pqt,l(tl,tp) and therefore the index 4 cannot be improved by proposition 34.11..See Ex 34.11. for two interesting reformulations. As was already mentioned, the proof will use a trick involving tensor stability - this time with respect to the right-tensor norms A,. The idea is condensed into the following lemma which shows how to improve weak inequalities.

u:,.

LEMMA 1: Let Zl c .C(K",K " ) be a set of operators and A , B : % two functions such that the following three conditions are satisfied:

(1) For every e > 0 there is

a c(e)

-

[0,oo[ be

> 0 such that

A ( T )_< c(€)n'B(T)

-

for all T E .C(Zi", K " ) n2l .

(2) [T@T : Kn@ K" = Kna K"'] E ZI whenever T E %. (3) There are constants a and b such that for all T E ZI A(T)25 a A(T 18 T ) and B(T €3T ) Then A(T)5 a b B ( T ) for all T E ZI

PROOF: Fix E

< b B ( T ) 2.

.

> 0. Then for each T E C(K",K") in 2l

A(T)25 a A ( T @ T ) 6 ac(c)n2'B(T €0 T )

< abc(e)n2'B(T)I ,

hence A(T) 5 d m n ' B ( T ) . Replacing C ( E ) by d m and iterating the argument, leads to A(T)5 (ab)c:=i 2-' ( c ( E ) ) ~ - '. 'n . B ( T ) . Taking first k -+

00

and then

E -+

0 shows the result.

The reader may have noticed the similarity between the proofs of lemma 1 and proposition 34.10..

466

94. Tensorstable Operator Ideals

- and this is the desired weak estimate. Now apply lemma 1. 0 Littlewood's result was that I E v 4 / 3 , 1 ( t 1 , t 4 / 3 ) (see 34.11.) -the theorem shows that this operator is even (4,1)-mixiig which means: Every absolutely 4-summing operator T : 1413 F is absolutely l-summing when restricted t o e l . The 2,--local technique lemma for operator ideals 23.1. shows that the theorem implies 2, o 21 C !2Jlq,l - or, by the definition of (q, 1)-mixing operators

-

Cyclic composition (see Ex 25.16.) gives the

COROLLARY: Let p, q E [l,m] and l / q = 11/2 - l/pl. Then O V q02 , c ??3y ~ 1 o ~ c , o V q c ~ o , , XP 0 21 0 2 , c J,f , 2 , o ff102 , c n,, ffp 0 210 EWc n,/ , ep0 21 0 ff, c nqf.

c,

?

For an application of this to the question of which absolutely q-summing operators factor through a Hilbert space see Ex 34.12.. &) and recently Bennett [9] and Carl [20] have investigated the norms Pr,#(I:t,, Carl-D. [21] the norms Mq,,(I : 1, t")for other indices u, u. The latter paper also gives further applications of the tensor product trick presented in lemma 1.

-

-

Exercises: Ex 34.1. Let (%,A)be a quasi-Banach operator ideal. If T163pT2 E rz( and T2 # 0, then TIE % and A(Tl)IIT'11 I A(T163pT2). Ex 34.2. (a) Show that 2, is metrically P-tensorstable for @ = gp,gp+,,dp, d;/, and that for these @ the @-tensor product of two 2zspaces is an 2;-space. See also Ex 23.17.. Hint: 34.5.. (b) 9 2 is strongly @-tensorstable for @ = dz and 9 2 . Hint: 34.5., corollary 1. (c) (21o is P-tensorstable, but not metrically P-tensorstable for the tensor norms p = d2 and g2. Hint: (b), D2 IK G ( L o~Lw)'eg and the tensor product trick 34.12..

$4. Tensorstable Opemtor Ideals

--

467

Ex 34.3. (a) Use factorization and the fact that L1 @, * respects subspaces to show F16pF2 is absolutely l-summing if TI E 3 and T2 E Sp1. that T16pT2 : E16pE2 D

-

(b) Use the factorization El co 2 t1 T16pT2 is quasinuclear whenever TI E 3 and

T2

F1 of nuclear operators to show that E 0%.

These results are due to Holub [117].

Ex 34.4. (a) For 1 5 p 5

00 the ideal C , is neither w2 nor wi-tensorstable. Hint: , $ C, and for p = 00 Grothendieck's inequality, For 1 5 p < 00 use the fact that 2 tensor form, and 34.7..

(b) For 1 < p 5 2 the ideal 2 , of type p operators is neither w2 nor wi-tensorstable.

(c) For 2 5 q < 00 the ideal Cq of cotype q operators is not w2-tensorstable. However, it is not known whether or not C2 is r-tensorstable; recall from 34.7. the open question of whether or not t 2 6 3 n t 2 6 3 r t 2has cotype 2.

Ex 34.5. (a) Prove the following extension of Schauder's theorem due to Vala [273] : If T E C(E1, E2) and S E 2(E3, E4) are compact, then

is compact as well. Hint: Use the compactness of E16,,EI, that Hom(T, S") is compact.

TQPS'

E2&E4 to show

(b) Use the same type of argument to show that for 1 5 p < 00

if S E ?J$, and T E SppdUO'.

Ex 34.6. (a) Let 1 5 p 5 00. If a Banach space E contains all 5 uniformly compleG = E @, G holds topologically, then G is finite dimensional. Hint: mented and E Use 29.5. to show that idG E 9,. (b) There are no infinite dimensional Banach spaces E, F and G such that EBCF&G = E @, F @= G holds topologically. Hint: & r-* E F = E @, F.

Ex 34.7. Let (%,A)and ( % , B ) be two quasi-Banach ideals such that 2l

c % and

3 B(.) _< cA(.). If both ideals are /3-tensorstable and for all T E 2 B(T)2 5 B(T6bpT) and A(T6pT) then

c

5 A(T)2 ,

can be chosen to be 1.

Ex 34.8. Show that V r , is~ not E-tensorstable for 2 5 r < 00. Hint: This result even holds for all r €11, 00[

E space(!&,l). which will be shown in the next exercise: 12

468

94. Tensorstable Operator Ideals

Ex 34.9. (a) Show that

Hint: Use 4.6. or Ex 8.l.(a) and $, @A,, (b) Use this and 32.4.(c) to prove that

= qjm.

= ?&,p. Hint: (c) Assume that l / q + 1/r = l/p. If !&,p is 6-tensorstable, then Use the first estimate in 32.12. and lemma 1 of 34.12.. (d) Use (c) and 32.12. to show that '&,I is not c-tensorstable for all 1 < r < 00. Ex 34.10. (a) Deduce from Kwapieri's result 34.11. and Ex 11.21. that

for p, r, s E [l,001 such that 1/r 5 1/s - Il/p - 1/21. (b) Show that this equality also holds for 1 5 p 5 2 5 s I00 and r := p's/2 by interpolation between the case p = 1 (trivial) and p = 2 (little Grothendieck theorem). Hint: Proceed as in 34.11.. The second result is taken from Bennett [lo]. Ex 34.11. Let p,q E [l,001, l / q = 11/2 - l/pl and T E 1 2 ( e 1 , e p ) . (a) For every probability measure p on Bf;there is a probability measure v on Bf,

for all x E e l . (b) For each weakly summable sequence (2,) in L1 there is a sequence (An) E l,i and a weakly q-summable sequence (yn) in lpsuch that Tx, = Any, for all n E N. In particular, for every unconditionally summable sequence (zn)in .f!1 there is a sequence (An) E t4/3 and a weakly 4summable sequence in e d / ~such that Z n = Any, for all n. Hint: 34.12. and section 32. Ex 34.12. Show that for p, q E [l,m]

(every absolutely q-summing operator on Lp factors through a Hilbert space) if and only if l / q 2 Il/p - 1/21 . Hint: 2 2 = '.= I( ) 21$ o 2-)*.

95. Tensor Norm Techniquesfor Locally Convez Spaces

469

35. Tensor Norm Techniques for Locally Convex Spaces

The final section shows that tensor norm techniques can be successfully used for the study of problems in the theory of locally convex spaces. As in the case of Banach spacea, tensor norms (different from E and T) and local Cp-techniques help one to better understand various phenomena concerning the projective and injective tensor product of locally convex spaces - such as the investigation of topological-geometric properties, Grothendieck’s (DF)-problem and the negative solution of the “problhme des topologies” given by Taskinen. Moreover, the so-called tensor norm topologies serve to classify locally convex spaces. Tensor norm topologies were introduced by Harksen in his dissertation 1979 (see also [102]). We assume that the reader is familiar with the basics of the theory of locally convex spaces (see Jarchow [123], Kothe [152]).

35.1. The topology and the uniform structure of a locally convex space are given by the family cs(E) of all continuous seminorms on E - or by the family U ( E ) of all absolutely convex closed (or open) neighbourhoods of zero. For p E CS( E ) denote I C ~:

E

-

E/kerp ; Ep := (E/kerp,p) ; Ij(kp(x)) : = p ( z ) .

Ep is a normed space which sometimes is also denoted by Eu if U := {x I p ( z ) 5 1) and one wants to point at the neighbourhoods instead of the seminorms. It is clear that it is enough to consider bases &(E) of U ( E ) or bases cs,(E) of cs(E) (this means: for every p E cs(E) there are a q E cs,(E) and c 2 0 such that p 5 cq ). If p 5 cq, then there is a natural map I C ~ :, Eq ~ Ep, and the completion of a separated locally convex space E can be represented as the projective limit of the net ( & p ) p E e s , ( ~ ) :

-

-

E = proj Ep . PE~J.(E)

The strong topologies on the dual E‘ and the bidual E” will be indicated by EL and E f ; the space E“ endowed with its natural topology of uniform convergence on equicontinuous subsets of El will be denoted by EF. 35.2. Given a tensor norm a (on NORM) and locally convex spaces E and F, define for p E cs,(E) and q E cs,(F) the seminorm p @a q on E 8 F by

P @a q(z; El F ) := a ( . p 8 ‘ E q ( ” ) ; Ep, Fq) for

I

E E 8 F. The family

I

{P @a 4 P E cso(E),Q E c s o ( F ) )

470

35. Tensor Norm Techniquesfor Locally Convez Spaces

defines a locally convex topology on E @ F (notation: E F and E G a F for the completion), which is certainly independent of the special bases cs,(E) and cs,(F) which were taken. This so-called tensor norm topology was introduced by Harksen. For cr = B or T it coincides with the usual injective or projective topology on the tensor product of locally convex spaces. In many respects the tensor norm topologies behave very similarly to the tensor norms on normed spaces.

PROPOSITION: Let a be a finitely generated tensor norm. (1) If E and F are separated, then so is E @a F. (2) Mapping property: If 9i C C(Ei,Fi) are equicontinuous, then { T I 8 7 ' 2 :El @a

E2

-

FI @a F2

IZE g i }

is equicontinuous. ( 3 ) Embedding lemma: The natural map

E @ F~CE @ F,"~ is a topological embedding. (4) E @ ,, F, + Eaa%' is a dense topological embedding which in particular means that E&F = E&f' holds topologically. ( 5 ) Extension lemma: If D c ( E @,, F)' is equicontinuous, then the set

D" := {'p^l'p € D ) of

right-canonical extensions is equicontinuous in ( E

F,")'

The assumption of a being finitely generated is not needed for (1) and (2); the statements (3) and (4) also hold for cofinitely generated tensor norms.

PROOF: For 0 # z E E @ F it is easy to see with corollary 2.3. that there are seminorms

c ', @ tcq(z) # 0; this implies (1). (2) is obvious from the definition of the seminorms p @a q. For (3) observe that for U E U ( F ) the dual of Fu is isometric to the Banach space [Uo] c F' (see 3.1. for this notation) and that

p , q with

(F")uoo

[U']'

A (Fu)"

is an isometric embedding. The embedding lemma 13.3. for normed spaces shows that

hence the conclusion.

35. Tensor Norm Techniquesfor Locally Convex Spaces

471

It follows from 13.4. (with an obvious notation) that the mapping

is dense and isometric - and this implies (4). (5) also follows directly from the normed case since every equicontinuous D C ( E B aF)' factors through an equicontinuous subset of some ( E u QDa Fv)'. 0

-

It is easy to see that the trace t r E : Ei @= E K is continuous if and only if the locally convex space E is normable. Therefore using the trace requires more precautions than in the Banach space case. Nevertheless, under certain circumstances the trace can be defined on the space %(E, E) of all nuclear operators (i.e., for complete El those operators which factor through a nuclear operator between Banach spaces):

where the inductive limit runs over all p E c s ( E ) and Banach discs B c E; see [44] for details concerning this topic.

35.3. For the study of right-projective tensor norms (to be done in 35.6.) and traced tensor norms a little lemma is helpful. For a given surjective map Q : F + G and a seminorm p on F, define the quotient seminorm pQ on G to be

-

LEMMA: Let Q : F G be continuous and surjective. Then Q i s a topological surjection if and only if for every p E c s , ( F ) there is a q E cs(G) such that the canonical map FP

-

is defined, a metric surjection and ker ti.,

G,

c kerpQ C G.

PROOF:If Q is topological, then q := pQ is continuous on G - and this gives the conclusion. Conversely, it is enough to show that pQ 5 q. Note that (p")". = q on G,. Take q(z) < 1. Then (PQ)"q(.q(4> = C(ti.,(.)) < 1 1 hence there is y E ker K , c kerpq such that pQ(z PO(, Y) PQ(Y) < 1. 0

+ +

+ y) < 1. It follows that p Q ( z ) 5

Recall the definition of a traced tensor norm a @G p from 29.1.. Its fundamental property also holds for locally convex spaces; the following result is taken from [41]:

472

35. Tensor Norm Techniques for Locally Convez Spaces

THEOREM: Let CY and p be two tensor norms (on NORM), G a normed space and 6 := (Y @G p. Then for all locally convez spaces E and F the tensor contraction

is a topological surjection.

PROOF:For pi E c s ( E ) and pz

E cs(F) consider the following commutative diagram:

For p := (pi @,9 11 I l ~ i8% ) (11 IIG @a pz) and q := p1 86 p2 the lower tensor contraction C,$' is a metric surjection - and, by the lemma, it remains t o show that pcQ(z) = 0 for all

+

= (kerpl @ F) ( E @ kerpz)

z E ker(rcp, @ rcpa)

Decompose z = (u', u ) E G' x

G with

.

+ Cy='=m+l Z j 8 yj according to this equation and choose

zj @ yj (u', u) = 1. Then

i=l

and

n

pCQ(z)5 P ( ~ u5)C P l ( z i )

II~'IIII~IIP Z ( Y ~ ) = 0

*

i=l

35.4. Let a and p be finitely generated tensor norms with Q accessible. For a given locally convex space E, under which conditions is the operator

continuous for all locally convex spaces G 7 It is enough to check this only for Banach spaces G - or even just G = Cz a Johnson space: Apply lemma 17.16. t o

However, using traced tensor norms, continuity of this latter mapping means that

Z q , p E 9' where 1)-p@CY*

35. Tensor Norm Techniquesfor Locally Convez Spaces

473

(see theorem 29.4.). Since a*= (% o %*)* by 29.8., one obtains

PROPOSITION: Let a and p be finitely generated tensor norms, a accessible, with associated maximal normed operator ideals % and 23, and E a locally convex space. Then

-

is continuous for all locally convex spaces G _(Or only C2) if and only i f for every p E c s ( E ) there is a q E c s ( E ) such that Zq,p : E, Zp is in (% o a*)*,the operator ideal associated with (p @J a*)*. If Ctl is a quasi-Banach operator ideal, call a locally convex space E to be in space(%) if for all p E cs,(E) there is a q E c s ( E ) such that iE,,p E %. For Banach spaces this coincides with the notation which was used 90 far. For locally convex spaces the books of Jarchow [123], Junek [133] and Pietsch [214] give much information about space(%). Assume % to be maximal and normed and take a to be its associated tensor norm. Then !2 a = (a*)* = ( E @ a*)*= (a*@ T * ) *

-

(see 29.8.), hence the proposition implies the

COROLLARY: Let Sf be a maximal normed accessible operator ideal with associated tensor norm a. For each locally convex space E the following statements are equivalent: (a) E is in space(%). (b) G@Ja. E = GI& E (topologically) for all locally convex spaces G (or only G = (22). (c) G

E = G c&,

E (topologically) for a11 locally convex spaces G (or only G = Cz).

For the equivalence (a)

H

(b) accessibility of % is not needed.

35.5. Some examples: The spaces in space(%) are usually called nuclear spaces. Since for all 1 5 p < 00 there is an n E N with qpo ..o !JlP c % (n-times composition, see 11.7.), it follows that

space(%) = space(qp) = space(JP) = space(3)

-

€or all 15 p < 00. To be in apace(3) means that E = * BXE (topologically) since 3 r, which is Grothendieck’s original definition of nuclear spaces. See also EX 35.2.Ex 35.4.. The locally convex spaces E in space(EP) have the property that for all p E cs,(E) there is a q E c s ( E ) such that N

474

95.

Tensor Norm Techniques for Locally Convez Spaces

factors through some L p ( p ) . We do not know whether or not each complete locally convex space E E space(Cp) is the reduced projective limit of a net of C;-spaces (the Banach spaces in space(Cp)) - except in the case p = 2 : The spaces in space(&) are exactly the Hilbertizable locally convex spaces. The Cp-local technique reads as follows:

REMARK:Let (1) If f o r T

a and /3 be tensor norms, /3 finitely generated.

E C(F1, F2) (locally convez spaces)

is continuous, then

idEgIT: E @ p F1

-

E B a FZ

i s continuous f o r each locally convex space E E space(CP).

(2) If id : tp@ ~ p tq-+

tq is continuous, then

tp

id:E@DpF - E a a F is continuous f o r all locally convex spaces E E space(Cp) and F E space(&).

PROOF:(2) is a consequence of (1). For (1) take for U2 E U(F2) a U1 E U(F1) such that i 4 , @ Tul,u, : tp€ 3 (F1)u1 ~ 4, @ a (F2)u,

-

-

is continuous. Then Ex 13.7. gives that e, can be replaced by Lp(p). Thus, if the operator iiv,,~,:bv1 &, is pfactorable, factorization into the bidual gives

1 and hence the result. 0 In [44], 4.4., it was shown that if a’ i s totally accessible, then all locally convex spaces E E space(%) have the approzimation property. Banach spaces in space(%) even have the bounded approximation property (see 21.6.) - but for general locally convex spaces (and a = T ) this is already false for Frtkhet spaces since there are nuclear Frechet spaces without the bounded approximation property. Since w; is totally accessible, locally convex spaces in space(2,) have the approximation property.

35. Tensor Norm Techniquesfor Locally Convez Spaces

475

35.6. Corollary 35.4. showed that E E space(Cp) if and only if

GmCE = G B W ,E or G@"; E = G @ wE for all locally convex G or only G = Ca (see also section 23). Since woo = w; = / T (see 20.14.), it follows, as in the case of Banach spaces, that:

E E space(&) i f and only i f E 8- = E @*\ E E space(Ze,) i f and only i f E mC* = E @c/

\E

and

.

Thus it will be important to know that injective norms respect subspaces topologically and that projective norms respect quotients topologically also for locally convex spaces. The following result is due to Harksen [loll, I1021 :

-

THEOREM: Let a be a tensor norm (on NORM) and E , F, G locally convez spaces. (1) If a is right-injective, then idE @ I : E

@a G

is a topological embedding whenever I : G

L,

E ma F

F is.

( 2 ) If a is right-projective, then

is a topological surjection whenever Q : F

+G

is.

PROOF:(1) follows from the fact that the sets V n G with V E U ( F ) are the typical neighbourhoods of G and

for all U E LL(E). For ( 2 ) lemma 35.3.will be applied. Take a typical seminorm p = p1 @a pz on E @a F. 1 Then, by the lemma, there is a q E cs(G) such that ker K , C kerpf and FPa + G , is a metric surjection. Since

is a metric surjection, it remains to show (again by the lemma) that each element z E ker(Kpl @ Ic,) satisfies pid@Q(z)5 E for all E > 0. Write z

=

m

n

i=l

i=m+l

Czi@ Yi + C

Zi

@ yi E ker(npl @ K , ) = kerKp, @ G

+ E @ ker

K,

476

35. Tensor Norm Techniques for Locally Convez Spaces

and choose elements Ji E F such that Q(gi)= yi; since ker tcq C ker p28 one can achieve p2(gi) 5 C E for i = m 1, ...,n with a constant c 2 0 to be chosen in a moment (1).It follows that

+

i=l m

n

i=l

i=m+l

The theorem generalizes the well-known (and simple) facts that E respects topological embeddings and A topological surjections. The characterizations of space( C,) and space(&,) given above allow one to extend the characterization 23.5. of f!! and .C&spaces from Banach to locally convex spaces; this result is due to Hollstein, [lo71 and [lo91 :

COROLLARY: Let E be a locally convex space. (1) E E space(&) if and only if E @= respects topological embeddtngs. ( 2 ) E E space(2,) if and only if E @c respects topological s u j e c t i o n s .

-

PROOF:It follows from the theorem that both conditions are necessary. For the sufficiency take G = Cz a Johnson space and look at the diagrams

These results have many applications to the extension and lifting of operators and to the theory of vector valued functions. 35.7. For the duality theory of E@
-

holds topologically; the mapping property implies that the identity is always continuous. A locally convex space is said to have the countable neighbourhood property if for each sequence (pn) in cs,(E) there are a p E c s ( E ) and cn 2 0 such that pn 5 c , p . (DF)-spaces E enjoy this property (this is easy: look at Ut c Ei which is a Fr6chet space).

35. Tensor Norm Techniques for Locally Convez Spaces

PROPOSITION: Let E and F, be locally convez spaces and the countable neighbowhood property, then

n=l

CI:

477

a tensor norm. If E has

n=l

holds topologically.

PROOF: The typical semadormon $, Gn is C r, with r, E cs(G,). It has to -e shown that for pn E cs(E) and qn E Cs(Fn) there is a p E c s ( E ) and q E CS($, F,) such that

Observe first that for all rm E Cs(F') the space (Fm)r, is l-complemented in the space ($, F,)c~., hence for each p E cs(E)

<

for all C, r, E E @ @ , F,. Choose pn c,p according to the countable neighbourhood property and define q := C,"=l2"cnqn. Then

m=l

00

W

m=l

m=l

00

m=l

An inductive limit ind,,F,

n=l

n=l

is (topologically) a quotient of the direct sum

F,,

80

holds topologically if E has the countable neighbourhood property and one of the following three conditions is satisfied: (1) The kernel of the canonical map F, ind,F, is complemented (following De Wilde such inductive limits are said to have a partition of unity - for example Schwartz' space 9(Q) of test functions on Q c R" has such a partition; see PBree Carreras-Bonet [207] for details). (2) a is right-projective. (3) E & . = E%/* . These results are due to Harksen [loll, [lo21 (see also [SS]).

@r=l

-

478

35. Tensor Norm Techniques for Locally Convez Spaces

For a = 6 this implies, e.g., that each (DF)-space E in space(2,) satisfies the relation E BCind F, = ind E @c F, - and this is even characteristic for the 2&-spaces among the Banach spaces (see Ex 35.11.). To obtain E&ind F, = ind E&eFn one needs additional assumptions (see Hollstein [lo71for information in this direction). 35.8. If E is a Banach space and F = ind F, an inductive limit of Banach spaces such that E @a ind F, = ind E @a F,, then E @a F is quasibarrelled (i.e. bounded sets in the strong dual are equicontinuous). There is a good amount of literature investigating topological-geometric properties isuch as quasibarrelled, bornological, barrelled) of E @a F and C ( K )BCF or C ( K ,F ) = C ( K ) & F. We want to show how traced tensor norms can successfully be used in this context. A pair (E, F ) of locally convex spaces is said to satisfy property (BB) (for bibounded) if for each bounded A c E B r F there are bounded sets B c E and C c F which “lift” A, i.e.

AcI’(B@C). If E and F are F’r6chet spaces, then compact sets in E&F can be lifted (this follows as in 3.4.) - but, in general, bounded sets not; this (deep) result will be treated in a moment. The pair ( e l , E) having ( B B ) is also known as property (B) of Pietsch of E. Endow the space c ,(E) of zero sequences in E and the space e l ( E ) of absolutely surnmable sequences in E with their natural topologies coming from the seminorms supp(z,) and Cp(zn) (for p E c s,(E)), respectively. Note that c0(&) = c,&E and el(,!?) = el&E hold topologically. It is easy to see that the space c , ( E ) has a strong dual which is topologically embedded into el (EL) :

the duality bracket being ((a&),

(2,))

:= x,”==,(z;, zn). Since for U E U(E)

(co(Ev))’= (c,6CErr)’ = el&E:

= l1([U0])

holds isometrically, the equicontinuous sets in (cO(E))’are exactly those which are in some unit ball BL,([UO])c el(EL) * Since every (zk) E !l(EL) is the limit of its finite parts, (c,(E))’ is dense in ll(EL). It also follows that (co(E)); = e i ( E i ) holds topologically if every countable bounded set in ( c , ( E ) ) ; is equicontinuous. This is the case, for example, if c ,(E) is quasibarrelled.

Let E be a locally convex space. Then the following statements are PROPOSITION: equivalent: (a) C ( K ,E ) is quasibarrelled for all compact sets K (or only one infinite compact K ) .

35. Tensor Norm Techniquesfor Locally Convez Spaces

479

(b) c,(E) is quasibarrelled. (c) G @e E is quasibarrelled for all C L - s p a c e s G ( o r only one infinite dimensional 20,-space G ) . (d) E is quasibarrelled and (.!I, Ei) has property ( B B ) .

PROOF:Using finite sequences (and the above remarks) it is easily seen that (b) and (d) are equivalent. Moreover, since co @e E is a large subspace of c,(E) (each bounded set in c,(E) is in the closure of a bounded subset of co E l see Ex 35.7.), c,(E) is quasibarrelled if and only if c, @e E is quasibarrelled (call this (b‘)). The crucial point in the proof is now the statement of Ex 29.9.(a): If F is an infinite dimensional 2L-space (hence contains the Pm uniformly complemented by 23.3.), then &/@‘F&/

(b‘)

fi

=Wl@F W

I N Wl

= &/

.

(c): Take F = co and G an ZL-space. Then

(G@e/

.!I) @r

(co @e/ E )

-

G @c/ E

is a topological surjection. Since G and co are f&-spaces, one can replace &/ by e in all three places. Now it is enough to note that X @= Y is quasibarrelled if X is normed and Y is quasibarrelled (take A c (X Y)‘bounded; then A is bounded in &(X,Yd) and hence A ( B x ) c Y{ is bounded = equicontinuous). (c) TC (b’): Assume the space GoaCE to be quasibarrelled for an 2Cg,-space Go.Since €1@G, &/ &/, it follows that the surjection N

(co @e/ Gb) @r ( G o @e/

E)

-

co

@c/

E

is topological, hence the same argument applies. Since C ( K ) & E is also a large subspace of C ( K ,E) (this again follows from Ex 35.7.), the equivalence with (a) follows easily. 0 These results are due to Marquina-Sanz Serna [184], Mendoza Casas [190] and D.Govaerts [48]. In his paper Mendoza also proved that for barrelled spaces E the space C ( K ,E) is barrelled if it is quasibarrelled. Hence, the proposition also gives a characterization of barrelled C ( K ,E ) (see also Ex 35.11.). For more information about topological-geometric properties of C(0, E) (when SZ is completely regular) see Schmets [252].

-

35.9. Let E and F be Frkchet spaces; then Baire’s category theorem implies that every continuous operator E F‘; factors through some [Uo]with U E t l ( F ) ,hence 2 ( E , Fi) = (E @= F)’ holds algebraically. Grothendieck’s “probkme des topologies” ([92], I, p.43) is the question of whether or not the strong topologies coincide:

.&,(E,F,’)

( E a rF);

.

95. Tensor Norm Techniques for Locally Convex Spaces

480

The strong topology on C ( E ,Fl) coma from pairs (A, B) of bounded sets, the strong topology on (E Br F)‘from bounded sets in E @ r F. It follows that the canonical map (E F); &(El Fl) is always continuous and that the topologies coincide if and only if the pair (E, F ) has property (BB), defined in 35.8.. For separable E and F it is not difficult to verify that &(El FL) = (E ar F)6 if and only if & ( E , Fi) is a (DF)-space. Moreover, the “problkme d a topologies” is closely related to another one of Grothendieck’s problems namely, whether or not E @c F is a (DF)space if E and F are; see 35.10. below. The problem was solved in the negative by Taskinen [267] in 1986, see also [268]. He constructed a FrBchet-Monte1 space F such that (12, F ) and (F,F) do not have property (BB). The space FkrF contains t 1 - hence is not even reflexive. On the other hand it is easy to see that E k r F is FrBchet-Schwartz whenever E and F are: since compact sets are liftable, (ElF ) has (BB) in this case. We shall concentrate on the case in which one space is a Banach space.

-

PROPOSITION 1: If E is an C:-space, then for every Frtchet space F the pair (ElF ) has property (BB). PROOF:Since local technique will be applied, the case of E = 1 1 will be treated first. Topologically 1 1 & E = 1 l ( E ) , hence, for each bounded set A c t l ( E ) it is enough to find a bounded C C E such that A C 1,([a) is bounded: If ( p k ) is a basis of seminorms of F, put

choose

Ak

> 0 such that c(PZ1 C k X k 5 1 and define

c := SUp{Z

EE

I

m

.\kpk(Z)

5 I}

.

k=l

Obviously, C is absolutely convex and bounded. Then for c o w

n=l

n=l k = l

c o r n

(2”)

EA 00

k=l

- and the first claim is proved. Now take E to be an arbitrary &:,A-space. It has the &bounded approximation property (see 21.6.), hence there is a net (Tq) of operators Tvf 5 ( E ,E) with llTvll5 X converging pointwise to i d E . The very definition of c!,Aspaces gives operators

481

55. Tensor Norm Techniques for Locally Convez Spaces

It follows that

1 1 4o T911 5 2Aa and the net (4o T , ) @ i i d ~: E 8=F

-

ti % F

is equicontinuous. Now take A C E 8=F bounded; then

A, :=

U{ ( 4 T9)@ id^(.^) I z E A } 0

9

is bounded in tl 8=F, hence contained in I'(Bt, 8 C) for some bounded C C first part of the proof. Since S,(Bcl)C BE, it follows that for all z E A

The fact that

F by the

T9@ i d ~ ( z converges ) to z concludes the proof. 0

This result is due to Taskinen [269], but the present simple proof by local technique is taken from Bonet-D.-Galbis [12]; see Ex 35.6. for more applications of this technique to the problem of lifting bounded sets in E 8=F. Astonishingly enough, a converse of the above result is also true. The following is a slight generalization of Taakinen's famous original example (see [269]):

PROPOSITION 2: For every Banach space E which is not an 2 : -space there is a Frkchet space F and a countable set A c E @= F which is not contained in any set of the form

where C c F is bounded. The Fre'chet space F can be chosen to have the bounded approximation property and to be the reduced projective limit of a sequence of separable reflexive Banach spaces with the approximation property.

PROOF (taken from D.-F.-Taskinen [47]) : Note first that the C:-spaces E are exactly those Banach spaces for which E 8 = respects subspaces (with a constant A, see 23.5., corollary 5). Fix a Banach space E which is not an tt-space. Then this observation together with an argument as in the proof of 20.3.(1) shows that there are G, E FIN with subspaces M, c G, such that there exist z, E E 8 M, with

-

-

T(zn;EIGn)< 1 and 2 n 5 ~ ( z , ; E , M , ) 5 2 " .

Fix a continuous projection P, : G, IIYlln,t

It follows that

11 - [In 5 11 - Iln,b

G, onto M , and define for all n, k E M

:= IlPnYlln

+ klly - PnYlln

for y E G,

and llyll, = IIyll,,~ for y E M,. Notation:

1 Gn,k := (Gni II * Iln) G:,k := (Gn, 11 * Iln,t) I f o := & ( ( G ; , k ) n , k ~ ~ ) , H := ~ z ( ( G ~ , ~ ) *~ , E E N )

.

482

35. Tensor Norm Techniques for Locallu Convex Spaces

-

Clearly, Ho C H and 11 1 1 5~ 11 1 1 ~ 0 . If then %n,k := (idE 8 In,k)(%n)satisfies

(*I

I n , k : Gn,k L, H

%(%,,tiE , H ) 5 1 and

is the canonical injection,

%(%,,k;E , Ho)5 2"

- the latter since s(zn;El Gi,k)I 7r(zn;El Mn) 5 2". Now assume that for k,m,n E IV %n,L E

r(BE C3 (mBHo n ~ B H );)

then (by projection on the (n, k)-th component of H) there is a representation N

N

Everything is prepared for defining the needed space:

H," := J'm

H for n 2 { Ho for n < m

m

:= lz((H,")nEN) C I f N

n 00

F:=

F,.

m=l

F is a fitkhet space of the kind expressed in the statement of the proposition. Denote by J n the canonical injection of the n-th component into H" and by Qn the corresponding projection. The set

A :=

id^ 8 Jn(zn,k) I n, k E N } C E @ r F

35. Tensor Norm Techniques for Locally Convez Spaces

483

is bounded - this follows immediately from (*). Assume now that A could be “lifted”, i.e. there were cn 2 1 such that the bounded set

C:=

c~BF,,,C F m=l

satisfied A C

r(BE

8 C ) . Since

r(BE

8 U ) is a zero neighbourhood of E GDrF for

U := c ~ B Ff ,l Cn+lBF,+I, it would follow that

Propositions 1 and 2 give the

THEOREM (Taskinen): A Banach space E i s an f!y-space if and only if (ElF ) has property (BB) for all Frkchet spaces F. 35.10. Besides the Frhchet spaces the most important class of locally convex spaces are the (DF)*paces introduced by Grothendieck; they simulate the structure of the strong dual of a Frkchet space. A locally convex space E is called a (DF)-space if it has a countable basis for the bounded sets and every strongly bounded subset A C E‘ is equicontinuous if it is of the form A = u,”==,An with equicontinuous An. If E is metrizable its strong dual is ( D F ) and the strong dual of every (DF)-space is a F’rhchet space. It was one of the problems which Grothendieck left open in his study of (DF)-spaces whether or not E 8cF is ( D F ) whenever E and F are. As was already mentioned, this problem is intimately related with the “ p r o b l h e des topologies” just solved in the negative. It is in some sense dual (the proofs will show this) - so it is not surprising that the C&-spaces are the “good” Banach spaces for this problem. As before in 35.9. (proposition 1) the “typical” Zk-space c, will be treated first.

PROPOSTION: Let F be a (DF)-space. Then the space c , ( F ) of all zero s e p e n c e s in F is a (DF)-space as well.

PROOF:That c,(F) has a fundamental sequence of bounded sets is straightforward. Moreover, note that (c,(F))b c k‘l(F;) (topologically) by 35.8. - and that the equicontinuous subsets in (c,(F))’ are those which are bounded in k‘l([Uo]) for some U E LI(F).

484

35. Tensor Norm Techniques for Locally Convez Spaces

So take a bounded A = U n E U(F) such that

U,"=lAn C (c,(F));

with equicontinuous A,. Then there are

for all Since (k'~,F{) has property (BB), it follows that there is a bounded C

c

(zL)E An .

c F,'

such that

a,

mc(z:,)

for all (&) E A

I1

.

m=l

Take C n := C n U t . Then equicontinuous set D = I?

U,"=lC,,is equicontinuous since F is a (DF)-space. For the Ur=l C, it follows that mD

5 mc, I m c + mu:

which implies that C,"==, mD(zl,) 5 2 for all z' = (2;) E A,, and hence for all z' E A . This shows that A is bounded in t?l[D])and hence is equicontinuous. 0 This result implies (by what was said in 35.8.) that (c,(F)); = ll(F,') if F is a (DF)space. Moreover, since co BCF is a large subspace of c,(F) (see Ex 35.7.), the space co BCF is also a (DF)-space in this case. Now, as in 35.8., the traced tensor norm technique F) -0 E alF , (E 8Clel) B~ (co the fact that the Ir-tensor product of two (DF)-spaces is (DF) (see e.g. Jarchow [123], p.335, for a proof), and the fact that E & = E &I * whenever E is a CL-space give immediately: E aCF is (DF) if F is a (DF)-space and the Banach space E an CG-space (see also Ex 35.8. and Ex 35.9.). The converse statement is also true - a result due to D.-F.-Taskinen [47] :

-

THEOREM: For each Banach space E the following statements are equivalent: (a) E is an 2L-space. (b) For every (DF)-space F the space E BeF is a (DF)-space. (c) For every inductive limit F of separable reflexive Banach spaces with the approrimation property the space E&F is a (DF)-space. Note that the completion

of a (DF)-space G is a (DF)-space.

PROOF:It remains to show that (c) implies (a). For this it is necessary to show that E' is an 2;apace (by 23.2.). Assume not, then proposition 2 in 35.9. gives a Fr6chet space G, which is a reduced projective limit G := proj Gn c n

35. Tensor Norm Techniques for Locally Convex Spaces

485

of a sequence of reflexive separable Banach spaces with the approximation property and a countable A C El QDr G which is not liftable. Then

F := G; = indGL n-

; Fn := GL

holds topologically since G is distinguished and hence G; is bornological. This space F is as in (c), therefore E&F is a (DF)-space. (1) Consider the natural map J : E @ J ~F 4 &(El, F ) . Since for U E U ( F ) and tEEQDF

--

it follows that J is continuous. &(El, F ) is complete and therefore also the dual (E&F); is continuous. The continuity of the canonical mapping P : (&(El, F ) ) ; bilinear map E‘ x F{ (Eb(E‘,F ) ) ; gives that the natural map (note that ( E & F ) ; is a F’rhchet space by (c))

is defined and continuous. The diagram

shows that is injective since the approximation property of G , implies that the lower map is always injective. (2) The equicontinuous subsets D c (E&F)’ can be described as follows: There is a U E U(F) such that D is bounded in the Banach space

Since G = F,‘ is reflexive, U o is weakly compact, hence there is another bounded = equicontinuous Banach disc V o such that I : [U’] LI [Vo] is a weakly compact operator (see e.g. Jarchow [123], p. 215). 2Do 3 c 9l (see Appendix CS.),therefore D is bounded in

It is clear (elementary tensors) that

48 6

35. Tensor Norm Techniquesfor Locally Convez Spaces

T commutes. Now take the countable bounded A c E’ which cannot be lifted. This means, in particular, that A is not contained in any I’B with B c E’&fVo] being bounded. However, * ( A ) c (E& F): is countable and bounded, hence equicontinuous since E&F is a (DF)-space. The description of the equicontinuous sets in (2) and the injectivity of give a contradiction: E‘ must be an Cftzpace and E an CL-space. 0

@rx

*

The main work in the proof was showing that

holds -with a good description of the equicontinuous sets in ( E a t F)’. In essence this type of duality result is due to Grothendieck’s investigation of what he called Phillip’s property ([92], I, p.108) -nowadays referred to as the Radon-Nikodim property or the local Radon-Nikodfm property; for more information about the duality of E and ?r in the locally convex setting, see Collins-Ruess [34], D.-F.-Taskinen [47], and [40], [44] .

35.11. The theory of tensor norms represents a good part of the so-called “local” theory of Banach space (i.e. the study in terms of finite dimensional subspaces). The tensor norm techniques just presented for locally convex spaces may mark the beginning of a “local” theory of locally convex spaces.

Exercises: Ex 35.1. Let 24 and % be two quasi-Banach operator ideals. (a) If % C %, then space(%) c space(%).

(b) space(%) n space(%) = space(% o 23) = space(% o a). Clearly, space(%) is used here in its locally convex meaning. Ex 35.2. Prove: (a) For ( p , q ) # (GO, GO) space(!DP,*)coincides with the class of all nuclear spaces. The same holds for space(3p) = space(zP,l)and space(C1,p) if 1 5 p < 00. Hint: For the first statement use Kwapieri’s factorization theorem. (b) space(gP,() = space(&), all Hilbertizable locally convex spaces, if 1 < p , q 5 2. Hint: 12.8..

35. Tensor Norm Techniques for Locally Convez Spaces

487

35.3. Use the previous exercise to show that for a locally convex space E and (p, q ) # (1,l) the following are equivalent: (a) E is nuclear. E = G@.,E for all locally convex spaces G (or only G = (72). (b) G@u,,q (c) G @, E = G &;,* E for all locally convex spaces G (or only G = CZ). Ex 35.4. Show that space(Cp) n space(E,) = nuclear spaces if 1 5 p < 00 and

Ex

space($) n space(&) = nuclear spaces if 1 < p 5 00. Hint: For the first equality use that Cpo&, is contained in !J32if 1 5 p 5 2,and in Pp+, if 2 < p < 00 (see e.g. 23.10. and 24.7.). For the second look at 2102~ c (Cp,~C,)dua'. This result is taken from Pietsch [214],29.7.6.. Ex 35.5. (a) Let Q a be accessible and E a locally convex space. Then E * =E respects quotients isomorphically if and only if E * holds and if and only Hint: For the first equivalence use the ideas of if E E space(Odua' o (Odu4'o 2-)*)*. 35.6.and for the second 35.4.and 20.12.. (b) Let p = 2 or 00. Show that a locally convex space E is nuclear if and only if E@.,? respects quotients topologically. Hint: Check the equalities (C,o(C202,)*)* = (Cz0!4?2)*= !4?2 (1ittleGrothendieck) and (C~O(&O~,)*)* = (&09$)* = (21022)* = (C2/)* = / 9 z = ,pa' (Grothendieck's inequality, 20.12. and 20.15.). Note that the tensor norm w1 is right-projective. Ex 35.6. (a) Let E be an gl-space and F a Frkchet space. Show that if (lp, F) has property ( B B ) , then (E,F) has it as well. Hint: Use the technique from 35.9., proposition 1. (b) Prove that if (&, F) does not have ( B B ) ,then (tp,F) does not have (BB) for any p €11,00[ . Hint: 23.2.,corollary 2. These results are taken from Bonet-D.-Galbis [12]. Ex 35.7. (a) Let E and F be two locally convex spaces, F complete, E with the bounded approximation property and a a tensor norm. Then E& F is a large subspace of E& F, i.e. every bounded subset B of EbUF is contained in the closure of a bounded subset C of E @u F. Hint: Take an equicontinuous net (Tq)of finite rank operators on F. E which converges pointwise to idE, and look at C := UT,buidF(B) C E Note that, in general, K F = F is not a large subspace of P = K&, F.

-

-

(b) Show that c, 8, F is a large subspace of c,(F) for any locally convex space F. Hint: Finite sequences. (c) Let K be compact and F a locally convex space. Using partitions of unity, show that C ( K )@, F is a large subspace of C ( K ,F ) . Ex 35.8. Take a normed space E , a (DF)-space F and a finitely generated tensor F is a (DF)-space. Hint: Use the formula a/ = E @f- a from norm a. Then E

29.10..

488

35. Tensor Norm Techniques for Locally Conuez Spaces

Ex 35.9. Let El and

be (DF)-spaces. (a) If Eh QPc G is (DF) for all Banach spaces G (or only CZ),then El QPC E2 is (DF). Hint: E QP E = E . (b) If El, E2 E space(&), then El QPc E2 is a (DF)-space. Hint: GQPcEl: = GQPc/Eh and the previous exercise. Statement (a) can be applied to Hilbertizable bornological (DF)-spaces, which by a result of Hollstein [lll]have the property that their &-tensor product with each normed space is (DF). In [12] this was used to prove that pairs of Hilbertizable Frkchet spaces have property (BB)- a result stated but not proved by Grothendieck [92]. Ex 35.10. (a) Use the ideas of 12.9. to show that for normed spaces E and F the natural map ( E QPc l p ' ) @ x 4 QPr(fp) @3r F ) E @a,,q F E2

-

(zn)

(An)

8 (yn)

rw+

2

Anzn 8 yn

n=l

+

+

(with l/p' l/q' l/r = 1) is a metric surjection. (b) Extend (a) to locally convex spaces. This result simplifies the study of @m,,,-stability of classes of locally convex spaces. Ex 35.11. Use 35.7., 35.8. and 35.10. to show that for a Banach space E the following are equivalent: (a) E is an &&-space. (b) For every inductive limit ind F n of Banach spaces E aC ind Fn = ind E a t F n holds topologically. (c) E QPC F is quasibarrelled for each quasibarrelled F such that (41, F i ) has (BB). Hint: EBCF has a fundamental sequence of bounded sets if E and F are (DF)-spaces. Ex 35.12. Let T : E F be a homomorphism between locally convex spaces, i.e. a topological surjection onto its image. Then the followingoperators are homomorphisms as well: (a) idG QPr T whenever G is a locally convex space in space(&). (b) i d c QPc T whenever G is a locally convex space in space(&,).

-

Appendix A : Some Structural Properties of Banach Spaces

489

Appendix A: Some Structural Properties of Banach Spaces

This appendix collects some results from Banach space theory which may be unfamiliar to the reader. These facts are used in various places throughout the book without always calling this to the attention of the reader. References will only be given for those facts which are not in nearly every introductory book on Functional Analysis.

-

Al. A normed space is a pair (ElII.II) of a vector space E over the field K = R or CT of real or complex scalars, together with a norm 11 11 : E [0, m[. If it is clear what the norm is, then we write simply E for the normed space - and 11 1 1 for ~ the norm if necessary. If every Cauchy sequence in E converges, E is called a Banach space.

-

0

The notation BE (open unit ball), BE (closed unit ball), 2 ( E , F ) (continuous linear operators) and E' (dual space) is used (see list of symbols). Obviously, the reader should be familiar with basic examples of Banach spaces - such as: c o , . t p , C ( K ) , Hilbert spaces and Lp; for the latter see Appendix B. Moreover, we do not believe it is necessary to explicitly state the open-mapping theorem, the closed graph theorem and the Hahn-Banach extension theorem. However, in the last case the reader might like to be reminded of the real version in which sublinear functionals w are involved: R sublinear, i.e. for Let E be a real vector space and F a linear subspace, w : E all 2, y E E and X 2 0 :

-

-

-

and (o : F 42 linear with (o W ~ F Then . there is a linear extension (2, : E R of (o with (2, 5 w . It is worthwhile to recall that this version of the Hahn-Banach theorem has powerful applications even in the case where F = (0) ! A mapping Z E C ( E , F ) is called an isometry or a metric injection if 11Zzll~= 11tll~ for all t E E. A surjective mapping Q E L(E, F ) is called a metric surjection if

1

1

for all z E E, i.e. 11 I ~ J P is the quotient norm of 11 l l along ~ Q. Notation: L+ and -W to indicate a metric injection or surjection, respectively. For every Banach space E there are natural mappings

490

Appendiz A: Some Structural Properties of Banach Spaces

Obviously, BE can be replaced by any dense subset of the unit ball and BE, by any norming subset of BE,. In particular, for every separable Banach space E there are mappings 1

1

el----*

E k e , .

The natural metric injection of a normed space into its bidual E”

will be denoted by K E . If ICE is surjective, E is called reflexive. Hilbert spaces H are reflexive - and in the complex case the reader should be careful not to mix up the bilinear duality bracket (., . .) with the scalar product (.I -) (linear in the first variable, antilinear in the second variable) when identifying H with H‘.

-

-

A2. A separating dual system ( E ,F) is a pair of vector spaces E and F (over K) with a bilinear map :E x F K such that for each 0 # x E E there is a y E F such that (2, y) # 0 and for each 0 # y E F there is an z E E with (2, y) # 0. The topology on E of pointwise convergence with respect to F is denoted by a ( E , F) - and, by symmetry, a(F, E) on F : the weak topologies of the dual system. (E,u(E, F))’= F holds with the obvious identification. For A c E define the absolute polar A’ of A by (a,

- 7 )

I

A’ := {y E F I(a,y)I 5 1 for all a E A} and analoguously for B c F. Polars A’ are absolutely convex and a ( F ,E)-closed; the Hahn-Banach theorem implies that

holds for the bipolar Aoo of A (bipolar theorem). The most important example of a separating dual system is (ElE l ) if E is a normed space - the duality bracket being (t,d) := d ( z ) (this notation was already used in Al.). a ( E ,E‘) is called the weak topology on E and u(E’,E) is sometimes called the w e a k - t o p o l o g y on El; note that on E’ there is also the weak topology a(E’, E“). Though they are quite different (if dim E = co) the norm and the weak topology on a normed space E have two things in common: the convex closed sets and the bounded sets: (1) A convex subset A c E is norm-closed if and only if it is a ( E ,El)-closed. ( 2 ) A C E is norm-bounded (i.e. sup llAll < co) if and only if it is a ( E ,E’)-bounded (i.e. sup I(x’,A ) [ < 00 for all 2’ E E l ) . ( 3 ) Let E be a Banach space. A c El is norm-bounded if and only if it is u(E’, E ) bounded. ( 2 ) and ( 3 ) are refered to as the Mackey theorem or the uniform boundedness principle. It is important to know that the unit ball BE’ = I?$ of the dual space is always a( El, E)-compact (Banach-Alaoglu-Bourbaki theorem).

Appendiz A: Some Structural Properties of Banach Spaces

491

A3. Compact convex sets with respect to the norm or a weak topology - more generally: in a locally convex space - enjoy many interesting properties.

LEMMA OF KY FAN:Let K space and ff C {f : K

#8

-

[--00,

be a convex, compact subset of a topological vector

m[

I f is concave, upper semicontinuous }

f E ff there is an 0 f o r all f E ff.

a convex set such that f o r each

an

to E

K such that

f(to) 2

t

E K with f(x) 2 0. Then there is

See Pietsch [214], p.40, for a proof.

-

A4. A basis ( 2 , ) of an infinite dimensional Banach space E is a sequence of elements of E such that for each t E E there exists a unique sequence (a,) of scalars such anzn.The coefficient functionals ti : 2 a, are well-defined and that t = C,"=l = 6n,m. For an linear and one can show that they are continuous. Clearly, (z',,~,) orthonormal basis ( e n ) in a Hilbert space the coefficient functionals are e', := (. le,) and hence Ile,ll = llebll = 1. Results concerning the existence of bases and related questions are treated in section 5 . Clearly, finite dimensional Banach spaces E always Such a basis can even be chosen with have a finite algebraic basis (z,):=~.

- in this case, (tn)ZGis1called an Auerbach basis (see [177], I, l.c.3., for a proof).

-

A5. If the complex Banach space A also has a multiplication A x A A which makes A an algebra and satisfies 112 . yll 5 1 1 ~ 1 1llyll, then A is called a Banach algebra. Examples: C(E,E ) , l , , L m ( p ) , C ( K ) with their natural multiplications. If, in addition, A which is antilinear and satisfies for all t,y E A there is an "involution" * : A (a)''t = t , (b) (zy)' = y*t* , (c) 11tt*1)= 1 1 ~ 1 ,1 ~ then A is called a C*-algebra. Commutative C*-algebras with a unit for the multiplication (such as l m and L a @ ) ) are isomorphic (with norm and algebraic structure) to a space C ( K ) where K is compact (Celfand representation theorem). K is the weak-

-

*-compact subset of BA,consisting of all non-trivial, multiplicative, linear functionals on A.

A6. A real Banach lattice E is a real Banach space E with an order reflexive and antisymmetric) which satisfies (1) z 5 y implies t

5 (transitive,

+ z 5 y + z for all z E E, and Xz 5 Xy for all A 2 0.

(2) The minimum 2 A y exists for all t,y E E .

(3) 1.1 5 Iyl implies that

Ilzll 5 llyll

(where 1.1

:= z v (-2)).

492

Appendiz A: Some Structural Properties of Banach Spaces

A complet Banach lattice E is a complex Danach space which has a real linear subspace E, such that (1) E, @ iE, = E; the unique decomposition of t is written as z = Re(z) + i Im(z) and E, is called the real part of E. (2) E, has an order such that (with the induced norm) E, is a real Banach lattice. (3) 1.1 := sup{l Re(ei'z)l 0 5 t 5 2r) exists in E, for all z E E. (4) 1z11 I1.21 implies 11z111 5 1 1 ~ ~ for 1 1 all ZI,ZZE E. Basic references for Banach lattices: Schaefer [244], Schwarz [256], and Lacey [164]. A closed, linear subspace F of a (real or complex) Banach lattice E is called solid if t E F , y E E and IyI 5 1.1 implies y E F. In this case, the quotient E / F is a Banach lattice under the natural ordering: x ,< 0 if t 5 y (see Schaefer 12441, p.85 and p.137). The dual space El of a Banach lattice has a natural order given by d 2 0 whenever (z',t) 2 0 for all z 2 0. With this order El is a Banach lattice. The canonical embedding K E : E 4 El' preserves the order.

I

A7. The most common examples of Banach lattices are L, and C ( K ) . A (real or complex) Banach lattice is called an abstract Lp-space if for all t ,y E E with 2,y 2 0 and x A y = 0 the relation

IIZ + Yll' = 11~11' + IlYll' Ib + Yll = m=4ll4L IIYII)

(if l I p < o o ) (if p = cu)

holds. For p = 00 these Banach lattices are also called abstract M-spaces, for p = 1 abstract L-spaces. In this latter case the definition is equivalent to

for all t , y E E with c , y 2 0 (see Schwarz [256], p.98). It is obvious that a complex Banach lattice is an abstract L, if and only if its real part is. The following representations hold: THEOREM: (1) For 1 5 p < 00 every abstract Lp-space is linearly isomorphic (with norm and order) to some L p ( p ) ,where p is a Borel-Radon measure on a locally compact space. (2) Every abstract M-space with 4 unit (i.e. an element e E BE such that 111 5 e for all x E B E ) is linearly isomorphic (with norm and order) to some C ( K ) , where K as a compact space. For p = 1 and p = 00 these results are usually attributed to Kakutani, for 1 < p < oo to Bohnenblust and Nakano. Proofs can be found in Schaefer [244] (p.104, 114, 138 and 149) or Lacey [164] (p.59 and 135). In particular, these results show that a Banach lattice E is an abstract Lp-space if and only if its dual E' is an abstract L,'-space. For p = 1 even more is true:

Appendiz B: Integration Theory

493

THEOREM: Let E be a Banach space such that E' is isometric to an abstract M-space with unit. Then E is isometric to an abstract L-space, hence to an L l ( p ) for some Borel-Radon measure. This result is due to Grothendieck [91]; more recent references for a proof are Schaefer 12441, p.150 and Lacey [164], p.95-96 and 99.

Appendix B: Integration Theory

In this appendix a summary of the measure and integration theory which is used throughout the book is given. A n outline of the Daniell-Stone integration theory will be presented - the details can be round in many textbooks; we follow [71], where the reader will find all proofs of the following statements on real-valued integration. The necessary modifications for the complex case will be indicated. At the end of this appendix the basics on Bochner integration will be given.

Extension Procedure and the Dasic Theorems

€31. Let R be a non-empty set and C a vector lattice of functions R + R which is Stonean, i.e. whenever f E 2. min{l,f} E C A complez (or signed) Daniell junctional on C is a linear functional is u-continuous, i.e.: If fn E C and fn 1 0 pointwise, then Iimp(fn) = O

I.(

: 2 + (T which

.

If p is real-valued, it is called a real Daniell functional - and if it is positive (i.e. 0 5 f E C implies p(f) 2 0), it is simply called a Daniell functional. If p is a complex Daniell functional, then

IPl(f) := fJUP{IP(h)lI 0 5 lhl I f,h E 21 Icrl(f) := I/W+)- IPl(r-1

if05f E C iffEC

defines a Daniell functional: The total variation of p . Clearly,

I/l(f)lIIPl(lfl)

for all f E C.

494

Appendis

B: Integration Theory

If p is a real Daniell functional, then its positive and negative parts

are (positive) Daniell functionals and p = p+

- p-.

Examples: (1) If Q is a locally compact topological space and C := C,,(Q,R),the space of all real-valued continuous functions on Q with compact support, then, by Dini’s theorem, every positive linear p : C,,(Q,R) R is a-continuous. Moreover, if p : C,,(Q, R) -+ C is linear and of bounded variation, i.e. Ipl(f) < 00 (defined as before) for all f 2 0, then R,e(p)+, R e ( p ) - , Im(p)+ and Im(p)- are well-defined positive linear functionals and hcnce a-continuous; it follows that p is a-continuous as well. These p are sometimes callcd (signed) Radon measures (see the fundamental theorem B3. below). (2) Let C be a (non-empty) ring ofsubsets of 52 (i.e.: A U B , A n B , A\B E C whenever A, B E C). Then 6 ( C ) is the Stoncan vector lattice of all step functions

-

n m=l

where m E N ,am E R, and X A , is the characteristic function of A , E E.An additive function p : C C i s called a complex (or signed) measure if it is a-additive, i.e. -+

m=l

m=l

whenever An,U,OO,lAm E C and the A, are pairwise disjoint. If p is positive (i.e. p(C) C [O, 0 0 [ ) , then p is a-additive if and only if its “linearization” pL : 6(E) -+ C n m=l

is well-defined and a-continuous; this equivalence holds for arbitrary additive p if C is a local a-ring (i.e. a ring closed under countable intersections). The terms real measure and measure (:= positive measure) will be used.

B2. It is the aim of measure and integration theory to extend p to a larger class of functions which satisfies good convergence theorems. Let p : C + R be a (positive) Daniel1 functional. Then LL := {U : Q -+I - O O , ~ I] there exist fn E 13 : fn u pointwise} and p’ is well-defined. A function / : 0 -+ [-m, m] is called p-integrable (in symbols: f E & ( R , C , p ) =: 2 1 ( Q , p ) =: C , ( p ) ) if for every E > 0 there are U, -v E LL with v lf suand

p’(u - v )

5E .

Appendiz B: Integration Theory

495

In this case,

is the p-integral of f . A complex-valued function is p-integrable if Re( f ) and Im(f ) are p-integrable,

J f d p := J h ( f ) d p + i J Im(f)dp .

This integral is linear, monotone and a-continuous on & ( p ) (more precisely, on the Stonean vector lattice Ci ( p ) of real-valued integrable functions), hence a Daniel1 functional. The extension procedure applied to this functional p : 2{ + R gives & ( p ) again, nothing new. The convergence theorems: (1) BEPPOLEVI THEOREM (MONOTONECONVERGENCE): I f f n E & ( p ) is an increasing sequence such that s u p s fndp < 03, then lim,,+, fn is p-integrable and n+w lim

Jfndp =

J A fndp ~ ~. I

(2) FATOULEMMA: Lei fn E & ( p ) be non-negative functions with s u p s fndp Then the junction liminf,,, fn is p-integrable and the inequality

/

l i m 2 f fndp 5 lim inf n-w

J

< 03.

fndp

holds.

(3) LEBESGUEDOMINATED CONVERGENCE THEOREM: If f n , g E & ( p ) are such that Ifnl 5 g and f := lim,,,, fn exists poiniwise, ihen f i s p-integrable and

B3. A function f : R + [-oo,oo]is p-measurable (in symbols: f E M(p) or simply 9Jt)if for all 0 5 h E & ( p ) the function md(f,h,-h) := ma{-h,min{f,-h},min{f,h}} (the “middle” of f , h, and -h) is p-integrable. This definition (due to Stone) says that a function is measurable if its graph looks “locally” like the graph of an integrable function. Complex-valued functions f are p-measurable if Re(f),Im(f) E M. The trivial criterion f p-measurable,g E C l ( p ) ,If1

5g

TC

f p-integrable

Appendiz B: Integration Theory

496

gains its importance from the fact that M is very large: It is stable under the formation of pointwise limits and suprema of sequences and for f1 , ...,f,, E M the function

is also p-measurable whenever cp : [--00, mIn .-* [--00, co] is continuous (more generally: Borel measurable). M is 80 large that for the Lebesgue measure it is not possible without using the axiom of choice - to construct a non-measurable function. If C is a a-algebra on R (i.e. a local a-ring with R E C, hence closed under countable unions and intersections), then a function f : Q --* [-co,m] is called C-measurable whenever [f L 4 := 1. E R I f(.) 5 a) is in C for all LY E R. For a topological space R the smallest a-algebra which contains all open sets, is called the Borel a-algebra %(a);Borel measurable means %(a)measurable. If p is a Daniell functional, then the p-measurable sets Cp := { A C 0 I XA E m(P)}

form a a-algebra. It follows that f : $2 -+ [--00, m] is p-measurable if and only if it is C, -measurable. A function f : R [--00, m] is called p-null if f E &(p) and J Ifldp = 0; a set A is p-null if XA is p-null, and it follows that f is p-null if and only if the set [f # 01 is p-null. One says that two functions f and g are p-almost everywhere equal (= p-a.e.) if the set [f # g] is p-null. Note that C,, and the local a-ring -+

) 0 and B C A , of p-integrable sets are p-complete in the following sense: If ~ ( x A = then B E I,, and p ( x ~ )= 0. A subring Co of I,, is called p-dense if for every pintegrable set B there is an A E Co with ~ ( X A A E = ) 0. p is called finite if R is p-integrable and a-finite if R = UF=lA,, with A,, E I,,. If p is a Daniell functional on a Stonean vector lattice 2,then

c ( A ) :=

XAdp

defines a measure on the local a-ring I,,. Since a non-negative function f : Q is the pointwise limit of the increasing sequence (f,,)of functions

-+

[0, m]

Appendix

B: Integration Theory

497

the convergence theorems imply the

FUNDAMENTAL THEOREM OF ring Co c I , is p-dense, then

DANIELL-STONEINTEGRATION THEORY: If the

THE

= .el(% VO), 4

l:l(Ql

with equal integrals. This means that actually every Daniell functional comes from a measure on a certain ring of sets, e.g., the local a-ring of integrable sets. This is why we always simply speak of a measure space ($2, p ) or (0, C,, p ) if we want to draw attention to the pmeasurable sets. Note that C, is always p-complete and that p is only defined on I,, (and has finite values) and not on all p-measurable sets. For Radon measures (i.e. poHitive liricar functionals on Coo(f2))this theorem will have a particularly interesting consequenw (see B9.). For real Daniell functionalslreal rriawures p define 1:,(p) := WPI)= 1:dp+) n el(p-),

and in the complex case

(Obvious definitions for f : R

+ C ) . It

follows that

whenever f is p-integrable.

B4. If (Qi, Cqi, pi) are two measure spaces and 1:the Stonean vector lattice of all step functions on R1 x Rz, n

498

Appendix

B:

Integration Theory

defines a Daniel1 functionaI/rneasure on C.It is called the tensor product of p l and p 2 . If f :RI x R2 + C i s such that the function

is p1-almost everywhere defined and p1-integrable, one says that the iterated integral exists.

J J fdpadpl

FUBINI THEOREM: Iff E El (p1 8 pz), then Both iterated integrals erist and

Conversely,

FUBINI-TONELLI THEOREM: I f / : S 2 , xRz 3 C (or -+ [-a, w]) i s p1@p2-measurable, [f # 01 c S21 x R2 is a-finite (wilh respect t o p1 8 p z ) and one of the iterated integrals of If1 erists, then f is p1 8 pa -inlcgrable. None of the assumptions in this last result is indispensable. (A set A c R is called a-finite with respect to p if there arc An E I,, with A c An.)

u:=l

The Lp-Spaces

B5. Let (S2,p) be a measure space and 0 < p < 00. A function f : 0 -+ K ( = R or a;3 is called pintegrable if it is p-measurable and l f l P is p-integrable; in symbols: f E tP(Q p ) =: t p ( p ) . Since every integrable function 52 -, [-00, 031 is almost everywhere equal to a real-valued function, for p = 1 this slight difference in the definition will cause no confusion. 2,(p) is the space of p-measurable functions f : R K which are uniformly bounded outside a localp-null set A, (i.e. a set A such that p ( A n B ) = 0 for all p-integrable B ; in general there exist local p-null sets which are not p-null). Define, for f E Cp(p),

-

llflloa

:= inf( r 2 0

I [If1

> r] local p - null}

-the latter is called the essential siipremum of the function f and is sometimes denoted by ess-sup lfl. The functions in f!,(/r) are called p-essentially bounded.

HOLDERINEQUALITY:I f p , q E [l,oo]satisfy l/p X&), then f ‘ 9 E & ( p ) and

+ l/q = 1

g E

It/ . Ylll I

.

llfllPll~ll~

and

f E

.Cp(p) and

499

Appendiz B: Integration Theory

This allows one to show that 11 -lip is a seminorm on .Cp(p)for 1 I p _< LP(QII1) := L,(P) := .cp(P)

/ {f I llfll,

00,

and therefore

= 01

(or briefly: L,,) with 11 - 1, is a norrried space. The convergence theorems show that L, is a Banach space - even a Bariach lattice with its natural ordering. Note that Ilfll, = 0 is the same as f being a p-null function (for p < m) or a local p-null function (if p = 00). The fact that the elements of Lp are classes of functions should not be forgotten: If necessary the notation f or j'" for the class containing f will be used. For 1 5 p < 00 the space L, is an order complete Banach lattice (but for p = 00 in general not: see B6. below), even more: every norm-bounded increasing net in L, has a supremum. For p < 00 the triangle inequality rcads as follows

The sum can be replaced by an integral - this means: CONTINUOUS TRIANGLE (OR MINKOWSKI) INEQUALITY: Let (nilpi) be measure spaces, 1 p < 00 and f : sZ1 x sZz -+ [-00, 001 (or a' )PI @ pz-measurable such that [f # 01 is a-finite. If an the following inequality the right iterated integral exists, then the left one does also exid and

<

holds.

+

B6. For l/p l / q = 1 Holder's iricquality gives a natural separating duality bracket between L,(Q p ) and L,(R, p ) : whenever (replace

-

by

yoc if p

f E L,, 5 E L ,

or q = oo),hence an embedding

Since

-

holds, the unit ball of L, is norrning for the norm 11 llP of the Banach space L,; in particular, @ is an isometry. For 1 < p < 00 this isometry is even onto

L,U& P)' = L,(Q, P )

*

500

Appendix B: Integmtion Theory

This follows from the Radon-Nikodim theorem below. In particular, all Lp are reflexive (1 < p < m); this is not true for p = 1 or 00 if L1 is infinite dimensional. Moreover, the dual of L , is strictly larger than L1 if L1 is infinite dimensional. The Radon-Nikodim theorem is the appropriate tool for determining the conditions under which Li = L, holds: For this, take E a Stonean vector lattice of bounded functions (this is not really a restriction by the fundamental theorem B3.) and p and v (positive) Daniell functionals = measures on C. The measure v is called p-absolutely continuous if every v-measurable p-null set is a v-null set. v is called p-singular if there exists a u-measurable p-null set N such that

for all f E C (and then for all f E C,(v)).If go 2 0 is a locally p-integrable function (i.e. f . g o E C l ( p ) for all f E C),I h n

W ):= Jfm+

for all f E C

is p-absolutely continuous; nuch a go is called a p-density of v.

SEGAL’S LOCALIZATION THEOREM: Let C be a Stonean vector lattice of bounded functions on a set 2 ! and p a Daniell junctional on C. The following statements are equivalent: (a) L,(p) is order complete. (b) Ll(p)’ = L,(p) isometrically. (c) Every Daniell junctional u 5 p on C (i.e. v(f) 5 p(f) if 0 5 f E C ) has a p-density. (a) The RADON-NIKOD~M THEOREM holds for p : Every p-absolutely continuous Daniell functional u on E has a 14 - density. The main part of the proof iH to h o w that the Radon-Nikodim theorem holds for finite measures - the rest is to pius in L , from the increasing net (gp)coming from the p-integrable sets A to all of R; this is why measures with L , ( p ) being order complete are called localizable. c fitiibe and Borel-Radon measures (see B9. below) are localizable, but there are noii - locirlimble measures. IR proving this theorem one A ~ N OgdtH

LEBESGUE’SDECOMPOSITION TIIWREM Let: p and v be Daniel1 functionals on a Stonean vector lattice of bounded junctions and let Iu(f)l 5 c i f f E C and If I 5 1. Then there is a unique decomposition (v,,,vd) of Daniell functionals on E such that v = v,

+ v,

Appendix B: Integration Theory

501

where u,, is p-absolutely continuous and u., is p-singular.

B7. When is a measure localizable? A measure space (Q, C,, p ) is called strictly localizable if it admits a p-decomposition 9, i.e. a set of pairwise disjoint p-integrable sets K such that for every p-integrable set A of positive measure there is a K E 9 with p ( A n K) > 0. It follows that a\ U 3 is a local p-nu11 set and a function f is p) p-measurable if (and only if) all functions fXK with K E 9 are p-measurable: (0, is decomposed into the family (K, ~ I K ) of finite measure spaces. It is easy to see that every strictly localizable measure is localizable.

-

Maharam's theorem says that every finite measure p admits a linear, multiplicative, positive mapping Am : Ih$-4p) .e,(Q,p)

, -+ L, (i.e. X,(f") c f"") and which is a right inverse of the quoticnt mapping 2 such that X,(ifoc)= 1. Such a Am is called a (multiplicative) lifting. For p < 00 there are no liftings (even without mulliplicativity) for the Lebesgue measure. It is obvious that the existence of liftings f'acilit,fites the analysis of a measure enormously. It can be shown that a measure space (fl,/r) is strictly localizable if and only if it admits a lifting. Localizability in some sense gives thc possibility of decomposing L , into finite measures, strict localizability givm ttrc possibility of decomposing 2,. Since integrable functions vanish p-a.e. outsidc a (z finite set, it is always possible to decompose the Banach space L l ( p ) : REMARK:For every measure space ( 0 , p ) there is a family ( 0 , , p , ) of finite measure ( I E I ) such that h ( p ) = 4(l;Ll(p'))

spaces

holds isometrically.

For the proof take a maximal family 3 = {Q,li E I} of p-integrable sets such that i 1 # ~2 (Zorn's lemma), observe that for every p-integrable A (and then also for every u-finite A ) the set of indices p(flbl n $Ih2) = 0 if

{L

E I I p ( A n R , ) > 0)

is countable and that

A=

u{

A flL2,

I p ( A fl Q , ) > 0)

p-a.e. .

This clearly shows that the linear tilapping

LIW*P)

f

-

4(1;&(0&9PIn'))

(fXn,)'€r

502

Appendix B: Integration Theory

is isometric - and onto anyhow.

Borel-Radon Measures and the Riesz Representation Theorem BS. Assume now that the (positive) Daniell functional p on the Stonean vector lattice 2 is not only u-continuous but even s-continuous: This means that for every pointwise decreasing net fr 1 0 lirnp(f7) = 0 . 7

One can now follow the same extension procedure as in the u-case - just replace increasing sequences in the definition of LI by increasing nets. This way one obtains L1‘ and (with the same definitions as bctfore) the spaces C l ( p ‘ ) and m(,ur)of pr-integrable and pr-measurable functions ( “Hourhaki extension procedure”). The integral p’ is ucontinuous on 2 1( p ‘ ) but not in general r-continuous. The Daniell extension procedure applied to p r on 2 { ( p r ) (finite valircd functions) gives nothing new. Clearly, & ( p ) c & ( p r ) , with

Moreover, for every f E where; in particular,

.Cl(p’)

thcw is an tP(P)

E & ( p ) with f

fo

= fo p’-almost every-

= tP(PT)

holds isometrically for 1 5 p < 00. Since L m ( p T ) is always order complete (see the following theorem), one can show that L,(,u) # L,(p‘) may occur.

K ~ L Z O W THEOREM: ’S p r is strictly localizable. It follows that L , ( p r ) admits a lifting, Ll(p’)’ = L , ( p T ) , the space Z,,(pT) is order complete and the Radon-Nikody’m theorem holds for p’ (and pr-abaolutely continuous measures v on IPr).

B9. Certainly the most important, examples of s-continuous Daniell functionals are Radon measures, i.e. positive linttar functionals IL :

-

c:,,(Q, R)

R

(Q locally compact) - a connoqiirnw of Dini’s theorem. In this case, lower semicontinuous, bounded from below, [u < 01 relatively compact It follows that all Bore1 sets are p r measurable and the measure

503

Appendiz B: Integration Theory

on the local a-ring %,(SZ)

of all relatively compact Borel sets is regular, i.e.

v(A) = sup{v(K) I K c A compact) = inf{v(V)

1 V 3 A open, v-integrable}

,

Measures on %,(n) are called Borel measures and hence Y is a regular Borel measure. Regularity and the fundamental thcorem (B 3.) imply that C,(p') = &(v) (with equal integrals): There is a one-to-one correspondence between positive linear functionals on Co,(52,R)and regular Borel measures on %,(a).This is why these functionals/regular Borel measurea are labelled Borel- Radon measures. If p : Coo(Q, K) ---* K is K-linear and of bounded variation, i.e.

I

Ipl(f) := sup{Ip(h)l 0 Ilhl If, h E C O O ( Q ,

W }<

for every 0 5 f E Coo(n, R), one obtains a Borel-Radon measure 191 : Coo(Q, R) 4 R from the definition IPl(f) := lpl(f+)- Ipl(f-) (and obvious extension to Co,(Cl,6') if K = 6').Clearly, if 0 5 f E Co,(S2,R). Considering p = (Rep)+-(Rep)- +i(Imp)+ - i(Imp)-, the Radon-Nikodim theorem (applied to (Rep)+ 5 lpl and so on) gives a locally 191'-integrable function go such that r for all f E Coo(n, E l ) (the polar decomposition of 9). Clearly, p defined by d p := godlplT is a signed Borel measure which i R rcgular (i.e. 1p1 is regular).

Itr~sz'REPRESENTATION THEOREM: rfC2 zs locally compact and 9 : C,,(n, K )+ IK is K - l i n e a r and of bounded variulion, then there is a unique signed regular Borel measure p on C l such that

for all f E Coo(Q,I") .

Conversely, every signed Borel measure p defines such a p - but note that there are non-regular Borel measures. For Banach space theory the case of compact SZ is the most important: If p is a regular signed Bore1 measure and 'p its associated linear functional

(cp,f)= on the Banach space

Jn f d p

C(n,K ) ,then IIPII II/ll(Q) = IpW) = IIPII

504

Appendix B: Integration Theory

(by the definition of Ipl). Denoting by M ( R , K ) the space of all K-valued regular Bore1 measures p with norm 11p11 := lpl(Q), one obtains the

COROLLARY: If 0 is compact, then C(Q,K)’ = M ( Q , K ) holds isometrically. B10. The following result is quite important for the theory of Banach spaces:

PROPOSITION: If ( Q , p ) is a measure Bpace, then L l ( p ) is I-complemented in its bidual. PROOF: Since L1 ( p ) is an abstract Lspace, Kakutani’s representation theorem (Appendix A7.) shows that p may be twumed to be a Borel-Radon measure on a locally compact Q, hence L1(p)’ = Lm(p). ‘Cake cp E L l ( p ) ” = Lm(p)’ of norm 1; the functional fE P 4 f ) := klf)

w,>

has, by definition, a p-absolutcly conhuous total variation [pol: The Radon-Nikodim theorem (for p applied to (Repo)+ ctc.) gives a locally p-integrable function go such that

cp”(f)= for all f E G(I,). Since

J fsodcr

/

I f s o 4 4 s llfllL&)

9

1. Clearly, the class go representing it follows that go is p-integrable and Ilioll~l(r,,l yo is unique, and hence Pp := io(Jcfines a projection

-

1’ : f&)”

of norm

L1(p)

5 1 .0

Bochner Integration B11. If ( Q , p ) is a measure spam and E a (real or complex) normed space, then a function f : S l --+ E is called a p---.ulcpfunction if it has a representation

with

E E and p-integrable Ak c $2; in this case, the integral n

Appendix B:

Integration Theory

505

is well-defined. f : R -+ E is called p-measurable (or: strongly p-measurable) if for every p-integrable A c 52 there i R a sequence (f,,) of p-step functions such that

for p-almost all w E A. It is easy to see that on every u-finite set f : 52 E is the a.e.-limit of a sequence of step functions. The following criterion is of fundamental importance: -+

THEOREM: Let (R,p) be a measure space, E a Banach space and F C E’ a subspace such that F n BE’ is norming f o r E. For each function f : R -+ E which vanishes outside a u-finite set the following statements are equivalent: (a) f is p-measurable. (b) f is p-essentially separably valued (i.e. there is a separable subspace E, C E with f ( w ) E E, p-a.e.) and Borel measrrable ( i e . f - ’ ( B ) is p-measurable f o r each Borel set B C E ) . (c) f is p-essentially separably valocd and u(E,F)-measurable (with respect to p ) , i.e. (d, f(.)) is p-measurable for all 2’ E F. (d) For every E > 0 there ezinta a 14 measurable, countably valued function f t : R -+ E E p-a.e. with /Ifr(.) - f(-)II The equivalence of (a) and (c) is called the Pettis’ measurability criterion. u ( E ,E‘)measurable functions are labelled weakly measurable - and u(E’,E)-measurable functions R -+ E’ weak+-measurable. There are many references for this theorem: First observe that it is obviously enough to prove it for finite p ; Dunford-Schwartz [59], I11 6.9. and 6.11. show the first three equivalences (for F = E’, but the proof is easily adapted to F c E’ as stated). For Pettis’ criterion see also Yosida [277], V.4., and Diestel-Uhl [55], p.42; the proof in this last reference even yields (d).

B12. For 1 5 p < 00 the p-measurable functions f : R -+ E such that the function 11f ( . ) I & is p-integrable, form a vector space zP(plE). Its quotient with respect to the p-null-functions (i.e. 1 1 f ( - ) l l ~ = 0 p-a.e.)

(or briefly: L,(E)) equipped wihh

is a normed space - even coniplctc! if E is a Banach space. It can be seen (the proof of Yosida [277], V.5, for p = 1 can cwily be modified) that f : R E is in & ( p , E) -+

506

Appendiz

if and only if there is a sequence p-integrable and

B: Integmtion Theory

(I,,)of p-step

functions such that 11fn(-)

- f(.)IIg

is

r

In particular, the (equivalence classes of) step functions are norm-dense in L p ( E ) .For p = 1 the integral

is well-defined. The functions in E1(p,E) are called Bochner integrable, those in .Cp(,u,E) Bochner p-integrable. As an immediate consequence of the scalar case one obtains Lebesgue's dominated convergence theorem: If (fn) is a sequence of p-measurable functions R + E which converges pointwise to a function f : R + E and if there is a g E & ( p ) such that 11fn(w)II 5 g(w) for all w E R, then f is Bochner integrable and

For p = 00 define E,(p, E) to be the space of all p-essentially bounded, p-measurable functions R -+ E and the normed space

L,(,uI E ) := C,(p, E ) p C

llfloclloO

with :=ess-supllf(.)llE; this space is complete if E is. Take p , q E [1,00] such that l / p + l / g = 1 . I f f E Cp(p, E) and g E .C,(p, El), then the function

(f(*),g ( ' ) ) E , E ' is p-integrable,

II(f,g)lll 5 Ilfllpllgll, by Holder's : h,(Pl E')

5

-

inequality and the mapping

(Lp(P,E)Y [i--/(f(w),

dW))P(dw)l

is even an isometry (into); to prove this, first use step functions. The question of under what conditions ip is surjective will be treated in D4..

B13. It is obvious that for T E E(E, F) and f E &(p, E) the function T 0 f : R is also Bochner integrable and

In particular, this implies that

-+

F

507

Appendix 8: Integration Theory

for all f E & ( p , E ) .

B14. I f f : R + E is such that ( d ,I(.))is p-integrable for all z’E E’, then for each p-measurable A C R the funcbionltl ,

ia linear, hence in El*. If this functional (for all such A) can be represented by an element X A E E Q E’*, then f is called o ( E ,E’)-integrable or P e t i i s integrable and

=n =:

J,f (w)Cl(dw)

is called the Pettis integral (or weak- or o ( E ,&‘‘)-integral) of f. Bochner integrable functions are Pettis integrable and the Bochner integral of f X A is I A . For a separating duality system (E,F) it is obvious how to define the notion of o ( E ,F)integrability. In particular, it is clear what weak-*-integrable means.

B15. The unit ball of L m ( p ) is norming for L1(p). Even though L l ( p , E ) ’ is, in general, much larger than & ( p , E’), this is also true in the vector valued case: If Co c I,, is a p-dense ring, then the F:’-valued CO-step functions are dense in L p ( p ,E‘) (for 1 5 p < m). Since Lq(P,E ) C

L L A P , E”) C

L (Lp(cI, E”’

(l/p+ l / q = l), this implies that tlw set of all E’-valued Co-step functions in the unit ball of L,(p, E’) is norming for L q ( p ,E) - and so is the whole unit ball. For finite measures p this argument applim also for q = 1. The following result shows that this is true for arbitrary measure8 and gives another, more convenient formulation of this fact:

VARIATION LEMMA:Let ( R , p ) be i s normed ond

n measure space and Co c

I,, u p-dense ring. If E

f E & ( p , E ) , then

-

The right side might be denoted by V ( f d p ) - the variation of the vector measure A .f,A f d p *

PROOF:Clearly, V ( f d p )_<

Il/llt,(~ Conversely, ).

first take a Co-step function

508

Appendix C: Representable Operators

with pairwise disjoint Am E Co. Thcn n

V(fOd~1)I I I ~ ~ I I L ~ ( = E)

C

AAm)llzmll

I~

( f o d .~ )

m=l

If

IIf

f

E C i ( p , E ) is arbitrary and

E

> 0, then there is a CO-step function

fo

with

- fol(tl(q I E . It follows that

and, for a representation of j o as hefore,

Appendix C: Representable Operators

This appendix deals with operators L1(p) 4 E (for finite measures p ) which have a representation Tf = J’ fgdp for some bounded Bochner integrable function g E L l ( p , E). The strong version of the Dunford-Pettis theorem will be proved. Since we use some results about the projeclive tensor norm A this is in some sense an appendix to Chapter I. In contrast to Appendices A and B in this appendix complete proofs will be given.

Grot hendieck’s Charactarizat ion

C1. Throughout this appendix thn measure p will be finite. If E is a (real or complex) Banach space, an operator ‘I’ E C(L t ( i 2 , p ) , E) is called Riesz representable (briefly: representable) if there is a 11 -essc!rilially bounded, p-measurable function g : R + E such that

Appendix C: Representable Operators

509

for all f E L l ( i - 2 , ~ )That . g is p-essentially bounded means (see B12.) that

- measurability implies that g is Bochner integrable. The function g is called a Rieszdensityof T - and it is clear that g iR necessarily a weak density, i.e. weakly measurable and (TlI !J) =

J

f ( W ) ( S ( W ) ,b ) l . r ( d w )

for all f E L l ( i - 2 , ~ and ) y' E E l . For E = K every operator = functional is representable by the Radon-Nikodim theorem and the density is the Radon-Nikodfm density; the study of reprmntable operators will serve to understand vector valued versions of the Radon-Nikodjrn thorem.

PROPOSITION: Let T E Z(Ll(Q,p),E ) for a finite measure p and a Banach space E . (1) If T is representable, then the range of T is separable. (2) If g is a Riest-density of T , then

In particular, the Riesz-density of a representable operator is unique p-a.e.. ( 3 ) If g E L l ( p , E ) and Co c I,, a /'-dense algebra (see Appendix B3.) such that

for all A E Co, then g is p-esnenlially bounded and a Riesz-density for T. (4) If p is purely atomic, their T in always representable. ( 5 ) If T is representable with denaily g and S E 2 ( E ,F ) , then S o T is representable with density S o g. ( 6 ) If g represents T,then g(w) E irn(T) p-a.e..

PROOF: Since the measure p is finile, every p-measurable function g has p-a.e. values in a separable closed subspace E,, of E, hence (1) follows. To see (2) and (3) choose a$ E BE' such that {zk} is norming for E,. The hypothesis of (3) gives

510

Appendix C: Representable Operators

and this implies that [(zk,g(w))l5 IlTll for p-almost all w E R. Since {zk} is norming, llgllm 5 IITII. It follows that the operator

is well-defined, continuous and coincides with T on the dense subspace of Co-step functions: T = To which proves (3). Clearly, IlTll 5 Ilglla, - and (2) follows as well. ' 7 = 0. That g is unique a.e. is a consequence of 1191 - gzllm = [IT 1

-

If p is purely atomic, then there are pairwise disjoint atoms An

c R with

m

n= 1

It is easily seen that 00

is a density. ( 5 ) is obvious, and (6) follows from the observation that

is representable and has the p-a.e. unique density

Q o g = 0 by (2) and

(5). 0

The interpretation of T being representable as a sort of Radon-Nikodim theorem (and the density as a sort of derivative) is supported by the following observation: If C is a finite partition (Al, ...,An) of R by /I-integrable sets and

then the following holds:

COROLLARY: T : L l ( p ) -, E as representable if and only if the net ( g z ) e is a Cavchy net in L 1 ( p , E). It is enough to take the partitions in a p-dense subalgebra Co of I,,.

PROOF: If (gz) converges in L1(pL,E) to g, then

511

Appendix C: Representable Operators

for all A E Co and property (3) of the proposition shows that g is a density for T. Conversely, if g is a density, then

the conditional expectation of g with respect to C. An easy calculation shows that for all h E L I ( p , E )

IIE(~lC)llhIllNlL.1

-

,

and it is immediate that E(hlC) 11 in L1(p, E) if h is a Co-step function: These two facts show that the net (E(h1C))e converges to h in L1(p, E) for all h E & ( p , E). This ends the proof. 0 C2. It i s clear that every nuclear operator T : L l ( p ) -+ E is representable: If

n=l

n=l

-

and gn E Ll(p)' = L , ( p ) , thcin g ( w ) := gn(w)yn defines a density. On the other hand, consider tlie opcrator T : L1[0,1] c, of Fourier coefficients: 00

If g were a density of T or of I Z ~ ,0'1' : L1 necessarily follow that g(2)

L,

then (looking at components) it would

= (exp(%int)), E ern

a.e.

.

But this function is not a.e. separably valued. So neither T : L1[0,1]-, c, nor tee, o T : L1[0,1]-+ Loo is a representable operator; another argument is: g ( t ) 4 c, as it should be almost everywhere as a density by C1.(6). Note that c p is isomorphic to cf. C 3 . The operator T E C ( L l ( p ) ,E ) having a density

means that T corresponds to a ccrbain nuclear operator L,(,u) the following result holds:

-+

E. More precisely,

THEOREM (Grothendieck): Lcl (i2,p) be a finite measure space, E a Banach space and T E C ( L I ( p ) ,E ) . Then the frdlowiiig statements are equivalent:

512

Appendiz C: Representable Operators

-

(a) T i s Riear representable.

T

I

L1(p) E is nuclear. (b) The opemtor T o I : L , ( p ) If p is a Borel-Radon measure on a compact space 0, then these statements are equivalent to:

- T

I

(c) The operator To I : C(Q) L1(p) I n both cases: If g is a Riesz-density of T , then

where I is the canonical mapping I,,

E is nuclear.

- L1 or C

L1.

PROOF:Clearly, (b) implien (c). For the converse denote by Z I , : C(Q)--* L, the canoniciil mappings and by

DL : C(S2)'

-

Ll(Q,/.) ;

: L,

4

L1 and

:= V,

the mapping which assigns to cacti signed measure v E C(Q)' its /.-absolutely continuous part va by Lebesgue's (It-cornposition theorem (see B6.); clearly llDLll 5 1. Moreover, observe that I : L , 4 I , , is a( L, l L1) - a(L1, L,)-continuous and I' = t c o ~I. ~ It follows that

DL o Zi o ICL, = idLl . If T o I o I , = C,"=lvn&xn is a nuclear expansion with E C(Q)'and tnE El then the above decomposition of ( T o I)' gives

since v,,

n=l

which implies that T OI = C:=p=I D',(vn)&tn

and

hence N(T o I ) 5 N(T o I o l o ) - - and even equality since the reverse inequality is obvious. It remains to show that (a) iti equivalent to (b) (and the statement about the norm). If g = C,"=lg n B Z n E Ll& Is' = I,I ( p , E) represents p , then

513

Appendiz C: Representable Operators

and hence N(T o I) 5 i n f C z = l llgnII llznll = J 1 1 g 1 1 ~ d p by 3.3.and 3.5.. Conversely, the operator T o I : Lm + E is c(L,, L1) - c ( E ,El)-continuous: so if it is nuclear, then it can be represented as m

z:=,

+

with gn E L1 such that IlgnllL,llznll 5 N ( T o I)(1 E ) by Ex 3.34. since L1 is gnBZn E & ( p , E) 1-complemented in its bidual (see 1310.). The function g = satisfiea for all A E I, , [XA = gdp I

c,”==,

,

hence it is a density for T by propHition C1.(3). Since

As an application, take a finite measure p which is not purely atomic. Then the map Lm(p) 4 L l ( p ) is not compact (simulate Rademacher functions as on [0, l]),hence, by (b) of the theorem, idLl is not representable. C4. Another example of representable operators:

PROPOSITION: Every operator T : I , * ( p )-+

el

i s representable ( p a finite measure).

PROOF: Since Ll(p)’ = Lm(/i),it i‘ollows that there are g n E L , ( p ) such that for all f E L I

,

where en = (&,m)m as usual. It rcniains to show that g := C:=l gn&en is a density. If PN is the N-th expansion operiitor of the unit vector basis and g N := P N g = (91, . . . , g ~ 0, , ...), then IlgN(w)ll 5 ~ ~ ! ~ N +for l ( all w )20.~ ~Since, by the variation lemma (see B15.),

k=l

514

Aplwndix C: Representable Operators

Beppo Levi's monotone convergence theorem implies that 11g( -)I1 = lim,v+oo llgN(-)ll is integrable, and therefore g i H I3ochrier integrable. Since

by Lebesgue's dominated convergence theorem, proposition C1.(3) implies that g is a density for T. The following result, due to Lewis and Stegall [170], says that the operators are typical representable operators.

L1 -+.

el

THEOREM: T : L l ( p ) -+ E i s representable if and only if it admits a factorization T = S o R through A!,. In this casr, l'Z = inf llRll llSll over all such factorizations. PROOF:One direction is obvious hy the proposition above. Take T with a density g and recall that 11g11m = IlTll. Then, by the characterization B11. of measurability, there are gn : R + E countably V H I I I C ~ and measurable such that g = C,"=lgn pointwise,

If gn =

m

X A ~ , *@3 Zk,n wit>hpnirwise disjoint Ah,,, (for fixed a),then

define a factorization as desired. Cl

The Dunford-Pettis Theorem

-

C5. It was shown in 3.3. that every operator T : L l ( Q , p ) + F' has a weak-*-density g :R

F', i.e. (1) (g(+), y) is measurable for cvery y E F and

(2) ( T f ,Y) = J f ( 4 ( s ( w ) , y)/4duj) for a11 f E Lib) and 31 E F(weak version of the Dunford -1'ebbin theorem); moreover, g ( w ) E (TBL,)" and hence, in particular, Ilg(w)IIp I 117'11 for all w E R. If p is finite and g a.e. separalbly valucd, then g is p-measurable (by the Pettis criterion, B11.) and hence Bochner ittLegrahle. This easily implies (see corollary 3.3.) the following result:

515

Appendu C: Representable Operators

PROPOSITION: Let p be a finite measure. (1) If F’ is separable, then e u e y operator L1(p) -+ F‘ is representable. ( 2 ) Every weakly compact operator L l ( p ) + E with separable range is representable. This result has some nice structural consequences: There is an operator L1[0,11 + co which is not representable (see C2.),hence the separable space co cannot be isomorphic to a dual space. In the same way, Cdfand’s famous result that L1(0,1] is not isomorphic to a dual space follows. C6. To get the full Dunfortl-Petlin theorem, two more results are needed which are both of interest in their own right.

LEMMA:If T E c ( L ~ ( pE) ,) :a reprmentable and D C L1(p) relatively weakly compact, then T D is relatively comparl. PROOF: (a) If g is a density for TIthere are step functions gn which converge almost everywhere to g and satisfy llgnIlm 5 llglloo = llTll. For each measurable A c R define TAf

:=

fgdp

and

Tn,Af :=

fgndp

-

Then the T n ,are ~ finite rank operators and hence compact. For each 6 > 0 there is, by g uniformly Egoroff’s theorem, a measurable A c SZ with p(SZ\A) 5 6 such that gn on A. It follows that

and hence TA is compact for this A C a. (b) Every weakly compact Riil)mt / I c L1(p) is uniformly integrable (see for example [70],p.108), which means: For ewry E > 0 there is a 6 > 0 such that for all f E D

whenever p ( B ) 5 6. Choose, for Then

R

given

E

> 0, such

a number 6

5 E and A

as in (a).

for all f E D,which easily irriplien that the set TD is precompact, hence relatively compact. 0

Note that (for finite measurm p ) the relatively weakly compact subsets D c L1 ( p ) are exactly those which are norm- bounded and uniformly integrable (e.g.[70], p.108).

516

Appendix C: Representable Operators

PROPOSITION: Let fll be an operafor ideal, p a finite measure and F a Banach space. If every T E %(L,(p),F) wilh separable range is representable, then every operator T E fll(Ll(p),F ) has sepamble range and therefore is representable. PROOF:Take T E % ( L l ( p ) ,F ) . To show that i m T is separable it is enough to prove E I,,}is relatively sequentially compact. Take Bn E I,, and let Co be the that (TXB(B a-algebra generated by {&In E OV}. The Co-measurable functions in L1(p) form a separable subspace and the operator To : L l ( p ) --c F defined by

(the conditional expectation) is in 24, has separable range and is hence representable. Since the unit ball of L , is weakly compact in L1, it follows from the preceding lemma that the sequence (To x ~ ,,= T X B , ) n E N has a convergent subsequence. 0 C7. This proposition implie8 the

-

THEOREM OF DUNFORD-PETTIS (strong version): Let /r be a finite measure and F a Banach space. Then every weakly cornpact operator T : L1(p) F as Riest representable. PROOF: Apply proposition C5.(2)

10 fll := (10 in

proposition C6.. 0

Note that the weak version implies that the Riesz-density g of T takes its values in TBL, p-almost everywhere. Note also that representable operators need not be weakly compact: Take a quotient map L I[0, I] A e l . The lemma in C6. implies

COROLLARY 1: If T : L1(p) -+ F is a weakly compact operator and D C L , ( p ) is a weakly compact subset, then 7'D is compact. This property of L1(p) is called the Dunford-Pettis property. If K is compact, then C ( K ) has the Dunford-Pettis property as well; for more information about this property see Diestel's survey [53].

-

COROLLARY 2: (1) If T : L1(p) L l ( p ) is weakly compact, then T2is compact. (2) L1 ( p ) has no rejletive, infinite dimensional, complemented subspaces. It follows that the closed span Rad of the Rademacher functions (see 8.6.) is not complemented in L1[0,1]; set-! ~ I H O1.5.. C8. Finally, as an application, a result (also due to Grothendieck) about integral and nuclear operators:

Appendix D: The Radon-Nikody'm Property

517

THEOREM: If T E C ( E ,F ) i a inlegral and U E L(F, G ) weakly compact, then U o T is nuclear and N ( U o T ) 5 IlUll I(T). In short: ID o 3 c !? and I N 5 11 11 o I . PROOF:U o T admits a factorization

(see 10.5.). Since U" o S is weakly compact, it is representable by the Dunford-Pettis theorem, and therefore U" o S o I is nuclear by Grothendieck's characterization C3.. It follows that U o T = U" o S o I o R is nuclear and

by the estimate given in C3.

COROLLARY: If U E C ( E ,F ) is weakly compact and T E 2 ( F , G ) integral, then U'oT' i s nuclear and N(U' o T') 5 I ( T ) ~ ~ In U ~particular, ~. KG o T o U is nuclear; if E' has the approximation property, lhen 7'0 U is nuclear and N ( T o U ) 5 I(T)IIUII. The last statement follows from 6.9..

In the following appendix, which deals with the Radon-Nikodim property, a more systematic approach to this method of argumentation will be presented in order to obtain more information concerning the relationship between nuclear and integral operators.

Appendix D: The Radon-Nikodgm Property

The study of those Banach spaces which allow a certain vector valued RadonNikodjm theorem for vector measures to hold goes back to the work of Dunford, Pettis and Phillips in the late thirties and to Grothendieck's "propriCtk de Phillips" in his thesis [92], p.104. Nowadays, the standard reference is the book of Diestel-Uhl [55] (and its appendix [56]).Their point, of vicw is quite measure theoretic. Since for our purposes the operator theoretic a ~ p e c t arc , ~ more important, we orient this short introduction

518

Appendix D: The Radon-Nikody'm Property

in this direction. This appendix is based on the results of Appendix C and contains proofs of all the results which are given.

Basic Properties and Exarnplc?~

D1. A (real or complex) Banach space E is said to have the Radon-Nikodim properly if for each finite measure p every operator T E I!(Ll(p), E) is representable. Since every operator T : L1(p) -* E defines a (a-additive) vector measure M on the set I,, of p-integrable sets by M ( A ) := T ( X A ) > it is not difficult to see that E h w the Radon-Nikodfm property if and only if (for every finite p ) E-valued measures of a certain type (p-absolutely continuous and of bounded variation) have a density with respect to p; this is shown in Diestel-Uhl [55], p.63 - we do not need this result. In other words, the above definition is equivalent to the one given in Diestel-Uhl's book. The situation is the same as in the scalar case (see Appendix B6.) where the scalar Radon-Nikodfm theorem (for p ) is equivalent to the statement that Ll(p)' = fJhJ(p): this operator theoretic aspect of the RadonNikodim theorem will be iniportarit for us in the vector valued case; see D4. for more information about this point. It is not necessary to check for all finite measures p in order to know that a given Banach space has the Radon Nikocljm property.

THEOREM: A Banach space E has the Radon-Nikody'm properly if and only if e v e r y operator T : L1[0,1] 4 E (Lebesgue measure) i s representable. The following proof of this well-known result is essentially taken from Botelho [13].

PROOF:Clearly, the condition is necessary. Conversely, take an arbitrary finite measure p on Q and decompose it into its continuous and purely atomic parts ( Q c , p c )and (Q5,pa) : L l ( Q , p) = Ll(Qc,C(C) @ Ll(flZaip4) .

By proposition C1.(4) each operator T E .C(L,(p), E) has a density on the atomic part one may assume p to have no atoms (and to be normalized). If L , ( p ) is separable, then t l ( p ) is isomorphic to L1[0,1] due t o a theorem of Carat h h d o r y (and Halmos-von Neurnann, see Brown-Pearcy [18], p.181) and therefore each T is representable by thc t!--factorization theorem of Lewis and Stegall C4.. If L l ( p ) is arbitrary, then corollary C1. will be applied. One has to show that for T : L l ( p ) -* E the net (9:) is Cauchy: this is the case if (gZm)converges for every increasing sequence (C,) of p a r t i h n s . Given such a (Cn),take the p-complete calgebra C generated by UC,,;then L l ( E , p ) is separable and, by what wasjust proven, - hence

Appendiz l3: The Radon-Nikody’m Property

the restriction of T to &(C, p ) is representable. Since the algebra C, corollary C1. gives that ( g z J n is Cauchy. El

519

U C,

is p-dense in

D2. The following results will be used frequently when dealing with the RadonNikodfm property:

PROPOSITION 1: Let p be a finite measure and E a Banach space. ( 1 ) If E has the Radon-Nikodfim property, then every operator T : L1(p) -+ E has a separable range. (2) Every T : L1(p) -+ E which faclors through a space with the Radon-Nikody’m property is representable. (3) Let I : L m ( p ) c, L1(p) be the canonical embedding for also I : C ( K ) L t ( p ) if p is a Borel-Radon measure on a compact K ) and led E have the Radon-Nikody’m property. Then T o I is nuclcar f u r all T E Z ( L l ( p ) ,E ) and N(T o I ) 5 llTll 11111. -+

This is immediate from what was shown in Appendix C about representable operators particular Grothendieck’s characterization C3.. The Lewis-Stegall theorem C4. reads (with a slight extension) as follows:

- in

PROPOSITION 2: A Banach space E has the Radon-Nikody’m property if and only if for each measure p (finile or nol - or: only p = A the Lebesgue measure on [0,1]) each operator T : L l ( p ) -+ E faclors fhrough some l l ( r ) .In this case,

- the infimum taken over all such factorizations. One may take natural metric surjection .fr(D,y) --y E .

r = BE and R

the

PROOF: For finite measures this is just C4. (and the result in D1. about the Lebesgue measure; note that L1[0,1]ie separable). For arbitrary p recall from B7. that

for some index set I and finile mewiires p,. This easily gives the desired factorization through . f l ( I ; l l )= !,(I x N).Tho fact that every operator l l ( r )-+ E is “liftable” (see 3.12.) to some operator CI(I‘) l l ( B ~* ) E implies the result. 0 -+

Another characterization is alno quitc useful:

PROPOSITION 3: For each Banach space E the following statements are equivalent: (a) E has the Radon-Nikodim properly. (b) All closed subspaces of R have fhe Radon-Nikody’m property.

520

Appendiz D: The Radon-Nikody’m Property

(c) All separable closed subspaces of E have the Radon-Nikodim property.

PROOF:That (a) implies (b) follows from C1.(6) and that (c) implies (a) is a consequence of proposition C6.for IZL := 2.0 D3. The following Banach spaces E have the Radon-Nikodim property: (1) E = .tl (see C4.) or more generally: Every Banach space with a boundedly complete amemI n E aV} bounded implies the convergence of anen; inspect basis (i.e. the proof of proposition C4. with the equivalent norm supn 11 C:=l cr,e,ll).. (2) E = Ll(I’), where r is infinite (proposition 2 in D2.). (3) E is a separable dual space (thin follows from the weak version of the Dunford-Pettis theorem C5.). (4) E reflexive (this is the Dunford-Pettis theorem, strong version, C7.).

{x:

x?

The following spaces E do not have the Radon-Nikodim property: (1) E = c, (see C2.). (2) E = &,(r),where r is infinite (by D2. since c, c &(I?)). (3) C(K)whenever K is an infinite compact space (same argument). (4) E = L1(p) whenever p is not purely atomic (see C3. if p is finite - but the general case is easily reduced to this). (5) M ( K ) := C(K)‘,the space of signed measures on a compact space K for which there exists a measure p which is not purely atomic (since L1(p) C M ( K ) isometrically). It follows, in particular, that neither the Radon-Nikodim property nor its negation is inherited by quotients (take el (r)+ c, in the first case and L1[0,1] +I!. in the latter) and that E having the property iR not related to E’ having the property (consider C,

,el, ew 1.

In Diestel’s and Uhl’s book [55], p.217-219, the reader can find a list of almost 30 equivalent formulations of the Ratloti-Nikodim property and a list of spaces with or without this property.

D4. It was observed in B6. that the Radon-Nikodim theorem for a (finite) measure p simply means Ll(p)’

= L,(p). For the vector valued case recall from €312. that

I

Lm(p, F ) := { g :R -+ F p-essentially bounded, p-measurable }/

-

(equivalence classes with respect to llgllw = 0). Now 3.3, the universal property of the projective tensor norm, and Appendix B12. imply that

Obviously, equality holds if and only if every operator L 1 ( p ) -, E is representable.

Appendiz D: The Radon-Nikodkm Property

PROPOSITION: Let E be a Banach space, p E [l,oo[ and

l/p

521

+ l / p = 1. Then

holds isometrically for all finite measures p (or only the Lebesgue measure on [0,1]) if and only if E’ has the Radon-Nikody’m property.

PROOF: If this equality holds, take T : L 1 ( p ) + E’ and let pT E ( L , ( p ) @= E)’be its associated functional. Since

is continuous, it follows that ,&o I E L , ( p , E’) : There is a g E L q ( p ,E‘) C L1(pl El) such that, in particular, ( T X A , z)= (h’0 11X A @ ) = L ( d w ) i z ) k ( d w )=

(L

gdP1 z) I

sA

hence T X A= g d p and it foflows, by C1.(3), that g is 8 Riesz-density for T. Conversely, take ‘p E L p ( p ,E)’ and define (as in the scalar case) an E’-valued measure M by ( M ( n ) z) , := (PIX A @ ) This measure (we omit the details Nince the result is not needed; see Diestel-Uhl [55], p.61) is of the type which has a density g E L l ( p , E ’ ) (see Dl.). For the sets Bn := { w E C4 I Ilg(w)IIE) 6 n) it follows that cpn defined by

coincides with ‘p for step functions f which are zero outside B,. This implies that IbnII I IId (in L p ( p , E ) ‘ ) and hence llgIIqdp)”q = IlV’nll I IId; Beppo Levi’s monotone convergence theorem and the theorem of Banach-Steinhaus show that the function g belongs t o Lq(plE‘) and that g represents cp . 0

(sB,

Pietsch Integral Operators D5. Integral operators T : E

-+

F are characterized by their factorization

Appendiz D: The Radon-Nikodim Property

522

(see 10.5.). In general, there is no such factorization with S taking values in F (see

D9.below for an example). If this can be achieved, then T is called Pietsch integral (or strongly integral). The class

PI(T) := inf{llRllll~llllsll

p3 of all Pietsch

integral operators together with

I T :E A L , ( ~ ) L

I

S

~1(p)

~ , finite p

1

is easily seen t o be a Banach operator ideal.

PROPOSITION : (1) 'J1 C y3 C 3 c

and all these inclusions are strict; moreover,

Pi 5 1 5 P I 5 N . ( 2 ) If F is A-complemented in F", then

p3J(E, F') = 3( E , F ) and PI(T) 5 AI(T) . In particular, P3(E,GI) = 3( E , C') holds isometrically. ( 3 ) If F has the A-extension property, then

P 3 ( E,F ) = 3( E , F ) = P l ( E ,F ) and PI(T) 5 APl(T) . In particular, T3(E,F) = 3( E , F ) = y l ( E ,F ) hold isometrically if F = too or L , ( p ) , (4) If T E 3, then the dual T' E y3 and PI(T') 5 I(T). (5) If T E 3(E,F ) , then ICF o T E y3 and PI(ICF o T ) = I(T).

PROOF:The only non-obvious statement in ( l ) , (2) and (3) is 'JJ3 # 3 # : this follows from Ex 11.5. and an example which will be given in D9..If T E 3, then T' f 3 (by 10.2.), hence (4) follows from (2). Since I(KFo T) = I(T) (the regularity of the operator ideal 3, see 10.2.), (5)is also a consequence of (2). 0 In general, T 4 '!433 if T' E '$3 : this would imply that P 3 = 3. For the same reason, the ideal 'J33is not regular.

D6. The Grothendieck-Pirtnch factorization theorem (see 11.3., corollary 2) shows that:

PROPOSITION: If K is compact and F a Banach space, then P l ( C ( W ,F ) = J(C(K),F) = W(C(K), F) holds isometrically.

In particular, the same relation holtlti for E = L , ( p ) (where p is an arbitrary measure).

Appendix D: The Radon-Nikody’m Property

523

The Radon-Nikodfm Property and Operator Ideals D7. The key to understanding the role of the Radon-Nikodym property in the theory C3.,which states that an operator

of Banach operator ideals is Crothendieck’s theorem T : L l ( p ) + F is representable if and only if

ToI:

C(I0

or Lmb)

is nuclear; moreover, N(T o I) = Riesa-density of T.

- I

J llglldp 5

T

L1(p)

Ilglloop(S2) =

F

IlTll IlIll whenever g is a

THEOREM: F has the Radon-Nikndkm p r o p e d y if and only if p 3 ( E , F ) = % ( E , F ) E.

for all Banach spaces

This is a consequence of t h n following proposition, which contains some interesting details concerning this result.

PROPOSITION: (1)If F has the Radon-Nikodim property, then the relations

Pi(C(K),F ) = 3((31<),F ) = VTJ(c(K), F ) = g(C(I
% Y E ,F ) = %(ElF ) E ) hold isometrically. ( 2 ) IfPTJ(E, F ) = % ( E , F ) /or E = C([O,11) (for all Banach spaces

or

E = L,[O, 11, then F has the Radon-

Nikoddm property.

PROOF: To see (2) take T E .C(L,[O, 11,F) (by theorem Dl.).Then

- -

T o I : t , [ O , 11

-

L1[0,1]

F

(or T o I : C[O,11 L1[0,1] F) is Pietsch integral, hence nuclear by assumption. Grothendieck’s theorem C3.gives that T is representable. For (1)one only has to show that ‘$,(C(K),F) A % ( C ( K ) ,F) by D6. (and the fact that L,(p) A C ( K )for some K) : ‘litkeT E Pl(C(K),F).Then, by the GrothendieckPietsch factorization theorerri,

524

Appendiz D: The Radon-Nikody'm Property

Since S is representable, T = S o I is nuclear and

and hence equality holds. 0 The result D5.(2) implies

COROLLARY 1: F' has the Radon-Nikodim property i f and only i f J ( E , F') = % ( E ,F') for all Banach spaces E . In this case, the equality holds isometrically. A nice application is

COROLLARY 2: If F c G and G' has the Radon-Nikoddm property, then F' has it well.

PROOF: Take T E J ( E , F') = (E C&F)'. Since E @c by the Hahn-Banach theorcrri there exists

satisfying Q o f' = T (for Q : G' hence T is also. 0

-it

as

F c E & G holds isometrically,

F'). It follows that the operator

f' is nuclear and

D8. This was the characterization of the Radon-Nikodfm property in terms of Pietsch integral = nuclear for the range space of an operator. For the domain space one has the

THEOREM: For each Banach space E the following statements are equivalent: E' has the Radon-Nikodim property. (b) !J3J(E,F ) = %(ElF ) f o r all Danach spaces F (or only F = L1[0,1]). (c) J ( E , F') = %(ElF') for all Hanach spaces F. (a)

In this case, all the relations hold isometrically. This result, in essence due to Grolhendieck, was formulated more or less in this way by various authors and got ila final form (removing an assumption about the approximation property) from Alencar [I] only in 1985.

PROOF:Assume E' to have factorize as follows:

lhe

Itadon-Nikodfm property, take T E P 3 ( E , F ) and

Appendix D: The Radon-Nikcdy’m Property

525

Then D7. implies that Tl E 3(Lm(p),E’) = %(L=(P),E‘) and

Since Ll is 1-complemented in its bidual, it follows that

A

which implies (b) and also (c) since 3 ( E ,F’) ‘$3(E, F’). The fact that L1[0,1]is 1complemented in its bidual alw yields that (c) implies ?3(E, L1[0,1])A %(ElL1[0,11) - and it remains to verify that this implies (a). In order to apply theorem D1. take T E C(Ll[O, 11, E‘). Since T

ToI:L,[O,l]Cf+L~[O,l]-

is the dual of S :E

E”

T’

L , [0,1]

I

E’

L1[0,1]

and I is Pietsch integral, it follows that S is nuclear and hence also ToI : Grothendieck’s characterization C3.gives that T is representable. 0

COROLLARY: I f t h e dual E’ of a Banach space E has the approzimation property, then E’ has the R a d o n - N i k o d ~ m properly if and only i f 3 ( E ,F ) = %(ElF ) for all Banach spaces F ( o r only F = L1[0,1]). In this case, ihe equality holds isometrically.

PROOF:By the theorem the condition is clearly sufficient. For the converse take T E 3 ( E ,F), then

T’E 3(F‘,E‘) = n(F’, E’)

and N(T’) = I(T’) = I(T)

(by D7., corollary 1). Since N(T) = N(T’) thanks to the approximation property of E’ (see 5.9.), the result follows. 0 The example in D9. will show that the assumption that E‘ has the approximation property is indispensable in this corollary. For absolutely 1-summing operators there is no such characterization since Lewis and Stegall [170] proved that V , ( E ,F) = % ( E ,F) for all Banach spaces F if and only if E‘ is isomorphic to some .t,(l’), see 23.9..

526

Appciatlix

I): The Radon-Nikodkm Property

D9. Take the Figiel-Johnson example (see 5.2.(5)) of a Banach space E with the approximation property but without the bounded one - and with E' separable: this implies that E' has the Radon-NikodJim property. It is shown in 16.8.(with the help of D7.)that there is a non-nuclear T f 2 ( E , E) with nuclear dual T'. Theorem D8. shows that T cannot be Pietsch integral as well; since T E 3 if and only if T' E 3, it follows that:

PROPOSITION: There is a Banach space E with an integral operator T E 2 ( E ,E ) which i s not Pietsch integral, but T' E 9l c p3 c 3 and E' has the Radon-Nikodim property. This observation is due to Alencar [I]. Note that by corollary D8. the dual E' of any Banach space E which admits such an operator does not have the approximation property.

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540

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List of Symbols

545

List of Symbols

1. General Symbols

set of natural numbers set of integers set of real numbers set of complex numbers =Ror67 := K" := {z E RU{-oo,oo}l a 5 z 5 b} for --oo 5 a := {z E R I a < z < b} for -00 5 a 5 b 5 00

5 b L 00

conjugate exponent of p E [l,001 satisfying l/p+ l/p' = 1 real and imaginary part of z E C class of all normed spaces class of all Banach spaces class of all finite dimensional Banach spaces := { F c E I d i m F < 00) := { F C E 1 F closed subspace, dim E/F < oo} separating dual system, 490 duality bracket, 490 weak topology on E with respect to (E,F), 490 scalar product of a (pre-)Hilbert space Riesz mapping defined by (Rz,' ) H ' , H := ( * I z )H if z,y are elements of a lattice, 491

Let A be a subset of a vector space E: conv(A)

r(A) spanA mJ4

convex hull of A absolutely convex hull of A linear hull of A Minkowski gauge functional of A, 26

546

List of Symbola

:= (spanA, mA) if A is absolutely convex, 26 := {y E F I I(aly)I 5 1 for all a E A} absolute polar of

CAI A0

A

c E with respect to the dual system (E,F),490

Let E and F be normed spaces: closed unit ball {z E E

BE

I

11zll~5 1)

I

open unit ball {z E E llzil~< I} := { T : E -+ F I linear } := {T E L(E, F) I rank T < oo) := {T E L ( E ,F) I continuous }

:= {T E C ( E , F) I rank T < oa} topologies on e ( E , F), 58, 190 := L ( E , K ) algebraical dual of E := C ( E ,K) (topological) dual of E completion of E Banach-Mazur distance, 70 projection constant of E, 307

E and F are isometric identical mapping E -* E K E :E

1

E"

canonical metric injection defined by ( K E ( z ) , z')E",E' := ( z , z ' ) E , E '

PE : E'" f, E'

canonical restriction defined by (pE(x'"), Z ) E , , E := (,'Ii, K E ( z ) ) E , , , , E , ,

I,E:G&E

canonical metric injection of a subspace G of E into

QE : E f EIG 1

Q E :e i ( B ~ + ) E

IE : E

1 Q

Let T : E

IlTll NT) ker(T) im(T)

loo(B~,)

-

E

canonical metric surjection for a closed subspace G of canonical metric surjection ( E a Banach space), 489 canonical metric injection, 489

F be linear and continuous:

:= sup{llTzll~I z E BE} norm of T extension norm of T, 312 := {z E E I T z = 0) kernel of T := {Tx E F I x E E } range of T

E

547

List of Symbols

rank(T) T:E-F 1

T:E-F T:E+F

--

T:E-~*F T' : F' E' T" : Elt F

:= dimT(E) rank of T T injective

T metric injection, i.e. llTzll~= 11zll~for all z E E T surjective T metric surjection, i.e. l l y l l ~= inf{llzIIE I Tx = y} dual operator T'z' := 2' o T' astriction of T if T E !ZD(E,F) defined by T"(z")

:=

2. Sequence spaces and related spaces :=

ni,, E product

eiCI E direct sum := {(zi) E ni,, Ei I Il(zi)llp < := (xiC, IlzillpE,)l/P if 1 5 p < :=

W}

where

00

:= SUPiel 11ZillE.

:= .fp(I, K) :=

e,(q

:= tp({l,..., n}) := tp(nv,E), 90 91 92 := ((2,) E t,(E)J linh-, := c,(Hi)

1 1 ~ ~ =1 0)1 ~

:= II(1IZnllE)llp I 90, 107 :=s~p{ll((z',~n))llp llzlll~!5 I}, 91, 104, 107 ultraproduct of Banach spaces E, along an ultrafilter U,

I

226 ultraproduct of operators T, E C(E,,F,), 227 diagonal operator for a (finite or infinite) sequence A = (A,) of scalars, Dx((z,)) := (Anzn) 3. Spaces

space of continuous functions f : K + E, where K is compact and E a locally convex space := C(K, K )

548

COO(Q, K

List of Sgmbols

)

HP := H P ( 0 )

CP Rad

-

space of continuous functions f : R K with compact support (0locally compact) Hardy space on the unit disc D c 6' Johnson space, 71 closed span in L2[0,1] or Lz(D,p) of the Rademacher functions, 10, 97 Rad with norm induced from L,, 99

4. Locally convex spaces

P@'a P

continuous seminorms on E a basis of cs(E) neighbourhoods of zero in E a basis of U(E) := {z E E I p ( z ) = 0) kernel of a seminorm p := ( E l kerp,p) if p E cs(E) where p ( 6 ) := p ( v ) canonical quotient mapping for U E U(E)absolutely convex if p = mu 2 ( E , F) (continuous linear operators) with the topology of uniform convergence on bounded sets := & ( E , K) strong dual, 469 the bidual E f fwith the topology of uniform convergence on equicontinuous subsets of E', 469 469

5. Measure and integration theory

(0, 4,( Q ,c,,P I

1 / 4 9 P+JPW P LWP) P I @ P2 P7

c, IP

V) g(P)

measure space, 497 variation, positive and negative part of a measure/Daniell functional p, 493, 494, 497 real and imaginary part of p , 497 product measure, 498 Bourbaki extension of p , B8. set of p-measurable sets, 496 set of p-integrable sets, 496 space of C-step function, 494 := G(I,)

549

List of Syrnboln

&(Q, 2,p ) = 21(a,P ) = = Clb)

-

space of p-integrable functions f :0 K , 494,497 space of p-measurable functions, 495 space of pintegrable functions, 498 space of classes of pintegrable functions, 499 space of classes of &-step functions

498 498

LP(C(I

El, L,(E)

~ m ( PE, ) , J L ( E )

WdP) p-a.e.

class of the function f, 499 space of K-valued regular Bore1 measures, 504 space of (classes of) Bochner p-integrable functions 0 -+ El 505

57,88,506 507 = p-almost everywhere, 496

&A

characteristic function of a subset A c R defined by = 1 if z E A and XA(Z) = 0 if z 4 A. := {z E R I f(z) 2 cx) for f : R + [-.o,+m] obvious definitions Rademacher function on [0,1], 10 (discrete) Rademacher function on D, or D,96 Walsh function on Dn or D, 106

Dn := {-I, 1)" D := {-1,1)N

95 95

XA

XA(C)

If 1 4 [f < ffI,[a< f
Rademacher measure on D, resp. D, 95 Gauss measure on K" resp. K N ,100 Gauss function, 100 stable measure on K" resp. K N ,318

6. Bilinear mappings and tensor products

B d ( E , F ; G) BiI(E,F ) %iI(E,F ; G) %iK(E,F ) L ( E ,F * ) E' 8 F @-laL

= L ( E F, G) bilinear mappings E x F

-+

G,8,17

:= BiI(E,F ; K )

= C ( E & F, G) continuous bilinear mappings, 8,27 := %iK(E,F ; K ) = BQ(E,F ) = ( E gI F ) * , 9,17

= S ( E , F ) , 20 linearization of @ E %iK(E,F ; G), 26

550

Lint of Symbols

I

:= sup{ll@(z,Y)IIG t E B E ,y E BF} norm of E !Bi[(E,F ; G ) ,8 for T E L ( E , F * ) , 9 for 'p E Bil(E, F ) , 9 right- and left-canonical extension of 9 E !BiI(E,F ) , 12 tensor product of two vector spaces, 17 25,43 elementary tensor, 18 20 rank of z E E €3 F, 158 transposed of z E E 8 F, 21 for z E E' 8 F, 20 for T E S ( E , F ) , 201 trace on E , 21, 35 tensor norm a on E 8 F, 147 tensor product E 8 F equipped with a,147,469 completion of E F, 147, 469 tensor product of operators, 22 tensor product of sets, 26 tensor contraction, 378, 471 7. Norms on tensor products

Let a and P be tensor norms: transposed, 149 dual, 178 adjoint or contragradient, 178 finite hull, 149 cofinite hull, 149, 189 right-finite hull, 157 right-cofinite hull, 157 right-, left-, injective associate, 259 right-, left-, projective associate, 256 traced tensor norm along G, 379, 383 traced tensor norm, 384 equivalent norms: there exist c, d > 0 with ca

5 pP 5 d a

List of SVmbols

Projective norm: T

IT\ T\

Injective norm:

27 267, 362 265,306, 362

other associates of E and

&

\&/ €1 T:

46 267, 372 265, 306, 362

Sec.27

Other special norms: norm on L, 18 E coming from L, ( E ) ,see Index transpose of A,, 79 Laprestk’s tensor norms, see Index 158, Sec.28, 393 158, Sec.28 366, 375,393 Kilbert-Schmidt norm, 351 312 see Index see Index see Index table, 266 8. Operator ideals Let

((a, A)

be a (quasi-) Banach operator ideal: 108 injective hull, 112 surjective hull, 113 264 264 273 dual ideal, 114 regular hull, 299 maximal hull, 201 minimal kernel, 287

551

552

8*

2lA g m i n dual a m i n reg a i n j min %bur min

21min(E,F) = BmoZ(E, F) 80%

8o%cC

a@% (a 0 %)* = (a @ %)* 8 - 1 0 %, a 0 23-1

List of Symbols

adjoint ideal, 208 conjugate ideal, 221 297 298 299, 340 340 Sec.33 composition ideal, 114 cyclic composition theorem, 330 tensor product, 384 385, 333 quotient ideals, 332, 333

9. Special operator ideals

cotype q operators, 85 (p, q)-dominated operators, 209, Sec.19

= (9p,p#, Dp,pd)pdominated operators, 210 finite rank operators, 20 approximable operators, 47, 110 Hilbert-Schmidt operators, 134, 143 integral operators, Sec.10 267 = (Cp,l,Ip,l)p-integral operators, 209, Sec.18 compact operators, 110 K-convex operators, 118, Sec.31

= ('cp,q, Lp,4)min(p, q)-compact operators, 291 = (2,,Lp)min pcompact operators, 291 all continuous linear operators 111, 157,202, 313 Fourier operators, Sec.30 (p, q)-factorable operators, 208, Sec.18 337, 377,387, 392 337, 387, 392

337, 369,377, 388 = (ZPIP~, Lp,pj)p-factorable operators, 209

List of Symbols

553

2-factorable = Hilbertian operators, 116 238, 437,458 238, 458 ( q , p)-mixing operators, Sec.32

nuclear operators, 34

= (JP,

pnuclear operators, 291

absolutely (r, s)-summing operators, 105, 145 Pietsch integral operators, 522 Pietsch pintegral operators, 283 absolutely psumming operators, Sec.11 267 quasinuclear operators, 116 type p operators, 85 weakly compact operators, 110 10. Space ideals

116, 473

= C;-spaces, 40, 338, Sec.23 338, 339,437

space( C y ) space( 2 ; ' )

338, 339

space( Zip"'

274, 338,339

11. Relations between maximal normed operator ideals and tensor norms 2l-a

---

aini

2l.U'

a/ BP

/a a/ w;i*

BP4

&p',q'

\=PA/

3-a JP

gP

C-&

79',P'

202 263 263 264 210 209 368,374 204 209 211

\a

-

-

\ff

%dual

a*

at

a*

2L@%-a@/3

.e, .c,,,

--

264 206 208 384

209 208 gc'nj 8ur 7L 369,374 Cq~)rcg /3p,g 372, 374 (2,o 3gi)reg 6p,g 374, 376 PP $8 210 ff

(1:.

WP

QP,9

--

List of Symbols

554

12. Constants constants in the Khintchine inequality, 96 constant in the Lkvy embedding, 315,319

323 p-th root of the p-th moment of the Gauss measure 7,,, 145 complexification constant, 377 Grothendieck constant, 169 finite dimensional Grothendieck constant, 268 little Grothendieck constant, 139

Indez

555

Index ap,q,150, 153, 237, 242, 254, 279, 285, 326, 344, 360, 368,387, 414, 434, 442,444 absolutely p-summable sequence, 90, 92, 103, 128, 143, 322 absolutely psumming operator, Sec. 11,203,210,217, 221,232,271,285, 286,321, 322, 326, 332, 338, 342, 349,351, 370, 413, Sec.32, 441, 444,448,451,461,466, 467 absolutely l-summing operator, 128, 132, 138, 142, 272, 310, 342, 349, 362, 414, Sec.32, 522 absolutely 2-summing oprrabor, 134, 136, 143, 272, 274,342, 349, 362, 370,376, 413, Sec.32, 460 absolutely (r, s)aumming operator, 105, 145, 203, 322,426, 461, 462, 467,468 abstract Lapace, 38, 89, 492 abstract Lp-space, 492 abstract M-space, 492 accessible, right-, left-, operator ideal, Sec.21, Sec.25, 413, 415 tensor norm, Sec.21, 180, 253, 413 adjoint operator ideal, 207, 221, 278, 293,330, 333,446 adjoint tensor norm, 178, 207 Alencar, R., 5, 524, 526 Amemiya, I., 3 approximableoperator, 47,60, 110, 116, 165,240,287,313,412,415,445,447 approximation lemma, 160 approximation property, Src.5, 59, 64, 67, 219, 298,474 a-,Sec.21, 280 bounded, Sec.16, 59, 67, 279 bounded Q-, Sec.21, 280 X-bounded, 59 metric, 59, 193, 194 order p , 280

Aron, R., 125 associated bilinear mapping, 18 Auerbach basis, 491 averaging technique, 95 Banach, S., 315 Banach algebra, 491 Banach lattice, 491 Banach-Mazur distance, 70, 144, 360 Banach operator ideal, 109 pBanach operator ideal, 109 Banach-Steinhaus theorem, 14 Barton, T., 249 basis, 59, 158, 298, 358, 453, 457, 491 Auerbach, 491 unconditional, 358, 457 Beauzamy, B., 89 Beckner, W., 87, 89, 187, 346, 418 Behrens, L., 5 Bennett, G., 348, 466, 468 Beppo Levi monotone convergence theorem, 495 Bergh, J., 462 Bessaga, C., 94, 104, 315 Bessaga-Pelczyriski selection principle, 104 Bierstedt, K. D., 6 bilinear, 7 Banach-S teinhaus, 14 closed graph theorem, 14 continuous, 8 extension, 13 integral, 53, 55, 58 linearization, 18 nuclear, 9, 58 positive, 225, 228 separately continuous, 8 symmetric, 25 transposition, 11 bipolar theorem, 490 Bochner integrable function, 24,29,506 space of, 24, Sec.7, 29, 505, 507, 521

556

Bochner's theorem, 316 Bohnenblust, H. F., 492 Bonet, J., 477, 481, 487 Botelho, G., 518 Bourgain, J., 59, 86, 270, 285, 405 Brown, A., 315,403,518 Burkholder, D. L., 86 C'-algebra, 247,491 canonical decomposition of an opcrator into its finite parts, 231) canonical extension, riglit , I dk , 1 I, 14, 75, 161,470 Carathtbdory, C., 315 Carl, B., 5, 6, 348, 418, 427, 448, 453, 458,464,466 Castillo, J., 6, 105 Chatterji, S. D., 97 Chevet, S., 150, 186 Choquet, G., 316 Chu, C. H., 249 closed graph theorem, 14, 43 closed operator ideal, 110 cofinite hull of a tensor norm, 149, 160, 189, 191 one-sided, 157,165, 202 cofinitely generated tensor norm, 149 Cohen, P. J., 11 Cohen, J. S., 141, 150 Collins, H. S., 486 comp act operator, 40,45,48, 60, 92, 1 10, 165, 195, 326,445,447, 516 sets, 31, 33, 42, 159, l(i5 weakly conditionally, 189 ( p ,q)-compact operator, 291, 343,361, 451,458 pcompact operator, 313,466 compact extension properly, 45, 53 compactly approximable operator, 69, 412 completely continuous operator, 115 completely symmetric opcrator ideal, 114

Index

complexification, 176, 347, 361, 377 component of an operator ideal, 108 composition operator ideal, Sec.25, 114,217, 245, 278,300, 385, 393 normed, 115, 290, 386 conjugate operator ideal, 221 continuous functions, space of, 19, 38, 48, 50, 53, 89, 132, 156, 167, 171, 199,312, 479, 520 continuous trace, 300 continuous triangle inequality, 79, 499 contragradient tensor norm, 178 CotYPe Q Fourier, 402 Gauss, 399, 406 operator, 85, 89, 106, 111, 117, 203, 277, 349, 399,402, 403,408, 454, 167 space, 93, 103, 322, 324, 349, Sec.31, 417, 419 countable neighbourhood property, 476 crossnorm, 147 cyclic composition theorem, 330

dp, 152, 153, 178, 185, 238, 254,265, 271, 338, 350, Sec.27, 380,387, 389, 414 table for d z , 266 Ap, Sec.7, 82, 147, 184, 187, 233, 314, 339, 346, 369, 374, 380, 389, 418, 422, 424, 426,429 Daniel1 functional, 493 Davie, A. M., 174 Davis, W. J., 110 Day, M. M., 38 Defant, M., 5, 6, 86 Defant, U., 6 dense, p-, 496 density lemma, 80, 163 maximal normed operator ideals, 223 (DF)-space, 483 diagonal operator, 45, 145 Diaz, J. C., 280, 287 Diestel, J., 94, 105, 189, 505, 516, 517,

Indez

518, 520, 521 Dineen, S., 56 disc algebra, 60 discrete Rademacher functions, 96 ( p , q)-dominated operator, 210, Sec.19, 279,351, 360, 368,434 pdominated operator, 210, 309, 351, 360,382,429,466 double Khintchine inequality, 455 dual operator ideal, 114, 206, 264,278, 297,299 dual system, 490 dual tensor norm, Sec.15 duality bracket, 490 duality theorem, 179 Dunford, N.,505 Dunford-Pettis property, 516 Dunford-Pet tis theorem weak version, 30 strong version, 516 Duren, P. L., 57 Dvoretzky-Rogers theorem, 132, 143

Edwards, R. E., 84 eigenvalues, 459 elementary tensor, 18 embedding lemma, 162, 470 embedding theorem, 205 Enflo, P., 59 equivalent tensor normn, 153 exercises, 4 extension, + canonical extension lemma, 75, 161, 470 extension norm, 312 extension property, 13,38,41,273,300, 306 absolutely 2-summing operators, 143 + compact integral operators, 126 extreme point, 14 (p, q)-factorable operator, 208, Sec.18, 245, 279, 337, 344, 351, 360, 369, 434,442, 451, 458

557

pfactorable operator, 209, 230, 303, 307, 350, 351, 360, 362, 372, 454, 466 2-factorable operator, 116, 211, 230, 248, 250, 314, 344, 360, 370, 374, 376,400, 402, 409,411, 429,468 factorization theorem Davis-FigielJohnson-Pelczyriski, 110 factorization theorem pfactorable, 230 ( p , q)-factorable, 236 Grothendieck-Pietsch, 130 integral operators, 122 pintegral operators, 231 Kwapieri, 244 Maurey, 233, 241, 422 Pisier, 411 see also Sec.28 Fan, K., 491 Fatou lemma, 495 Figiel, T., 60, 86, 87, 110, 347 finite hull of a tensor norm, 149, 160, 189 one-sided, 157,165, 202, 251 finitely generated tensor norm, 149 finite rank operator, 20, 24, 47, 165, 198,427 Fourier cotype, 402 matrix, 56, 121, 345 operator, 203, Sec.30 series, 397, 407 transform, Sec.30, 44, 83, 90, 316, 511 type, 402, 405 Friedman, Y.,249 Fubini theorem, 498 Fubini-Tonelli theorem, 498 gp, 152, 153, 178, 185,232, 238,254, 265, 274, 276,281, 314,321, 338, 350, Sec.27, 372 376, 380, 387, 389, 414, 417,441 table for g2, 266, 274

558

Galbis, A., 481, 487 Garling, D. J. H.,137, 184,313 Gaudrey, G. I., 84 Gauss cotype, 399, 406 Gauss functions, 99, 105 Gauss measure, 100, 145 Gauss-Khintchine equality, 100, 145 Gauss type, 399,406 Gelbaum, B., 158, 358 Gelfand, I. M., 491, 515 Gelfand representation theorem , 491 Gilbert, J. E., 3, 5, 279 Gil de Lamadrid, J., 158, 358 Goodner, D. A., 38 Gordon, Y., 2, 5, 137, 145, 184, 186, 221,273, 285, 313,339, 364, 441 Govaerts, W., 479 Grothendieck, A., 1, 2, 3, 5, 31, 39, 58, 118, 125, 127, 129, 130, 139, 146, 147, 150, 166, 176, 182, 247, 267, 361, 382, 473, 479,483, 486, 488, 493,508, 511, 517, 524 Grothendieck constant Kc, 169, 173, 174, 176, 247,267, 415,428 finite dimensional Kc(n), 268, 274, 375,377 little K L G ,139, 173 Grothendieck’s inequality, Sec.14, 347, 376,462 abstract form, 414 matrix form, 172 non-commutative, 247 nuclear form, 292 operator form, 212, 267, 31 1, 342 proofs, 169, 175,413, 429 tensor form, 169,247, 267 Grothendieck’s notation, 364 Grothendieck-Pietsch dorriination theorem, 129 Grothendieck-Pietsch factorization theorem, 130 Grothendieck theorem, 212, 270, 310, 462 --+ little

Indez

representable operators, 511 spaces satisfying, -+ G.T. space G.T. space, 270, 429 Haagerup, U., 96, 174, 247, 248, 269 Hahn-Banach theorem, 489 for operators, 10 -* extension property Halmos, P. R., 315, 370 Hanson, B., 6 Hardy, G. H., 56 Hardy space, 57, 60, 285, 310 Harksen, J., 5, 469, 470,475, 477 Harmand, P., 5, 6 Hasumi, M., 38 Hausdorff-Young inequality, 405 Heinrich, S., 228, 240, 393 Helly’s lemma, 73 Hilbertian operator, + 2-factorable Hilbertizable space, 474, 486, 488 Hilbert-Schmidt norm, 351, 358,457 operator, 134, 143, 351, 361, 460 Hilbert space, 25, 88, 105, 134, 137, 144, 172, 238, 250, 304, 313, Sec.26, 457 characterizations, 141, 221, 311,370, 398, 400, 401, 405, 406, 407, 412, 460 Hilbert transform, 84, 86 Holder inequality, 498 Hoffmann-Jorgensen, J., 85 Hollstein, R., 476, 478, 488 Holub, J. R., 199, 448, 450, 451, 452, 467 Horowitz, C., 11, 14 inductive limit, 476 inequality continuous triangle, 79, 499 double Khintchine, 455 Grothendieck Hausdorff-Young, 405 Holder, 498

Indez

Kahane, 98 Khintchine, 96, 361, 455 Littlewood, 125, 462, 483, 466 Minkowski, 79, 499 injective Banach space, 13, 38, 306, 312,452 hull of an operator ideal, 112, 263, 340 operator ideal, 111, 388 tensor norm, right-, left-, Sec.20, 275,475 associate, 259, 260 tensor norm 6, Sec.4, 46, 78, Sec.6, 193, 265, 303, 306, 344, Sec.27, 380,414,425,452,467 coincidence with T,487, -+ Pisier space integral bilinear form, 53, 55 integral operator, Sec.10, 132, 142,193, 203,217, 245, 309, 351, 362, 386, 429,441,450, 517,522 a-integral operator, 204 pintegral operator, 209,216,231,271, 321, 332, 338, 349, 351, 368, 392, Sec.32, 436, 441, 451,486 integrating functional, 54 , 121 , 224 Iochurn, B., 249 isometry, 489 Iwaniec, T., 87, 347 Jameson, G. J . O., 37, 145, 174, 184, 428 Jarchow, H., 5,6,469, 473,484, 485 John, K., 56 Johnson, W. B., 2, 60, 77, 110, 135, 313,460, 461 Johnson space, 71, 76, 214 Junek, H., 473 Kaballo, W., 306 Kadec, M. J., 144, 314, 315 Kadison, R. V.,250 Kahane, J. P., 98, 107 Kahane contraction principle, 107

559

Kahane inequality, 98 Kakutani, S., 492 Kakutani (-Bohnenblust-Nakano) representation theorem, 492 Kalton, N. J., 452 Kandil, M. A., 398, 406 K-convex operator, 86, 118, 203,407, Sec.31 Kelley, J. L., 38 Khintchine inequality, 96, 361, 455 double, 455 Kisliakov, S. V., 270, 285 Kolzow’s theorern, 502 Konig, Her., 2, 115, 135, 174, 269, 300,356, 405,460,461 Kothe, G., 39, 197,469 Krickeberg, K., 348 Krivine, J. L., 166, 176, 269, 377 Kursten, K. D., 393 Kwapieh, S., 2, 5, 84, 136, 141, 144, 242, 314, 323, 336,338, 339, 350, 358, 369, 394, 400, 401, 402, 406, 457,461,463 Kwapieh’s factorization theorem, 244 Kwapieh’s theorem on the Fourier transform, 401 Cp-space, 304, 338 Cp-local technique, 163, 166, 301, 313, 474 C;-space, 40, 41, 47, 163, 165, 279, Sec.23, 320, 375, 406,466 Cf-space, 41, 260, 305, 308, 309, 314, 476,483 Cg-space, 51, 77, 171, 260, 305, 306, 308,312, 314, 428,476, 479, 484 Lacey, H. E., 38, 492, 493 Laprestk, J. T., 150, 208 large subspace, 479, 487 Lebesgue decomposition theorem, 500 Lebesgue dominated convergence theorem, 495, 506 left-tensor norm, 202 Leih, T., 3, 279

560

lemma approximation, 160 density, 80, 163, 223 embedding, 162,470 extension, 75, 161, 470 Fatou, 495 Helly, 73 Ky Fan, 491 local technique, 163, 301 Pietsch, 135 quotient, 82, 88 Riemann-Lebesgue, 400 variation, 507 Levi, B., 495 Lkvy, P., 314, 319 LBvy embedding, 319,348, 375 Lkvy’s theorem, 319 Lewis, D. R., 2, 5, 190, 221, 273, 310, 329, 364, 430, 432,434, 440, 441, 457,514, 525 Lewis-Stegall theorem, 430, 5 14 lifting Lm-, 501 operators, 13, 39,41,41,45, 273, 300 integral operators, 126 limit order, 355 Lindenstrauss, J., 1, 5, 10,41, 53, 60, 98, 144, 166, 270, 298, 300, 304, 306, 344,370,437 Lindenstrauss’ compactness argument, 76, 239 linear dimension, 315 little Grothendieck theorern, 139, 173, 213,272,428, 468 Littlewood, J. E., 56, 125,462,463,466 localizable measure, 500 strictly, 501 local technique -, 2,Lofstrom, J., 462 L6pez Molina, J. A., 280, 287 Losert, V.,3, 299 Lotz, H. P., 2, 3, 5, 146, 204 Loupias, G., 249

Index

Mackey theorem, 490 Maharam’s theorem, 501 Mankiewicz, P., 60 mapping property, 470 metric, 147 Marcinkiewicz J., 314, 315, 347 Marcus, M. B., 101, 107 Marquina, A., 479 Matos, M., 6, 312 Matziwitzki, I., 6 Maurey, B., 2,5,85, 101, 103, 105, 118, 135, 240, 314, 323, 400, 414, 415, 421,422,427,429,460,461 Maurey factorization theorem, 233,240 Maurey splitting theorem, 423 Maurey-Pisier extrapolation theorem, 42 1 maximal hull of an operator ideal, 201 maximal operator ideal, Sec.17, 288 density lemma, 223 McCarthy, C. A., 456 measure, 494 absolutely continuous, 500 Borel-Radon , 503 finite, 496 Gauss, 100 Lkvy, 318 localizable, 500 Rademacher, 95 signed, 494 singular, 500 space, 497 stable, Sec.24, 319 strictly localizable, 501 Meile, M., 5, 430 Mendoza Casas,J., 479 metric extension property, 13, 38 metric injection, 489 metric mapping property, 147 metric surjection, 489 Michor, P., 3, 299 minimal kernel, 287 minimal operator ideals, Sec.22, 327, 329, 332, 340, 342, 343, 430, 432,

561

Index

440,444 Minkowski inequality, 79,499 Minkowski gauge functional, 26 Mityagin, B. S., 127,285 mixed sequence, 423 (p, q)-mixing operator, Sec.32, 454, 464,468 multilinear mapping, 25 multiplication operator, 88, 142, 233, 240 multiplicity, 459 Mujica, J., 6 Murray, F. J., 1 Nachbin, L., 6, 38 Nakano, N., 492 Neumann, J. von, 1, 315 Nielsen, N., 60 p-norm, 108 normed operator ideal, 109 r n o r m e d operator ideal, 109 norming set, 46 nuclear bilinear form, 9, 58 nuclear operator, 34,35, 43, 45, 64, 68, 72, 76, 101, 111, 112, 115, 117, 120, 142, 193, 195, 245, 247, 300, 308, 310, 330, 351, 445, 4G0, 471, 512,517, 523 Q-, 291,297 dual, 66, 197, 200, 298,526 p, 291,326, 332, 340,343, 361, Sec.32, 436, 444,451, 466 nuclear space, 473, 486, 487 Oertel, F., 331 Ohazaki, Y.,343 open mapping theorem, 11 operator + absolutely p-summing + absolutely l-summing -+ absolutely 2aumming -+ absolutely ( r ,s)--8uriiming -+ approximable canonical decomposition, 239

compact compactly approximable, 69, 412 (p, q)-compact + p-compact completely continuous, 115 + cotype q diagonal, 145 -+ (p, q)-dominated -+ pdominated eigenvalues, 459 (p, q)-factorable p-factorable + 2-factorable + finite rank -+ Fourier -+ Hilbert-Schmidt + integral a-integral, 204 + pintegral K-convex + lifting + (p, q)-mixing multiplication, 88, 142, 233, 240 multiplicity, 459 nuclear -+ Pietsch integral -+ Pietsch p-integral -+ positive + quasinuclear Radon-Nikodim, 443 regular, 81, 90 representable, App.C, App.D -+ p-strong type P weakly compact -+ weak*-weak continuous operator ideal, Sec.9 + accessible, right-, left+ adjoint Banach, 109 p-Banach, 109 closed, 110 completely symmetric, 114 component, 108 -+

-+

-+

-+

-+

-+

562

+ composition

conjugate, 221 + dual + injective -* maximal -+ minimal non-accessible 413,415 normed, 109 pnormed, 109 -* product proper, 110 quasi-Banach, 109 quasinormed, 109 quotient, 332 + regular regular hull, 299 -+ tensor product characterization -+ surjective + tensorstable -+ totally accessible ultrastable, 393 Orlicz, W.,102, 103, 105 Orlicz property, 103, 462 Oya, E., 299 Paley projection, 86 parallelogram identity, 405 Parseval’s identity, 398 Pearcy, C., 315, 403, 518 Peetre, J., 405 Pelczyhski, A., 1, 5, 41, 67, 84, 86, 87, 104, 110, 127, 144, 166, 230, 270, 285, 300, 304, 310, 315, 344, 347, 351,358, 370,437,446,452, 457 PBrez Carreras, P., 477 Persson, A., 186, 332 Pettis integrable function, 507 Pettis integral, 507 Pettis’ measurability criterion, 505 Phillip’s property, 486, 517 Pietsch, A., 2, 109, 110, 111, 127, 129, 130, 132, 135, 137, 145, 300, 312, 323, 326, 345, 356, 357, 377,415, 459,460, 461, 473,478,191

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Pietsch integral operator, 522, 526 Pietsch pintegral operator, 283, 437 Pietsch lemma, 135 Pisier, G., 2, 5, 56, 59, 86, 89, 101, 103, 105, 106, 107, 118, 147, 166, 174, 240, 247, 248, 270, 344, 399, 407, 409, 411, 412, 413, 414, 417, 421, 457 Pisier’s factorization theorem, 411 Pisier, Maurey-P. extrapolation theorem, 421 Pisier space, 56, 59, 147, 412 Pitt’s theorem, 104, 197 positive bilinear functional, 225, 228 positive definite, 315 positive operator, 80, 88, 123, 129,216, 233,235 principle of local reflexivity, 74 operator ideals, 286 weak, 73 probikme des topologies, 479 product operator ideal + composition operator ideal projection constant, 44, 218, 307, 312, 452 projective tensor norm, right-, left-, Sec.20, 275, 475 associate, 256, 258 77, Sec.3, 33, 36, 78, Sec.6 192, 265, 303, 306, 344, Sec.27, 467 coincidence with &, 467 -+ Pisier space Prolla, J., 6 proper cotype, 85 proper operator ideal, 110 proper type, 85 property (B) of Pietsch, 478 property (BB), 478,483 Puhl, J., 355,357,415, 427 quasi-Banach operator ideal, 109 quasi-Banach space, 108 quasibarrelled, 478 quasinorm, 108

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quasinormed operator ideal, 109 quasinuclear operator, 116, 308, 434, 444,448,467 quotient formula, 333 quotient lemma, 82, 88 quotient operator ideal, 332 Rademacher functions, 10, 84, 95, 105, 107, 117,326 discrete, 96 Rademacher measure, 95 Radon-Nikodh operator, 443 Radon-Nikodh theorem, 500 operator valued measures, 4 1 Radon-Nikodim property local, 486 apace~,193, 194, 309,486, App.D tensor norm, Sec.33 Ramanujan, M. S.,448,453,158 rank of a tensor, 158 reflexivity, local 73, 74 tensor product and operator spaces, 196, 199, Sec.33, 439, 440,444 reasonable, 147 regular hull of an operator ideal, 299 regular operator, 8 1, 90 regular operator ideal, 119, 165, 200, 206,298, 299,336,375, 393,443 Reinov, O., 59, 281,283, 285, 299,340 Reisner, S., 277, 285 related operators, 143 representable operator , App. C , A pp .D representation theorem Gelfand, 491 Kakutani (-Bohnenblunt-Nakano), 492 maximal operator ideals, 203 minimal operator ideals, 290 Kesz, 503 respects subspaces, 36 respects quotients, 49 @sumk, 1,2,3, 127, 150, 182,217,267, 273, 299, 360, 364

563

Retherford, J. R., 2, 5, 135, 221, 306, 308,441,460,461 Riemann-Lebesgue lemma, 406 Riesz-condition, 352 Riesz-density, 509 Riesz-identification, 25, 352 Riesz-representable operator, -+ representable Riesz representation theorem , 503 right-tensor norm, 202 Ringrose, J. R., 250 Rivera Ortun, M. J., 190, 280, 287 Rosenthal, H. P., 304, 348, 360, 452 Rosenthal’s theorem, 189 rotating, 331 Rubio de F’rancia, J. L., 405 Ruess, W., 486 Sanz Serna, J. M., 479 Saphar, P., 2,5,150,186,271,275,280, 314,320, 339 Schaefer, H. H., 81, 492, 493 Schatten, R., 1, 2, 146, 147, 182, 353, 357 Schatten’s theorem, 353 Schmets, J., 479 Schur product, 325 Schwartz, J., 505 Schwarz, H. U., 293, 492 Shiga, K., 3 Segal’s localization theorem, 500 separating dual system, 490 sequence -+ absolutely psummable (q, p)-mixed, 423 -+ unconditionally summable -+ weakly psummable Sims, B., 228 Snobar, M. G., 144 space abstract L-, 38, 89, 492 abstract Lp-, 492 abstract M-,492 -+ cotype p

564

G.T., 270,429 Hardy, 57, 60, 285,310 -+ Hilbert Hilbertizable, 474, 486, 488 ideal, 116, 222, 279, 308, 453, 455, 473 injective, 13, 306, 312, 452 Johnson, 71, 76, 214 &-, 304, 338

2;2 : + 2s,-+

-+

measure, 497 nuclear, 473 Pisier, 56, 59, 147 -+ Radon-Nikodjm property Stonean, 38 T p - , 383, 393 -+ type P stable measures, Sec.24, 3 I9 stable type, 324, 326 Stegall, C. P., 306, 308, 310, 514, 525 Stein, E. M., 96 step function, 494 Stollmann, P., 6 Stonean space, 38 strictly localizable measure, 501 pstrong operator, 374, 437, 444 sublinear, 489 surjective hull of an operator ideal, 118, 263, 340 operator ideal, 112, 388 symmetric bilinear form, 25 symmetric finite dimensional spaces, 184, 189, 218 symmetric tensor norm, 149, 863 symmetry constant, 189 Szankowski, A., 59 Szarek, S. J., 60, 67, 348, 360 Takahashi, Y., 343 Talagrand, M.,103, 427 Taskinen, J., 480,481, 483, 484, 486

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tensor associated linear map, 22 elementary, 18 rank, 158, 188 tensor contraction, Sec.29, 155, 159, 472 tensor norm, 147 accessible adjoint, 178, 207 canonical extension -+ cofinite hull cofinitely generated, 149 contragradient, 178 dual, Sec.15 equivalent, 153 -+ finite hull finitely generated, 149 Hilbert-Schmidt, 351 --+ injective left-, 202 non-accessible, 413 -+ projective Radon-Nikodjm property, Sec.33 right-, 202 self-adjoint, 353 symmetric, 149, 353 topology, 470 -+ totally accessible traced, Sec.29 transposed, 149 tensor product algebraic theory, Sec.2 measures, 498 operators, 22 bidual, 222 injectivity, 66 operator ideals, 384, 393 tensor norms, 384 tensor product characterization of operator ideals, 203, 213, 215, 335, 381,384, 388, 390,392, 473 tensorstable operator ideal, Sec.34 metrically, Sec.34 strongly, Sec.34 -+

-+

Index

Terzioglu, T., 165 theorem Banach-S teinhaus, 14 Beppo Levi monotone convergence, 495 bipolar, 490 Bochner, 316 CarathQdory-Halmos-von Neu mann, 315 closed graph, 14, 43 cyclic composition, 330 Davis-Figiel-Johnson- Pelczyhski, 110 duality, 179 Dunford-Pettis, 30, 516 Dvoretzky-Rogers, 132, 143 embedding, 205 4 factorization Fubini, 498 F’ubini-Tonelli, 498 Gelfand, 491 Grothendieck, 212,270, 310, 462 Grothendieck, represerihble operator, 511 Grothendieck-Pietsch, 1 29, 130 Hahn-Banach, 489 Kakutani (-Bohnenblunt-Nakano), 492 Kolzow, 502 Kwapieri factorization, 244 Kwapieh, Fourier transform, 401 Lebesgue decomposition, 500 Lebesgue dominated convergence, 495,506 LCvy, 319 Lewis-Stegall, 430, 514 -* little Grothendieck Mackey, 490 Maharam, 501 Maurey factorization, 233, 210 Maurey splitting, 423 Maurey-Pisier extrapolation, 12 1 open mapping, 11 Pisier factorization, 4 11

565

Pitt, 104, 197 Radon-Nikodfm, 44, 500 + representation Riesz, 503 Rosenthal, 189 Schatten, 353 Segal localization, 500 type/cotype, 400 Timoney, R. M., 56 Tomczak-Jaegermann, N., 5, 437, 454, 461,462 Torrea, J. L., 405 totally accessible, Sec.21 operator ideal, 330, 342 tensor norm, 180 Tp-space, 383, 393 trace, 21, 24, 35,68, 183, 188,294,300, 47 1 continuous, 300 duality, 22, 24, 69, 76, 120, 177, 196, 207, 293, 312, 330 for operators, 64, 294, 300 traced tensor norm, Sec.29 transfer argument, 206, 211 transposition, 11, 23 type P Fourier, 402, 405 Gauss, 399, 406 operator, 85, 89, 106, 111, 117, 203, 277, 399,402,403, 408,454,467 space, 98, 324, 326, 419 stable, 324, 326 type/cotype theorem, 400 Tzafriri, L., 10, 60, 98, 304

Uhl, J. J., 505, 517, 518, 520, 521 ultrapower, 227 ultraproduct Banach spaces, 226 C*-algebras, 238 operators, 227 ultrastable operator ideal, 393 unconditional basis, 358, 457 unconditionally summable, 93, 103,

566

105, 132, 322, 360 uniform boundedness principle, 490 uniform crossnorm, 147

Vala, K., 467 Valdivia, M., 158 variation lemma, 507 Virot, B., 81 Voigt, J., 5, 6

wp,152, 153, 155, 249, 2G5, 271, 274, 276, 280, Sec.23,320,326, Sec.27, 372, 376, 380,393,414,425, 475 table for w2, 266 Walsh functions, 106, 409, 410 Walsh matrix, 176, 463 weakly compact operator, 31, 110,445, 516,517 weakly conditionally compact, 189 weakly measurable function, 505 weakly p-summable, 90, 92, 103, 104, 124, 128, 143, 155,322, 360, 423 weak-*-measurable f u n c l h , 505 weak retraction, 44 weak topology, 490 weak-*-topology, 490 weak*-weak continuous operator, 24, 45,48,295,299 Wloka, J., 476

Yosida, K., 505 Zalduendo, I., 125 Zygmund, A., 314, 315, 317

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