Operator Ideals (North-Holland Mathematical Library 20)

North-Holland Mathematical Library Board of Advisory Editors:

M. Artin, H. Bass, J. Eells, W. Feit, P. 3 . Freyd, F. W. Gehring, H. Halberstam, L. V. Horrnander, M.Mac, J. H. B. Kernperman, H. A. Lauwerier, %'.A. J . Luxemburg, I?. P. Peterson, I. 31. Singer, and A. C. Zaanen

VOLUME 2 0

NO RTH- H O LLAN D P U B L I S I I I N G C O M P A N Y AMSTERDAM * N E W YOXK * O X F O R D

Operator

4LBRE;CHT PIETSCH Friedrich SchiEler University of Jeriu German Democratic Republic

1980

NO RTH - H O L L A N D P U B L I SIII NG C O M P A N Y A M S T E R D A M . NEW Y O R K * O X F O R D

0 VEB Deut,scher Verlag der \Vissenschaften, Berlin 1979 Licenced edition of Sortli-Holland Publishing Company - 1980

All rights reserved. N o part of ihis publicaiiotz may be repioducetl, stored i n a retrieval .$ystem. or tranmitted, i i b nny f o r m or by any meuxs, rlectronic, mechnnicni, photocopying, recorditly 0, otherwise, without the prior $emission of the copyiight owner.

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:

Elsevier Sorth-Holland, Tnv. 52 Vanderbilt Avenue New York, S . Y. 10Oli

Library of Congress Cataloging in Puhlicatiori IMa Pietsch, Albrecht. Operator ideals. (Xorth-Holland mathematical library ; 20) Ribliogmphy: p. Includes index. 1. Linear operators. 2. Banacii spaces. (Algebra) I. Title. QA3292P5313 5 1 B’.71 794680 ISBN 0-444-8629XX

Printed in the German Ikmocratic Republic

3.

Ideals

Acknowledgements

The present monograph is the result of a11 intensive research work during the last teii years. Although most of the proofs are given in a new fashion I have been supported by many niathematicians. First of all the ingenious ideas of A. GROTHENDIECK qhould Fe mentiontd.

I would like to thank all my friends and colleagues for sending me books, reprints, and preprints as well as handwritten informations. Without this inestimable help this nionograph could have never appeared. Parts of the manuscript have been written when I was a guest of the Mathematical liistitutes of the Polish Academy of Science, of the University of Paris, and of the Viiiversity of Cambridge. Furthermore, I have also to thank lily Home University in Jena for a great deal of support. illiiiost all results presented in the sequel have been discussed in niy seminar. “Operatoreiiideale”. 011 these occasions I have learnt a lot from my own pupils. I n particular, I wish to thank each of them for reading large parts of the manuscript and the galley-proofs. The English translation WRS made by myself. Therefore I cnnriot, blame anyone else for errors which are to be found. On the other hand, many improvements Mrs. FRITSCH froiii our are due to several colleagues and especially to H. JARCHOW. Department of Foreign Languages has also read the whole draft. The English manuscript was typed by Mrs. GIRLICHand the original German version by Mrs. LOTZ.Moreover, the staff of “Deutscher Verlag c!cr Wissenschaften” did a lot of work to put this book into its final form. Last, but not least, I owe a great debt of gratitude to my wife a110gnve me nonnlathemativxl support during the long period of my writing.

Preface

In this monograph we present the theorj- of operator ideals which, beginning with is becoming more and more a special the fundamental work of A. GROTHENDIECK, branch of functional analysis producing results and problems of its own interest. On thc other hand, we would like to give a lot of remarkable applications to the spectral theory of operators, to the geometry of Banach spaces, and to the theory of stochastic processes. Last, but not least, we shall obtain a considerably large part of the theory of nuclear locally convex spaces within this new context. Let fj denote the class of all (bounded linear) operators between arbitrary Banach = spaces. An operator ideal 9l is, roughly speaking, a subclass of 2 such that U and 2 o o 2 = U. Important examples are the ideals of coinpact., nuclear, and absolutely summing operators. have I n their excellent monographs I. c. GOHBERGIM. G. KBEJNand R.SCHATTEN presented the theory of operator ideals on a fixed Hilbprt space. Even in this much simpler case it becomes clear that we cannot hope to get n complete description of all operator ideals which exist. Therefore we have to restrict oursell-es to special kinds of operator ideals having interesting properties and applications. In particular, we shall deal with the following aspects: - generation of operator ideals by general principles, -- construction of new operator ideals from given ones. - special properties of operator ideals, relationships between different operator ideals, extensions of operator ideals to a larger class of spaccs. ~- criteria for operators to belong to a given ideal, - miscellaneous applications. The organization of this monograph is tho following: I n the preliminary part we collect some general results which will be used later on. Since we need only a few deep theorems, a great deal of the treatise should be understandable for students graduated from a first course in functional analysis and measure theory. The abstract theory is presented in Parts T and 11. A t first the concept of an operator ideal is introduced. Then some well-known classical examples are given. Furthermore, we dcfine products and quotients of two operator ideals as well as (liveme procedures assigning new operator ideals to given ones. In Part I1 we deal with quasi-norined operator ideals. Many considerations in Part 11 are analogous to that in Part 1. We have of course to take into account soinc modifications due to the presence of quasi-norms. Additionally the concept of an adjoint norined operator ideal is studied. Moreover, wve deal with Bannch spaces having the (metric) approxiIllation property. In Parts III und I V we give the most intcresting examples. A t first various quasinormed operator ideals are generated by the help of so-called s-numbers. In this context we also develop the theory of operator ideals on the class of Hilbert spaces. I he quasi-normed ideals of absolutely summing, nuclear, integral, and factorable

+

< 1

8

Preiace

operators are treated in Part I V which is, from O W point of view, the heart of this nionograph. We also deal with diagonal operators in classical sequence spaces. Part V is devoted to soine xpplications in different branches of functional analysis and probability theory. Although the bibliography contains more than 500 itenis it is far from being coinplete. At the end of every chapter we describe the main streams of the developments. However, we do not refer to the exact sources of each single result. Finally, I would like to express rny hope that this monograph will encourage iiiariy young mathematicians to deal with the theory of operator ideals and its applications. Jeiia (GDR,)

March, 1977

ALBRECHTPIETSCII

INTERDEPENDENCE TABLE

25

CONTE NTS

PART 0

PRELIMNAICIES

Operators Families

21 21 21 22 23

B. B.1. B.2. B.3. B.4.

Special Operators

25

C. 0,1. C.2.

Special Banach Spa4e8 Classical &mach Spaces Abstract Lp-Spa oes Extenigion Property and Lifting Property Cartesian !-Product s of Ballad' Spaces Intermediate Spaces

31 31 32 33 35 35

D.3.

Operator8 in Hilbert Spaces. Hilbert Spaces. Functional s and Operators Spectral Representation .

37 37 37 37

E. E.1, E.2, E.B. E.4. E.5,

Diverse Lemmas R•esz's Lemma Dvoretzky's Lemma tAindenstraus-Rosenthal's Lemma Ky Fan's Lemma K.h inteitine's Inequality

39 39 39 39 40 41.

A.

A.1. A.2. A.3. A.4.

C.4. Ci

.

D. Dl. D.2.

Operators in. Banack. Spaces Banach Spaces Functionals

.

Finite Operators isomorphisms ................ .. Injections arid Surjections Projections

25 . _ 26 28

PART 1 THEORY OF OPERATOR J. 1.1. 1.2. 1.3. 1,4. 1.5. 1.6.

limns

Operator Ideals on Bauch Spaei4 Operator Ideals Finite Operators Approx Unable Opera turi Co III pact Operators Weakly Compact Operatera Completely Continuous Operators

45 45 46 4G 46 47 47

12

Contents

1.7. Unconditionally Summing Operators 1.8. Separable Operators 1.9. Kato Operator 1.10. Peiezyfiski Openators 1.11. Order Itelations beiAve.eii Operator ideals 1. 1 2. Notes 2. 2.2. 2.3. 2.4. 2.6. 2.7. 2.8. 2.9.

3. 3.1. 3.2. 3.3,

. . .. Banach Spaze Spi -e Idealg A-Factorablo Operators ......... Finite Dimensional Banaell Spaces Reflexive Banach Spaceq Banacli Spaces with Schlu. Property Banach Spz.Let.A with Pelezyfiski Property Separable ilanaela Spaces

Order Relationa between Space Ideals

1-8 48

50 59

53

54 5.5 57 ri1 58 58

Notes

Prodzwis flat? Quotients of Operatot Meals Products of Operator Ideals

Quotients of Operator Ideals

59 59 61

62

Notes

63 Operator fa1-. wth specia propertia8 4. Procedures 4.1. 114 4,2. Closed Operator Ideals 65 Radical 4.3. Symmetric Operator Ideals 4.4. 69 4,5. Regular Operator Ideals 70 4.6. Inieetive Operator Ideals 73 Surjective Operator Idcala 4.7. Minimal Operator Ideals 4.8. 76 Maximal Operator Ideal 4.9. 77 Not es 4.10.

araziicza Banath Spam 5. Ciooed Operator Id&t14 5.1. Operators in Sequence Spaece 5.2. Simple Banach Spaces 5.3. Non.Simple Bansch Spaces Closed Operator Ideals on Hilbert Spaces 5.4. • 5.5. Ni3te4, -

70 79 81 82 85 86

PART 2 THEORY OF QUALM-MIMED OPERATOR MEALS Quasi-Normal Operator 1def416rm Banach Spaces 6. Qiiiisi-Nornied Operator ideals 6.1. 6.2. p-wormed Operator Ideals Nuclear Operators' 6.3. 6.4. Integral Operators

g9 89 91 9'1 97

Contents

13 98

Absolutely Summing Operators Hilbert Operators Order Relations between Quasi-Normed Operator Ide-als Finite Nuclear NORM ..... Notes .

100 101 103

7.2. 7.3.

Products and Quotients 01 Quasi Norma Operezior ideals Products of Quasi-Normed Operator Ideals Quotients of Quasi-Normed Operator Ideals Notes

104 104 106 106

8.

Qua si 2forirted Operator Meats with Spetial Properties.

107 107 107 108 109 111 112 11.5 119 122

6.5.

45,7. 0.8. 6.9.

-

7.1.

-

-

S,1. 8.2.

Procedures Symmetric Quasi-Nortited Operator ideals Regular Quasi-Norined Operator Ideals 8.3. Injeetive Quasi-Normecl Operator Ideals 8.4. 8.5. Surjective Quasi-Normed Operator Ideals g.6. Minimal Quasi-_Normed Operator Ideals 8.7. Maximal Quasi-Normed Operator Ideals 8.8. Ultrastabie Quasi-Normed Operator Idea's Notes 8.9.

9.

Adioint Operator feleahr

9.1. 9.2. 9.3. 9.4. 9.5.

Adjoint.Normed Operator Ideals .. Representation of Funetionala Perfect Normed Operator Ideal8 Self- Ad jmnt Normed Operator Notes

10.1. 10.2. 10.3. 10.4. 10.5.

........

Banach Spaces with Approximation Property Approximation .Property Metric Approximation Property Consequences of the Metric ApproximAion. Property .. Countere !Ca mpIes Notes

..

123 123 126 128 129 129 130 130 130 132 135 141

PART 3 THEORY OF SEQUENCE

IDEALS

.5-Isrum62-8 of Operators in Bariack .8pamt 11.1. s-Numbers cif Operators 11.2, Approximation Numbers 11.3. Operators in Hilbert Spaces 11 4. I-filbert Numbers 11.5. Geifand Numbers 11,6. Kolmogorov Numbers 11.7. Symmetric s-Numbers ...... 11,8. Additive 8,Numbeni 11.9. Multiplioriar.re 8-Numberi, 11.10. Maximal 8-Numbers

1 1.

145 145 146 146 148 149 150 152 154 154 155

14

Contents

11,11. e- No mberq of Diagonal Operators 11.12, Relationships between a-Numbers . 11.13. Notes 12. 12.1. 12.2, 12,3. 12.4.

E•troyy Numtmrs of Operatorsii

158 ,, • ........ 165 166

Banach. Spaca

Outer and Inner Entropy 'Numbers

Entropy Numbers of Diagonal Operators Relationship between Entropy Numtpors and i-Numbers Notes ..... — .

13. Sequence Ideais on the Scalar Field 13.1. Sequence Ideals 13.2. Quasi-Normed Sequence Ideals 13.3. Procedures 13.4. Minimal Quasi-Norrned Sequence ideals 13.5. Maximal Quasi-Normed Sequence Ideals 13.6. Adjoint Normed Sequence Ideals 13.7. 8-Numbers of Sequences 13.8. Domination Property . 13-9. Lorentz Sequence Heals 13.10_ Sargent Sequence Ideals 13A1. OrIicz Sequence Ideals 13,12. Notes 14. 14.1. 14,2, 14.314.4. 14.5_ 14M.

15. 15.1. 15,2. 15.3. 154. 15.B. 15.6. 15,7.

.......

Opera -f o? 1 deals and Sequenue idvais ST-Operators Wp-Operators and b a-OperAt,on ea-Operators Limit Order Small Operator Ideals Notes

...

Operator Idea.18 on Hilbert Space!? ... Operator Ideals Proper Operator Ideals Operator- Ideals and Sequence Ideals Domination Proporty

e Operators Extensions of Operator Ideals

Notes

168 168 171 176 177

178 1. 78 179 180 181 182 183 184 185 188 188 189 189

190 190 192 197 ... 199 204 205 206 206 207 208 209 212 218 220

PART 4 BASIC EXAMPLES OF OPERATOR IDEALS

16. 16./. 16.2. 16.3. 16_4. 16,5.

Family Ideals on Ranach Spaeu Family Ideals Absolutely p-Sununable Families Weakly p-Sumrnable Families Mixed (a, p)-Sureariable ramifies .

Notes

.

223 223 224 224 225 227

Contents

15

228 Absolutely Summing Operatom 17. 9 2S 17.1. Absolutely (r, p, 0-Summing Operators 23,0 17.2. Absolutely (r, p)-Summing Operators 231 17.3. Absolutely r-Sum ming Operators 236 0-Dominated Operators 17.4. (p, 239 17.5. Special Absolutely (r, p, 0-Surnra ing Operators 240 17.6. Relatio-nships to Other Operator Idea& 9 41 17.7. Notes 18.

Nuclear Operators

18.1.

(r, p, 0-Nuc 1 ear Opera 'Lori

243

r- Nuclear Operators 18.3. p-Compact Operators

18.2

18.4. 18.5.

(p, 10-Compact Operators uelear Operators Special (r, p, Relation.ships to Other Operator Ideals

18.7. 18.8.

Strictly Nuclear Operators Notes

19.

Integral Op 6nTIOTS

252 253

,,,,,,,,,

19.1. (r, p, q)- integ ra I Operai org 19.2. r-Integrai Operat.icos 19.3. p-Factorable Operators 19.4. (p, q)-Factorable Operators 19.5. Multiplication Theorems 10.6. Division Theorems 19.7. Notes

21.1. 21.2. 21.3. 21.4. 2L5.

152

261 262 2E3 265 271 274. 276

277 21E

20. Nng Operatore 20.1. (s. p)-Mixing operntors 20.2 , Multiplication Theorems 20.3. Division Theorctna 20.4. Notes 21.

255 2 _

and Cotype Stable. Laws Operators of (8, »Type . Operators of Ultra (, p)-Type Type

Operators of (2, M-Cotype ates

22. Operators in L 1 Spaces 22.1. Operators in Hilbert Spaces 22.2. Lp Spaces 22.3. Bausch Spaces of (8, p)-Type and (2, p)-Cotype 22.4. Absolutely r-Summing and r-Integral Operalors 22..5. p-Dorninatcd and p..raetorable Operators 22.6. Absolutely (r, p)-Siimraing Opeyators. 22.7. Embedding Maps of 6'oboley Spaces 22.8. Note..

279 2'79 285 287 287 288

'fr91 294 996 298

-

-

299 :3,103 304 307

aa 320 . 322

Contents

16

23. 23.1. 23.2. 23.3. 23. 4 . 23.5.

323 323 324 327 329 331

LUST Pheory Absolutely i-,Sti/IIMillg Operators co-Nuclear Operators cc-Integral Operators Mu Itiplication Theorems No tes

PART 3

Ah_PPLIVA.TIONS

24. 24.1. 24 2 . 24.3. 24.4. 24.5. 24,6, 24.7.

Decomposing Operators Measurable Veetor PuntAions Radon-Nikodyru Operators Vector Measures Decomposing Operator. p-Decomposing Opeinatorg .............

25. 25, 1 . 25.2. 25.3.

Radonifyiqg Operators

Multiplication Theorems Notes

?3.4.

2:5. 25.6. 25.7.

26.1. 26.2.

26.3. 26.4. 26.5. 26,6.

26.8.

....

345 347

348 349 350 352 355 355

Notes

357

Rif:Az Thzory Ries zDecomposition .... Spectra/ Theory in Bartach Algebras

358 356

0-Isomorphisms Quasi -Compac t Operators Wen Operators! 0.Injections and 0-Surjection Ideals of Riesz Operators Notes

Fredhobit Thcorg Fredholm Predhol rn Detcrmi [La ii t.

28. .28.1.

Structure Theory of Ballad& Spar(A Printiple of Local Reflexivity Geornetrk Parameters Related to Operator Ideals Isomorphic Chankoteri7ations of Clansteal Banach Space , The Stru.ctitre of Certain Ideal Components

,

339

Probabilities on Topological Itausdorft Spaces Cylindrical Sets Cylindrical Probabi I iti p-Radunifying Operators Gaussian Cyliiidrical Probabilities Wiener Probability

27, 27.1. 27.2. 27,3_ 27.4. r

2g.2 28.3_ 28.4.

a2,5 335

Related Operator's Distribution of Cfirtraetevistic Values Notes

359 359 360 361

......... 363 2155 a66 387

3f2i 370 375

3:11i 382 383 383 386 388.

Conta.nta 28.5.

Miscellaneous Results

!S6 Notes Operator Ideate on Locally Uontrex 6paces Locally C-I'Dnvex Spaces 29.2. Uperator Micah and space Idoa 29.3. Injective. and Surjective Ideals 29.4. Cartedam idea's 29.5, Extensions of Operator Ideals 29,0. tirotnertalecu Space ideak 29.7. Nuclear Locally convex Spaces 29.8. Strictly Nuclear Locally Convex Spaces 29 9 :',Totes

29,1.

17 389 191 3R7S 393 395 396

196 397

4ut; 402 404405

igpitogu

406

Bibliography 1. Textbooks, Monographs, and Leeturc. Notes 2. Classical Papers 3. Papers 4. References added in Proof

409

409 413 415 437

Index of Axioms

439

index of Operator Ideals

439

inciez of Family icleal8

441

index of Scalar Sequence Idcale

Index of •rocedurgs

irdex of Exteasion& index of Symbols

index of Subjects

1L.Mr31. S.0. 11"

0.75/VW%

ix I,

441

442' 442 443 447

Part 0:

Preliminaries

-4.

Operators in Banach Spaces

J n this preliminary chapter the basic notations are established. Moreover, we collect

a few elementary facts which are used later on. For Banach space textbooks we refer t o [DAY], [DUN], and [TAU].

-4.1.

Banaeh Spaces

-4.1.1. The scalar field A/ is either the real field 9 or the complex field V?. A.1.2. In the following E , F , and G always denote real or complex Banach spaces. Their norms are M ritten as /I./). The class of all Banach spaces is denoted by L.

8.1.3. Every Banach space E is a linear topological Hausdorff space with respect to the n o r m topology. The closure of a subset X is denoted by 1.The closed unit bull of E is defined as IJ, := ( X E E : ljxil 5 1). 8.1.4. The identity map of E is denoted by I E .

A.1.5. Let N be a closed linear subset of B. Then iM becomes a Banach space with respect to the norm defined by restriction. Every Banach space obtained in this way is said to be a subspace of E. The embedding map from M into E is denoted by J 5 , or simply by J . A.1.G. Let N be a closed linear subset of E. Then EjN beconies a Banach space with respect to the norm defined by

,x(N)ll := inf { 115 - q,j,: r0 c N } , where x ( N ) denotes the equivalence class containing x. Every Banach space obtained in this way is said to be a quotient space of E. The canonical map from E onto EjN is denoted by Q", or simply by Q.

A.1.7. The collection of all finite diniensional subspaces and all finite codimensional subspaces is denoted by Dim ( E )and Cod ( E ) ,respectively.

A.1.8. Let E be an n-dimensional Banach space. Then we write dim ( E )= n.

-4.2.

Functionals

-1.2.1. All functionals a on a Banach space E are supposed to be bounded and linear. The value of a a t the element x is denoted by (2,a ) . A.2.S. The set E' of all functionnls defined on E becomes a Banach space with respect to t,he norm I

~~u~~ := sup ( / ( A ,a)l : z

E LTL).

This is the so-called dual Bnnach space of E.

22

Part 0. Preliminaries

A.2.3. The weak E-topology of E' is generatcd by the system of semi-norms p X ( a ):= sup {I(%, a)[: x E XI, where

X ranges over all finite subsets of E.

If the sequence (an)tends to a in this topology, then we write a

= E-lim

a,

n

A.2.4. The bidual Banach space E" is the dual Banach space of E'. If x E E , then a --f ( 2 , a ) defines a functional KEx on E'. I n this way we obtain the evrrluation map K E from E into E". Obviously lKExll = 11211 for ail z E E .

9.2.5. The weak E'-topology of E is generated by the system of semi-norms p A ( z ):= sup (](a, a ) /: a € A ] , where A ranges over all finite subsets of E'.

If the sequence

(2,)tends

to z in this topology, then we write x = E'-lim x,. n

Referring to this topology we shall use concepts like weak convergence, weak compnctness, etc. R e m a r k . Since K Eis one-to-one, we may identify E with a subspace of El'. Then the weak El-topology of E coincides with the weak E'-topology induced by E".

A.3.

Operators

A.3.1. All operators S acting between Banach spaces are supposed to be bounded and linear. The value of S a t the element x is denoted by Xx. A.3.2. The set 2 ( E , F ) of all operators from E into P becomes a Banach space with respect to the so-called operator norm

llSlj := sup (IJSzll:2 E U,} . The class of all operators between arbitrary Banach spaces is denoted by

2.

A.3.3. The weak operator topology on 2 ( E , F ) is generated by the system of seniinorms p x . s ( S ) := SUP { ~ ( S X h)/, : x E X,b E BJ, where X and B range over all finite subsets of E and F', respectively.

A.3.4. If S E e ( E . F ) . then we denote the null space and the range by N ( 8 ) and M ( X ) , respectively. Proposition. Let S E B(E, F ) . If Q denotw the quotient map from E onto Ea := E / N ( S ) and J denotes the embedding rnup from Fa := M(S) i&o F , then there

A. Operators in Barinch Spaces

23

e&ta a factorization

E-

Ls

,F 4

J

EO

+Po

SO

sicch that So6 B(E,, F,) is one-to-one and M(S0)= F,. Moreover, w e have liSoli = IISll.

A.3.6. For every operator S E @E, F ) the dual operator S’

(SZ,b) = (L, S’b) for all

2

f!(F‘,E’) is defined by

E E and b E F’.

It follows from the Hahn-Banach theoreill that IIS/1= ilS’ll. 6.3.6. The bidual operator S” is the dual operator of S‘. K e have the important formula S“lCc = KFS.

6.3.7. Finally, let us mention that ( K E ) K,, ’

A.4.

=

IEt.

Families

A.4.l. Let I be an index set. The set I(E, I)of all bounded families x E for i 6 I , becomes a Banach space with respect to the norm .T,

= (xz),where

1 1 ~ 1 1:= sup (lix,il:i E I } . The class of all bounded families in arbitrary Banach spaces is denoted by 1. R e m a r k . By choosing the above notation we want to emphasize the analogy with L (class of all Banach spaces) and B (class of all operators between Banach spaces). See also (3.1.3.

d.4.2. I€ s = (ci)and S E $ ( E , F ) , then we write sz := 2 = (z,) E I(E, I).

(GJ,) and

SX := (Sz,) for

6.1.3. Let n be any one-to-one map from I , into 1. Then \ye define a canonical injec:= (zZ+(&)), where xB := 0. The canonical tion J, from I(E, I,) into I(E, I) by Jn(x2,) surjection Q, from I(E,I ) onto I(E, I,) is given by Q,(s,) := ( L ~ ( ? ~Clearly, )). Q,J, is the identity map of l(E, I,) and J,Q, is a projection in I(E,I ) .

-!.*A. As usual. me denote the k-th utLit furnily by e, = ( e z k ) , where &,k is KronPcker’s symbol. Moreover, if M is any subset of I , then eAf = ( e ? ) ,with e , = 1 for f AMand ei = 0 for i pC M , is the characteristic family of M. ~

A.P.5. Let s ( I )be the system of all finite subsets i of I . Obviously ? ( I ) is directed tipwords with respect to the natural ordering.

6.4.6. A family (q), where LT, 5 E for i E 1, is called summable if the directed system of all finite partial sums converges in E. Then we write

24

P a r t 0. Preliminaries

where i 3 ( I ) . Further inforination about summable fanijlies can be found in [PIE] and [SAF,].

-4 family (zJ is said to be weakly suwzmubZe if a11 scalar fanlilies ( ( z t , a )with ) u K’ sunimxble.

a1e

W a r n i n g . -4 weakly suniinable family need not be summable in the weak topology.

-1.4.7. A family (x~), where i I, is called absolutely summuble if the scalar family (\lxtil)is surnmable.

R e m a r k . Every absolutely snriiinable family is also summable. However, as the Dvoretzky-Rogers theorems says, the conrerse implication holds only in finite dirnensional Banach spaces; cf. 17.2.7.

B.

Special Operators

I n this section we consider some special classes of operators in Banacli spaces and summarize their main properties. l o particular, the trace is defined for finite operators. We further introduce the concepts of a (metric) injection and a (metric) surjection. Finally, projections and complemented subspaces are considered.

B.l.

Finite Operators

B.l.l. An operator S S(E,F ) is called finite if the range N ( S )is finite dimensional. Then rank(S) is defined to be the diniension of M ( S ) . The class of all finite operators is denoted by

5.

R.l.3. Let no E E’ and yo E F . Obviously a0

0yo : z

-+

(2,UO)

yo

is a finite operator, and we have //aO (3yo\/= ~ ~ Ilyoi/. a o ~ ~

B.1.3. An operator S E S ( E , F ) is finite if and only if i t admits a so-called f i n i t e representution n

x =2’ui

0y;,

I

where a,,

...,a,, E E’

and y,,

...,yn C F .

n

B.1.4. Let S = ,r ni @ ziE g(E,3).Then 1 fl

trace ( 8 ):= 2 (xi,u i ) 1

does not depend

011

the special choice of the finite represention.

Lemma. If S, S,, S,

E g ( E . E ) and j. E .X, then

+

trace (8, 8,) = trace (8,)+ trace (&), trace (28) = 3. trace (8) nnd trace (S) = trace (8’). Moreover, we have trace (8L)= trace (LS)whenever S E 2(E.F ) and L g g(B’,E).

B.1.5. An operator J‘ E g ( E , F ) , where E and E’ are finite dimensional, is said to be eleni entaty .

26

Part 0. Preliminaries

B.2.

Isomorphisms

B.2.1. An operator T E B(E,F ) is called a n isowiorphism if there exists an inverse operator T-l E B(F, E). This means that T-IT = IB and TT-1 = I,. According to Banach's theorem a n isomorphism can be characterized as a one-to-one operator from E onto F . If /IT[/= [IT-li[= 1, then T is said to be a metric isomorphism. B.2.2. The Banach spaces E and F are (metricnlly)isomorphic if there is a (metric) isomorphism I E B(E, F ) .

B.3.

Injections and Surjections

B.3-I. Let T E B(E, B'). Then the injection modzilus is defined by j ( T ):= sup ('c 2 0: ]lTxl/2

t

11xil for a11 x E E l .

In particular, we put j ( 0 ):= 0.

B.3.2. An operator T E B(E, F ) is called an injection if j ( T ) > 0. Clearly, an injection can be characterized as a one-to-one operator from E into P with closed range. If llT/[= j ( T )= 1, then T is said to be a metric injection. This means that llTxli = [js/1 for all x E E .

B.3.3. P r o p o s i t i o n . The operator T E B(E,F ) i s an injection i f and only i f it admits o factorization 1' = JT,, where T oi s a n isomorphism and J denotes the embedding nmp from F , := M ( T ) into F. I n th.is case, j ( T ) = B.3.4. Let T E B(E, P ) . Then the surjection modulus is defined by

Q(T):= SUP ('C 2 0: T(U,) 2

ZUF).

I n particular, we put q(0) := 0.

B.3.b. For later use we prove the following Lemma. If T E

B(E,P ) , then

Proof. Denote the right-hand supremum by q(T).Obviously we have q(T)2 q(T'). To check the converse inequality we may suppose that H(T)> 0. Let y E UF and 0 < P < 1. P u t 'C := (1 - E ) H(T) and y1 := y. Choose inductively xl. x2,... E such that I/yk- t l T x k l /5

E~

a d

IIxA

5 IkjAI,

k- 1

wherc yk := y

---

2 zlTx,. Then [jxl[l5 jlylll 5 1. Moreover.

i=l [IZkIl

5 [llJk/J=

- zlTq-,lj

5 Ek-1

for k = 2 , 3. ...

B. Suecial Owrattors

27

m

03

1 1 ~ ~ 151 (1 - &)-I. Put z :== ,\7 g. Then we have

Hence 1

1 a3

y = 2’ t-lTxk = t l T z

and

llzl/ 5 (1 - E ) - ~ .

1

This proves that T ( U E )2 (1 - F ) rUF. Therefore p(T)2 (1

a(T).

- E ) ~

B.3.6. An operator T E S ( E , F ) is called a surjection if q(T)> 0. Clearly, a surjection can be characterized as a n operator from E onto F .

If llT[l = q(T)= 1, t,hen T is said to be a metric surjection. This means that T transforms the open unit ball of Id onto the open unit ball of P . B.3.7. P r o p o s i t i o n . The operator T E B(E, F ) i s a surjection i f and only i f it admits a factorization T = To&,where T , i s a n isomorphism and Q denotes the canonical map from E onto E, := E / N ( T ) .I n this case, p(T) = lIT~’l]-’. B.3.8. We now describe the duality relations between injections and surjections. P r o p o s i t i o n . Let 1’ E B(E, F ) . Then p ( T ’ ) = j(T) and

j(T’)= q(T).

Proof. (1) Let 0 < 7 < j(T).By definition ljTzll 2 t 1/xl1for all 2 c E. For every a E UE,, the equation (y, 6,) := (T-ly, a ) defines a functional bo on M ( T )with Ilboll 5 t l . Take an extension b E F’ such that llbll 5 rl.Then we have a = T‘b. Consequently T‘(UF,)2 zU,, and therefore q(T’)2 j(T). (2) Let 0 < t < p(T’). For x E E choose a E UEpsuch that l(z, a)l = Ilxll. Since l ” ( U p ) 2 tUE.,we can find b E U F ,with T’b = ta. Consequently

llTzll 2 ~ ( T zb)l ,

=

I(z, T‘b)\

= (2,7.)

=tl\~/\.

This proves that j ( T )2 q(T’). (3) By the same method we obtain j(T’)2 q(T).

a

(4) Let 0 < t < j(T’). Suppose that there exists y E UF with t y Y’(UE).Using a well-known separation theorem we can find b E F‘ such that I(q, 6)l > 1 and I(Tz, b)l 5 1 for all 2 E U,. Then

IIT‘bjJ= SUP {I(Tx,b)l

X E

U E ]5 1 < I(ty, b)j 5

t

IiblI.

This contradiction implies T(U,) 2 t U p . Finally, from B.3.5 it follows that

dT)2 j V ) . B.3.9. As a n immediate consequence of B.3.8 we formulate the following stetements. P r o p o s i t i o n 1. A n operator T E S(E,F ) i s a (metric) injection i f and only i f !l E’’ 2(F,E’) i s a (metric)’surjection.

P r o p o s i t i o n 2. An operator T E 2 ( E , F ) i s a (metric) surjectioTL if and d y i f

T‘ E 13(F‘,E’) i s a (metric) injection.

88

Part 0. Preliminaries

12.3.10. The next assertions can be derived from B.3.9. For other proofs we refer to [Ki)T, p. 2821. P r o p o s i t i o n 1. Let Af be a subspace of E. Then M’ and E ’ / X O ,

where

-Ifo := ( a E E’: (x.a ) = 0 for x E M ) , are metrically isomorphic.

P r o p o s i t i o n 2. Let N he a subspace of E. Then ( E / N ) ‘ and NO, where := ( ( I E E’: ( 5 , a ) == 0 for x c N ) , are metrically isomorphic.

NO

B.3.11. Finally, we prove two elementary inequalities which are very useful in perturbation theory. Lemma. Let T , A E C(E,F ) . Then

j(T

+ A ) 5 j(!i!’) + lIAl1

Clltd

y(!i!’

iA ) 5 y(T) f I$${.

Proof. We have

1lT.A 2 IITz Consequently j(Y’

a(!!’ B.4.

+ Ax]]- jAlrl/ 2 j(T + A ) ll2Il - IlAlI Ilxll.

+ A ) 5 j(!Z’)f \lA\l. Moreover, it follows from B.3.8 that

+ A ) = j(T‘ + A’) 5 j(T’)+ l]A’i = g(T) + I/AIl.

Projections

B.4.1. An operator P

c 2(E,E ) is called a projection if P 2 = P .

+

U.4.2. Let M and N be subspaces of E such that E = A! N and M n N = (0). Then E is said to be the direct sum of M and N . Obviously, every x E E admits a unique decomposition x = x , + ~ xN with x, E M and xN E N . By setting Px := xhf we obtain an operator P E l?(E,E ) which is called the projection from E onto M along N ; cf. [TAY, p. 2421. We mention that 1M and EjI? are isomorphic. Conversely, for every projection P t 2(E,E ) we know that E is the direct sum of M ( P ) and N ( P ) . B.4.3. A subspace M of E is called complemented if there exists a subspace N such that E is the direct sum of ill and N . Clearly. N is a complemented subspace, as well. We write E F if the Banach space E is isomorphic to some complemented subspace of F .

<

B.4.4. Lemma. A subspace A2 of E is complemented if and o d y if there are operator8 J E 2 ( N ,E ) and Q E 2 ( E ,M ) such that I , = Q J . In this case we may suppose J to be the canonical injectwn from M into E. P r o o f . If IAM= QJ, then P := JQ is a projection with 171 = BI(P). Conversely, suppose that 3 f is complemented. Then there exists a projection P C(E,E ) such that M = M ( P ) .Regarding P as a niap from E onto M we obtain a n operator Q E 2 ( E , M ) with I M = QJ, where J denotes the injection from M into E .

R. Special Operators

29

B.4.5. Using the same method we can check the dual Lemma. A subspuce N of E i s cornpleniented i f and only i f there are operators

J E a?(E/N,E ) and Q E 2 ( E , E / N ) such that IEi8 = QJ. I n this case we may suppose Q to be the canonical surjection from E onto BIN. B.4.6. Every subspace M of a Hilbert space I1 is coiiiplemented. If N := {z E H : (z, xo) = 0 for all z, E M ) , then H is the direct sum of M and N . The corresponding projection is called orthogonal. Here (. , .) denotes the scalar product in H . B.4.7. Lemma. Every finite dimensional or finite codimensional subspace of a n arbitrary Banach space i s complemented.

B.4.8. We now prove Auerbach's Lemma. Let E be n-dimensional. Then there exist xy, ..., xt such that \lz:l/ = 1 ) llutil = t , and

E and a:) ..., a," E E'

(zt,at) = F;,.

Proof. We consider a basis xl, ..., x, of E and set

..., a , E U E , . Since the function d is continuous, there are a:, ..., a: E UErfor which d attains its maximum a0 > 0. Therefore we can find xy, ..., x: E E such that d(a,, ..., a,) := (det ((xi,ak))i for a,,

n

2 (xt,up) x; = x, 1=1

for

i = 1, ..., n

Then {x:, a:) = E , ~ .It follows from

that det ((xt,a;)) det ((x;,a k ) )= det ( ( x i , a,)). Hence jdet ((xy, u,))l 5 1 for a,,

..., a, E

By setting a j := a and ak := a: whenever k i(z:,

UEt.

+ i we get

a ) / = ldet ((xy,cck))I 5 1 for a E bTE,.

Consequently llz'$ 5 1. Now 1 = (xp,a:) 5 IIxpII liarll l,a:ll = 1.

5 1 implies llzpil = 1 and

B.4.9. By the preceding leninia we now obtain an important improvement of B.4.7. Lemma. Let $1 be a n n-dimensional subspace of E . Then there exists a projection p E f!(E,E ) with M = N ( P )and ljPll 5 n. a:,

Proof. Let J denote the injection froin iM into E . Take xy, ..., xt t M and ..., a t E M' a s constructed in B.4.8. Using the Hahn-Banach theorem we can find

30

Part 0.Preliminaries

al,..., a,, E E' such that J'ai = a: and llaal[= 1. Put xi := Jx!. Then P :=

is the desired projection.

n

2 ai @ x 1

R e m a r k . I n 28.2.6 we will prove that there is even a projection P E e ( E ,E)with This is the best possible result.

&f = H(P)and ljPjj 5

B.4.10. Lemma. Let N be a n n-codimensional subspace of E. Then, given there emkts n projection P E 2 ( E , E)with N = N ( P )and l[Pll (1 E ) n.

+

E

> 0,

Proof. Let Q denote the surjection from E onto EIN. Take xy, ..., x,"E E / N and a;, ..., a," E (EIN)' as constructed in €3.4.8. Choose xi E such that &xi = xp and /lsili

1

+

n E.

Put ai:= &'a:. Then P := 2 ai@ xiis the desired projection. 1

B.4.11. Let El and E, be Banach spaces. Then the set El x E2 of all pairs (xl, z,), where x1 E El and x2 E E,, becomes a Banach space with respect to every norm I/(. ,.)I[ such that II(xl, o)ll = jlxl[[and l/(o, x,)l/ = /]xll/.Special norms are w 1 , z2)IL

+ llxzll

:= 11~111

and

lI(x1,

x2)Ilm

:= max

ll~Zll)*

( l l ~ l ~ l 7

The Banach spaces El x E, obtained in this way are called Cartesian products of El and E,. Put Q I h X d := 5 1 , &2(% %?I := Zz. JlZ, := (Xl, o), J2X2 := (0,Zz), Then P, := J,Q, and P, := JzQz are projections in El and P1P2= P2P,= 0. Pz = I,,,

P,

+

x E,

such that we have

6.

Special Banach Spaces

First we introduce the classical Banach spaces of scalar functions and describe their lattice-theoretical characterizations. Then the extension property and the lift,ing property of Banach spaces are defined. Finally, we construct the so-called Cartesian lp-product of Banach spaces. The theory of Banach lattices is presented in [SAFJ. More specific results about classical Banach spaces can be found in [LAC], [LIN], and [LIT]. We also refer to

[SED].

C.l.

Classical Banach Spaces

C.1.I. Let (Q,p ) he a measure space. If 1 5 p < 00, then LP(Q,p ) denotes the BanaGh space of all p-measurable scalar functiolzs f on D such that lflP is p-interable. The norm is defined by

ilfil,

:=

{jIf(w)lPdP(w))'/P-

Moreover, L,(Q, p ) is the Banach space of all bounded p-measurable scalar functions f on 9, and we set

\\fll,

:= ess-sup { / f ( w ) lw : E

QJ.

Actually the elements of LJQ, p) are equivalence classes of functions which coincide almost everywhere. R e m a r k , We use the same definition in the case 0 complete p-normed linear space.

< p < 1. Then Lp(9,p) is a

C.1.2. We write Lp[O, 11 simply, if the underlying rnewure space is the interval [o, 11 equipped with the Lebesgue measure.

c.1.3. Let I be a n arbitrary index set. The Banach space of all absolutely p - s u m d l e s ~ a l a rfamilies x = (ti), where i E I, is denoted by l p ( I ) .We put

Moreover, l,(I) is the Banach space of all bounded scalar families x = (ti)with the norm \lZllm :=

sup { / t i ] :i E

I].

We write I, and 1; simply, if the underlying index sets are (1,2, ...I respectively.

and (1, ...,n ] ,

C.1.4. If 1 < p < OQ, then the dual exponent p' is determined by l / p

+ l/p' = 1.

.>u *I.)

Part 0. Preliminaries

P t is convenient to put p’ = 00 if p = 1 and p’ = 1 if p = 00. Frequently the dual exponent will be denoted by p*. P r o p o s i t i o n . Let JI E Lp.(Q,p ) with 1

< p < 00.

Then

/f(.,, 9 b ) d P b )

i f ,s>:=

s2

defines a functional g on Lp(Q,p). The correspondence obtained in this wa y is CI metric isomorphism between Lpt(Q, p) and Lp(Q,p)‘. For a large class of measure spaces, the above statement remains true also in the casep = 1.

C.1.5. Let K be a compact Hausdorff space. Then C ( K )deiiotes the Banach space of 011 coiLtinuous scnlar functions f on K . The norm is defined hy

Ilfl ,

:= sup {lf(w)l: 0.J

e Kl

*

C.1.6. We write C[O, 11 simply, if the underlying compact Hausdorff space is the interval [0, 11.

C.1.7. Let I be an arbitrary index set. The Bunnclt space of all zero scdar f a n d i e s x = ( t 8 )where , i E I, is denoted by ~ ~ ( 1 We ) .piit /;xJl, := sup (IE J : i E I ) . Obviously c,(Z) is a subspace of Z=(I). We write co simply, if the underlying index set is { 1 , 2, ...).

C.1.8. Let K he a compact Hausdorff space. Then the so-called Borel 0-algebra B(K) is generated by the collection @ ( K )of all open subsets.

A Borel probability y of K is a measure defined on % ( K )such that y ( K ) = 1. Moreover, y is said to he regular, if p ( B ) = inf (p(G):B & G , G E 8 ( K ) ) for every B

E % ( K ) .The set of all regular Borel probabilities is denotcd by W ( K ) .

We now formulate Riesz’s representation T h e o r e m . Let

,LL

g W ( K ) .Then

( f , P’ .= J f ( 4 44m) K

defines a functional y on C ( K )with Jipil= 1 and ( f , p\ 2 0 whenever f 2 0. Conv6tseljj every such functional can be uniquely represented by a r e q d a r Borel probability.

C.2.

Abstract L,-Spaces

C.1.1. Let 1 5 p such that //XI

+

< 00.

An nbstract Lp-space is a (real or complex) Banach lattice E

x 2 J J P= l l z , / l P

+

//x211P

whenever

x

A

z q= o .

C. Special Banach Spaces

Every classical Banach space L,(S2, p ) natural ordering.

IS

an abstract /,,-space

33

with respect to tlir

C.2.2. For a proof of the follow~ingstatement we refer to [LAC', p. 1351. The spccial case p = 1 was first treated by S. KAKCTAXI [ 2 ] . The general result goes back to F. BQHNE~BLUST [I].

PI-o p o s i t i o n. Ever?/ ubstrlt r t L,-sprrc~ ie met ricnlly (I I rLd Zutfice) i.souiorpli ic lo L,,@, p ) for some measure spiice (Q, p). C.2.3. An nb.s,.trirctL m - s p m is ,z (real or complex) Banach lattice E siich that

+ xLll = max (I~.zJ, 1 1 ~ ~ 1 1 ) if

\In,

n, A .c2

= o.

Every classical Banach space C ( K ) is an abstract L,-space natural ordering.

with respect to the

C.2.1. The next result is proved in [LAC, p. 591 and [SAF,, p. 1041. See also S.KAKETANI

[3].

P r o p o s i t i o n . Euery abstract L,-spctce with unit i s nwtricnlly (and lattice) t s o morphic to C ( K )for some compact Hausdorff space K . R e m a r k . An element e E E is called a tinit

C.3.

if

Extension Property and Lifting Property

C.3.1. A Banacli space P possesses the extension property if for every injection J E .(3(Eo,E ) and every operator SoE i?(E,,3') there exists an extension S E e ( E , PI:

The metric extemwn property means that for every metric injection J E i?(Eo,E ) and every operator So E e ( E o ,F ) we can find S E e ( E , F ) with So = S J and ilXi1 = //So'].

C.3.2. Now the most iinportant examples are given. P r o p o s i t i o n 1. Let I b e u n y index set. Theit 1,(I) hus the metric e.dension property.

P r o o f . Every operator 8, E g(E,, I,(I)) can be written in the form S,zo = ((.co, a:)) foi. co E E,, where a: E EA. Moreover, i[Soll= sup {l\c('ll: i E I } . Let J C f?(E,,K ) be A metric injection. Then there are u, E E' such that a: = J ' a , and llu'l = llut,i. Nou Sx:= ( ( 5 ,a,)) for x E E defines the desired extension. For a proof of the following statement we refer to [SEI), p. 4521 3 I'ietcch, Operator

34

Part 0. Preliminaries

P r o p o s i t i o n 2. Let ( Q , p ) be any measure space. Then L,(B,p) has the metric extension property.

C.3.3. Let E be a Banach space. Then we put p n j

._ .-

zw(uEf)

and

JEx

:= ( ( 2 ,a)) for

2

E E.

Clearly JE is a metric injection from E into E i n j . I n this way every Banach space can be identified with a subspace of some Banach space having the metric extensio:? property.

C.3.4, The next result follows immediately from the preceding statements. Theorem. A Banach space has the extension property i f cc?donly i f it is ismnwphic to a complemented subspace of some Banach space l&).

C.3.5. A Banach space E possesses the lifting propeity if for every surjection Q E g ( F , Fo)and every operator So E B(E,F,) there exists a lifting S E B(E,F ) :

The metric lifting property means that, giren E > 0, for every metric surjection Q E g ( F , Fo) and every operator So E 2(E,F,) we can find S E g ( E , F ) with So= QS and IlfJIl 5 (1 IlfJoll.

+

C.3.6. P r o p o s i t ion. Let I be any index set. Then Zl(I)has the metric Eiftingproperty. P r o o f . Every operator SoE B(ll(I),Fo)can be written in the forin #,(ti):= CSiy: for ( t i ) E Zl(I),where yp E Fo. Moreover, llSoll= sup [/i$ll: i E I).Let Q E B(F,Fo) be a metric surjection. Then there are yi E F such that y: = Qyi and IlyilI 5 (1 ~)IIy;li. Now S ( t , ):= 2 f t y i for (ti)E Zl(f)defines the desired lifting.

+

C.3.7. Let E be a Banach space. Then we put Esur := 1 1 ( u E )

and

QE(tz):=

t z x for

(5,) E Zl(UE).

UB

Clearly QE is a metric surjection from Esur onto E. I n this way every Banach space can be identified with a quotient space of some Banach space having the metric lifting property.

C.3.8. The next result follows immediately from the preceding statements. Theorem. A Banach space has the Ziftingproperty i f and only i f it i s isomorphk to (a complemented subspace of) some Bannch space Zl(I). R e m a r k . Indeed, every complemented subspace of Zl(I)is isomorphic t o Zl(I,) for some index set I,; cf. [LAC, p. 1801.

35

C. Special Banach Spaces

43.4.

Cartesian Z,-Products of Banach Spaces

C.i.1. Lct ( E , )with i E I be a family of Banach spaces. Given 1 s p < co,we define the Cartes-inn E,-product Zp(Ei,I) to be the set of all families x = (xi),where xi E Ei for i E I , such that (llxill)E 7p(I).Obviously Zp(Ei,I) is a Banach space with respect to the norm

The definitions of lm(Ei,I ) and co(Ei,I)are analogous. We write lp(En)if the underlying index set is (1,2, ...). The canonical operators Jk and Qk are defined by J@k

:= (EikXk)

for

xk

E Ek)

where zik is Kronecker's symbol, and Qk(Xi)

:= xk

for

(Xi)

C.4.2. P r o p o s i t i o n . Let 1 (z,a) := (zi, ui)

p

E Zp(Eij I ) .

< co.If

(ai) Z,,(E:, I ) , then

I

defines a functional a on l,(Ei, I ) . The correspondence obtained in this way is a metric isomorphism between lp,(ES,I)and lp(Ei, 4'.

C.5.

Intermediate Spaces

C.S.l. Suppose that (Eo,E l ) is a couple of Banach spaces which are continuously embedded into some linear topological Hausdorff space U. The norms of Eo and E , are denoted by /I. 1, and 11, respectively.

+ El and Ed :=E , n El. Let := inf (Ilx,I],+ llzliil: xo + z1 = x,zo E E,, xl E El}

Put Et.:= E,

Il&and

for x E E,Y

IIXIIA := {IIxIIoj IlXIIiI for 5 E E d C.5.2. A proof of the following lemma can be found in [BER, p. 241, [BUT, p. 1651, and [TRI,p. 181. Lemma. Ez and E d are Banuch spaces with respect to the norms I~.IIcand I ] . ~ / A , respectively. c.5.3. A Banach space E is called a n intermediate space of {E,. E l } if Ed and the corresponding embedding maps are continuous :

E & EC

36

Part 0. Preliminaries

C.5.4. An intermediate space E of { E,, E l } possesses K-type 0 if

l[S:E

--I

Fil 5

11s:E',

----L

F1/'-0

11s:El --' 3''I

(K)

for every operator S E B(E',, F ) and all Banach spaces F , where 0 2

c.5.5. 1,eninia. .-In interrneditrtc spme inf

{B,

+ o1 \ \ n l [ ~ l + : ~ xl o -= s,

B,,,

5

1.

of {E,, El) htrs K-t?jpe 0 if rrnd only if

l~.c0\\,

for all x E E and

F)

E

E,, x1 E E,} 5 q-80; 11x1~

o1 > 0

P r o o f . To check the sufficiency of the above condition we consider a n opcrator S c B(E,, P).P u t oo:= l[S:E',3 Pi/ and o,:= IjS.F;, --f Pi!.Let x E IC and E > 0 be given. Then we can find x, E E, and x1 E E , such that x, n, = x and

-

l\Sx;l= I/S.r,

+ Sz,ll

5 o, /~.co/jo + o1 I~xl~', 5 (1

r c)

oh-'uf I I L I ~ .

This proves (K). Suppose that E is of K-type 8. By C.6.2. the set P:= E , f ti:, beconics a Banach space with respect to the norm

+ o1 / ! x ~xo~ t-~ .cl~ = : x , x0 E Eo, x , E E l ) .

j ; ~ , ':= ~ inf { o , j;xoljo

Let I be the identity map from E, into F . Thm we have 11:E;, --b 8 ' 1 5 oo and 111:El --f PI/5 o1. Hence 11T:E + PI1 5 ~;-~o!. This means that ll.ri/p 5 u: 'o; 'ix" for all 1: E E. So the required condition is fulfilled. R e m a r k . See also [TRI, pp. 61-62].

C.5.6. An intermediate space P of {F,,J',} possesses 6-type 0

11s:E --

PI! 2

11s:E --

Fol/l-8

if

11s:E + F,IIO

for every operator S i B(E, F A )and all Banach spaces FJ, where 0

(J1

5 0 2 1.

C.5.7. Lemnia. An inter me dint^^ space P of / F O ,PI)has J-type 0 i f and only i f llyl! I lIyll~--'llyif

for all y E F A .

Proof. The sufficiency of the above condit:on is evident. Its necessity follows if we apply (J) to the operator 1 @ y E B ( X , Eld). R e m a r k . See also [BER, p. 491 and [TRI, pp. 61-62]. C.5.8. We now give the main exanzples; [BER, pp. 106-1101, [BUT, pp. 181 -1871, and [TRI, p. 1281. P r o p o s i t ion. Let (f2, p ) be any o-finite nceasure space. Furthcrmure mppose thnt 1 5 p,, p 1 5 co nnd 1 ' p = (1 - O)/p, O/pl. Then Lp(Q,p ) ZF a n intermedhtr spuce of {Lpo(Q, p ) , L,Jf2, p ) } . iVloreover, L,(O, p ) hus K-type 0 and J-type 0.

+

R e m a r k . All Banach spaces Lp(f2,p ) are continuously embedded into the linear topological Hausdorff space L,(Q, p ) of all p-nieasurable functions.

D.

Operators on Hilbert Spwes

The ninin tools in the theory of operator ideals on HiIbert spaces are the spectral representation theorems described in the sequel.

For Hilbert space textbooks wc?refer to [AHI], [MAU,] and [RIR].

D.1.

Hilbert Spaces

D.l.1. In the following H and K always denote real or complex Hilbert spaces. Their scalar products are written as (., .). D.l.3. We denote arbitrary orthonormal familes by (xi) and ( ? I * ) , whereas ( e , ) is reserved to nmrk the ccoionicml orthonorinal basis in &(I). D.I.3. Every Hilbert space H caii hc identified with sonie & ( I ) .Then the dimension of H is well-defined Ly dim (a):=card ( I ) .Moreover, if H , is a suhspace or a quotient space of H , it follows that dim (H,) 5 dim ( H ) .

D.2.

Functionals and Operators

D.3.1. For every y E H the functional y* is defined by .G --f ( x , y). We have /[y*ll = IlyJ.

D.3.2. Let S E e ( H , K ) . Then there exists an adjoirit operator y) = (x,S*y) for all x E I1 and y E K . that (Sx,

D.2.3. An operator S E e(H,II) is called positive if S x E H.

=

S* E 2 ( K ,H )

S* and (Sx,x ) 2 0

such

for all

D.2.4. An operator c’ E 2 ( H , K ) is said to be purtiully isometric if VV*T’ = T’. Then we have IIVII 5 1. D.2.5. We now describe the so-called polcrr deconiposition. T h e o r e m . Let S c e ( H , K ) .Thenthereorenpositiveoperator 181= (S*S)1!2€2(f1, H) a partially isowetric o j x w t o r 77 E B ( H , K ) sirch that S = 1/15! (~d IS1 = V*S.

G:12d

R e m a r k . Obviously, we have S =- VS*V and S* = V*SV*.

D.3.

Spectral Representation

D.3.1. The next result can be proved by combining the polar decomposition theorem and the well-known spectra! representation theorem of positive operators [DUN, p. 911). Theoreni. Lct S E B ( H , 6 ) .TJLLZ theie ure (I Hilberl space L2(,C2, p ) , partially isometric operators X E E(Lz(Q,p ) , I€) attd Y E B(L,(Q, p ) , K ) as wcll u s (L diagonal

38

Part 0. Preliminaries

operator SoE B(L2(Q,p ) , L2(Q,p ) ) of the form S,f = sf, where s E L,(Q, ,u) and s 2 0, such that the following diagram conmutes: S

H

+K

yT!. D.3.2. We now state Schmidt’s representation theorem for approximable operators ; cf. 1.3.1. A proof of this important result may be found in [PIE, p. 1291. Theorem. Let S E 6 ( H ,K ) . Then there are orthonormal families (xi)and (yi) a8 well m a non-negative family (ai)E co(I)such that

S

=

2 asx;@ yi. I

D.3.3. Immediately from D.3.2 we obtain Schmidt’s factorization theorem which is a special variant of D.3.1. Theorem. Let S E @ ( H ,K ) . Then there aye partially isometric operators X E 2(Z2,H ) am? Y E B(l,, K ) as well as a diagonal qerator SQE f2(12, 12) of the form So(&) = (antn). where (a,) E co and a1 2 a2 2 .. . 2 0, such that the following diagram commutes:

Proof. Observe that in Schmidt’s representation theorem the index set I is at most countable. So, without loss of generality, we can assume that I = (1, 2, ...) and u1 2 a, 2 2 0. Then the partially isometric operators are defined by

---

W

W

X(5,)

:=

2 t,xn 1

and

Y(vn) := 2 my,, . 1

E.

Diverse Lemmas

I n this chapter we collect some import,ant lemmas.

E.1.

Riesz’s Lemma

E.1.L. The following statement is classical and elementary; cf. [SEC, p. 861 and [TAY, p . 961. Lemma. Let M be a subspace of E with M =+ E. Then, givcn 1 so E E such that Ilxoll = 1 srnd /lxo- xlj 2 -for all x E M . 1 -t 8

E

> 0, there exists

E.1.2. The next lemina due to M. G. KREJN/M.A. KRASXOSELSKIJ/D. P. MILMAN [ L ] is very deep. It follows from Borsuk’s antipodal theorem; cf. [KAT, p. 1991 and [ROL, p. 2911. Lemma. Let M , N E Dim ( E ) . Suppose that dim ( M ) < dim ( N ) . Then, given E

> 0. there exists xo E N

E.2.

such that lixo~l= 1 and //xo- zlj 2

1 for all x E H . -

I+&

Dvoretzky’s Lemma

E.2.B. For a proof of the following geometrical statenleiit we refer to [DAY, p. 801 ncd A. DVORETZKY/C. A. ROGERS [l]. Lemma. Let E be infinite dimensional. Then,given a natural number n and there exist xl, ..., z, E E such that llxl[l = -..= llxnll = 1 and

E

> 0,

E.2.2. We now forinulate the leinrna on almost spherical sections which goes back to A. DVORETZKY [l]. Simplified proofs have been given by T. FIGIEL[I], [3], V. D. MILMAN [2], and A. SZANKOWSKI El]. See also T. E’IGZEL/J. LIXDENSTRAUSS/V. D. MILMAN [I]. Lemma. Let E be infinite dimensional. Then, giveia a natural number n and there exists J E i?(l;, E ) sucli that 11412

E .3.

E

> 0,

5 I~Jzll5 (1 4E) jlx/j2 for all x E 7;.

-

Lindenstrauss Rosenthal’s Lemma

E.3.1. At this point we state a special version of the famous principle of local ref1en.idue to J. LINDENSTRAUSS/H. P. ROSENTHAL [I] and improved by w. B. JOHKs o ~ / HP. . ROSENTHAL/M. ZIPPIN[l]. In 28.1.3 we will present a simple proof which goes back t o D. W. DEAN[l]. city

40

Part 0. Preliminaries

I ~ e m m aSuppo,se . that B is finite diwensional. Lpt S E 2(E,3”’)n n d M E Dim (B”). Give?&F > 0, there is R E 2 ( E , F ) such that \\R!l(= (1 F ) IlSIl, (Rs, b) = (Sz,b‘ for all x E E ctnd b E &f (19 well K,Rz = Sx tcheneoer Sx E M ( K F ) .

+

E.3.2. L4s an iiniitediate conseqttcnce we obtain a lemma which is very crucial for the later investigations. Lemma. Suppose that K Oi s finite dinicnsional. Let A E e(E‘, Eh) m i d T E S(P‘,B‘). ‘/’hen,pkerL F > 0, there exists X E B(E’,, E ) such that IlXIl 5 (1 E ) 11-4 [Iand X’T = A T .

+

e(F:,,

Proof. P u t S := A’KEo and M : = M ( T ) . Choose X E E ) such that 11Xll 5 (1 $- F ) IISl] and ( X x o , a ) = (Sx,, a) for all xo E E , and a E M . Then 11Xll5 (1 E ) jlAil and (x,, X’Tb)= (KE,xo,ATb) for all xo E E , and b E F’. This proves that .V’T = A47’. Consequently S is the desired operator.

+

E.4.

Ky Fan’s Lemma

E.4.1. .I collettioii 9of real fririctions @ defined on a set h-is called concave if, h’ airen

el,..., G, E ,Fand

al,..., a n 2 0 such that

n

o1 =

1 , there is @ E 9 satisfying

1 I1

@(.I-) 2

C

‘ L ~ @ ~ ( Zfor )

all .z E K .

1

E.4.2. We now prove an iniportaiit mini-ntux theorem. Leniiiia. Let K be a comp,rct cotzz‘ex subset of u linear topological Hausdorff spacc, and let 9be (1 rmicave colleciiorz of lower semi-continuous conrex real functions @ 071 K . LYuppose that for every 0 E 9 theye exists x E K with @(x) 5 p. T J I c we ~ cun find .yo E K such that @(so) 5 p for crll @ F QFsintultaizeously. Proof. It follows from the lower senii-continuity that

A ( @ ,E ) := {.c E K : @(x) 5 ,o

+

6)

n

> 0. We now show that a11 finite intersections n A ( @ $F,) , 1 are non-empty. P u t P: = { (&, ..., 5,) E 9Zn: 5 p + F ~ and } let Q be the convex hi111 of all vectors (G1(x)... ., @,(x)) with x E K . Suppose that P n Q = 0. Applying a well-known separation theoren) wc can find (al.. . .. A , ) E 9%anti A E d such that closed for @ E 3 and

is

E

n j7

&

I

1 % j --

1,

1 11

5 %,El 5 ni 1

for all

(E,, . . .. tr,)E P ,

and I?

F iz I

3

.... F , ~ ) E (1.

Et 2 n for all

n

Since Lc, E 1’for 1. 5 9 , we h a w So there mists @ E 3 with IL

@(x)

2

>: 1

IJ

5 a. This implies

2 0. Hence

1a, = 1. 1

I2

?,cDL(.C) 2 a

iy,

2 ,r A t ( ? 1

+

t , ) >‘

&I

for all

1’

E

I<,

E. Diverse Lemmas

which is a contradiction. Conseqiiently

PnQ

m

i , == ,P &@t(zk) for all

i = I,

=f

41

0.Take (C1, . . ., t n )E P n Q. Then

..., n ,

k- 1 i7r

IIA

C

where q ,..., 2, E K and ill, ..., i, 2 0 such that belongs t o K . It follows from

= 1. Therefore . r : =

C ilkxhc

1

1

m

0f

E;

2 ‘i

2 &@t(Xk)

==

k-

2 Qa(5)

1

that x E A ( @ Le, I ) for i -- 1, ..., 11. I&

Since K is compact and all finite intcrsections (1A ( @ c c,) , are non-empty, there I

exists xo E I< such that xo E A ( @ ,E ) for all @ E 3 and whenever @ E 9.

E.5.

E

> 0.

Hence @(zo)5 p

Khintchine’s Inequality

E.5.1. By 8” we denote the set of all vectors e viously card ( P )= 2”.

= (E,,

..., E~,) with

E,

== f

. 01,-

E.5.2. For proofs of KIiintchiize’s inequaZit?y we refer to [RXC, p. 1311 and %\’G, [I]. volume I, p. 2131. See also A. KHISTCHINE Lemma. Let 0 (2-e

8.

l(x,e)lP

constarit cp such tlicct

5 cp 11z/I2 for crZl x E .Z:

)

R e m a r k . Suppose that cp is chosen as sniall as possible. Then, in the real a5 well a s in the complex case, we have for

0< p 5 2 ,

cf. TJ. &AGERI;P [I] and R. 31. G. YOUXG[I]. Using the absolute moments int,rodiiced in 21.1.2 (real line) we can write c p = for 2 p < 00. The classical proof yields c p 5 l / [ p / 2 ] 1 .

+

Il.5.3. We now state a converse Khintchine’s iriequality which follows from E.5.X. 1 his estimate was independently obtained by J. E. LITTLEWOOD [I].

r ,

Lemma. There is CI positive constant c, such fhut 11xi!25 c, (Pa 2 i(n,e\i)

for rtZZ

1E

2;.

P

R e m a r k . Suppose that cii is chosen as siuall as possible. Then s. ,J. SZAEEK [l] proved that c, = Moreover, it turns out that K?~i~!cJ~ine’s constant cK does not depend on the choise of the scalar field.

i?.

Part I:

Theory of Operator Ideals

Operator Ideals on Banaeh Spaces

1.

The first chapter starts with the funtianientnl definition of an operator ideal. This c o x e p t is a natural genera1iz:ition of the n-cll-known ring-thcoretical notion. We then introduce sonie spccral operator ideal3 which hare played an important role in the historical development of functional analysis. The finit? and approxiinable a t the beginning of this operators were studied hy E. GouRsAr and E. SCHXIDT rentury, and D. HILRERT introduced the completely continuous operators a t about the same time. The coiiccpt of compact operators goes back t o F. RIESZ’Swork circa 1918. The weakly compact operators were used for the first time hy S. KAKUTAXI and I<. YOSIDAin 1938. The ideal of unconditionally siiirnning operators was studied by A. YELCZY.G.SKT in the early 1960’s.Finally, strictly singular and strictly (1965), cosingular operators were crcated by T. RATO(1958) and A. I’ELCZY&KI respectively. The chapter concludes with the basic relationships between the operator ideals mentioned above.

1.1.

Operator Ideals

1.1.1. Let spaces.

11srecall

that 2 denotes the class of all operators between arbitrary Banach

An operator ideul

U(E,F):=

is a subclass of

2 such that the components

n e ( E ,F )

satisfy the following conditions :

(01,) I,x E N, where 3? denotes the 1-dintensional Banach space. (01,) It follows from S,, AS, E U(F,., F )that S , S, i % ( E , F ) . (01,) If 2’ E e ( E , , E ) , S E U ( E ,P ) , and R E 2 ( F , Fo), then RS7‘ E U(E,,F”).

+

R e m a r k . Operator ideals will be denoted by bold capital Gothic letters.

1.1.2. P r o p o s i t i o n . Lot 21 be linear spnces.

(111

npertrtor idriti. 1’Iiea nll coinpo?mts U(E, lJ) cue

Proof. By (01,)it reniains to show that S E %(I?, F ) and;. E X iinply 2.S E U(E. F). This follows from 2,s= (ZF) S I , and (O12). 1.1.3. We niention without proof an interesting fact.

+

p r o p o s i t i o n . Eoi every operator nlgebru 2(E,E ) let un ideul U(E,E ) { O } be given. Then tiierc exists n uniqite operntor ideul U with the components U(E, E ) if arid o d y if the following condition of COtILpU)C(tibilit!J holds: (V) If T E g ( F . JC), S E % ( E , E ) , W W ~ R € 2(;k:,If’!, the71 Rs7‘ E 2I(f”, F).

46

Pert 1. Theory of Operator Ideals

1.2.

Finite Operators

1.5.1. The class of all finite operators is denoted by 5. 1.2.2. Theorem.

5 b the smallest operator ideal.

Proof. Let a. E E’ and yo E F. Then

Consequently a. @ yo is contained in every operator ideal 9%. Since S E g (E , F ) can be written in the form n

S = C ai 0Yi, 1

we get S U ( E ,F ) . This proves that is omitted.

5

3. The verification of the ideal properties

Approxirnablle Operators

1.3.

..

1.3.1. An operator S B(E,3’)is called approximable if there are Sl,X,,. E g ( E , P) with lim \IS - S,jl = 0. n

The class of all approximable operators is denoted by 6. 1.3.2. Obviously we have the Theorem. Q b an operator ideal.

1.4.

Compact Operators

1.4.1. An operator S E B(E, F ) is called compact if the closed unit ball U , is mapped into a subset S(U,) which is relatively conipact in the norm topology. The class of all compact operators is denoted by a.

W a r n i n g . The phrase “completely continuous” will be w e d for another 0 1 ~ of s operators; cf. 1.6. 1.4.2. The following result is classical; [DUE, p. 4861. Theorem. 5l .is an operator ideal.

1. Operator Ideals on Banach Spaces

47

Weakly Coinpact Operators

1.6.

1.6.1. An operator S E B(E, F ) is called weakly compact if the closed unit ball UEis mapped into a subset S (U,) which is relatively compact in the weak F'-topology. The class of all weakly compact operators i s denoted by

m.

1.5.2. For a proof of the next result we refer to [DUN, p. 4831. Theorem.

is a n operator ideal.

1.6.3. Now a useful characterization is stated. P r o p o s i t i o n . An operator S E g ( E , F ) is we&y compacb i f and only i f every s e p m c e (SZ,~) with x, UE admits a weakly convergent subsequence. Proof. The assertion follows from the fact that compactness and seqmer;.tial iionipactness coincide for the weak F'-topology ; [DUK, p. 4301.

1.6.4. The next criterion is also proved in [DUN, p. 4821. P r o p o s i t i o n . An operator S 6 f?(E,F ) is weakly compact if and only i f the r a v e of S" is contained in the canonical image of F .

1.5.5. By the preceding result for every operator S E m ( E ,F) we can define a P)if S" is considered as a map from E" into E'.

.strongly bidunl operotor Sr E $(E",

Then the following diagram coinmutes:

Completely Continuous Operators

1.6.

1.6.1. An operator S E C(E,P ) is called completely continuous if every weakly conis mapped into a norm convergent sequence (Sxn). vergent sequence (zn) The class of all completely continuous operators is denoted by 8.

1.6.2. Obviously we have the Theorem. 8 is a n operator ideal.

1.6.3. P r o p o s i t i o n . Euery completely continuous operator maps weak Cuucl~ysequences into n o m convergent sequences. Proof. Let S E B(E, F ) , and consider a weak Cauchy sequence (xn).Suppose that (Sx,) is not a Canchy sequence in the norm topology. Then we can find subsequences (znJand (zm,) such that /jSz,, - 8zJ 2 E > 0 for i = 1,2, ... On the other hand,

E'-lifn (2, - zm,)= o iniplies 1e

[lSzn6 - Sx,,,ll = 0, which is a contradiction. E

4%

1 .i.

Part 1. Theory of Opcmtor Ideals

Unconditionally Summing Opera,tors

1.7.1. An operator S E B(E, F ) is called unconditionally summing if every weakly swnmable sequence (x,) is mapping into a sequence (Sx,) suminable in the noriii topology. The class of all unconditionally sumriling operators is denoted by U.

Remark. The phrase “unconditionally suinniing” is deduced froni the fact that a,

A Y T ~ ( ,IS ~ ) iiorni convergent for every perinritation

z.

1

1.7.2. Obviously we have the Tht.orem. % i s un operator d e a l . 1.7.3. The following characterization is proved in [ROL, p. 2iOl.

P r o p o s i t i o n . A n operator is E g ( E , F ) i s unconditionally summing if and onhy if the product SX is not an injection for trn?y S E B(co,E).

1.8.

Separahle Operators

1.8.1. An operator S E !$E, F ) ist called sepcinrble if there exist xl, .u2, thai Sxl, Sx2, . . . are driiie in t h r range of S.

... E

E such

The class of all sq)arable operators is denoted by X.

1.8.3. The quickest proof of the following result can be obtained using 2.7.3. T h e o r e m . X is an operrotor i h ! .

1.9.

Kato Operators

1.9.1. We start this section with a Lemiiia. Let S E e ( E ,P) 11~1vet h e property that SJA5is not (in injection for a n y finite codirnemionul subspuce 1%’ of E . ’I’hen,given E --- 0, there exist9 tin infinite dimensional subspace M , such that SJ, ir vppozimnblc rrnd ljSJ,I 5 F , w l m e J, denotes the injection froin MP~ n t oE.

P r o o f . We inductively conrtruct sequences of elemelits x1,xz,. . . E E and functional~a,, f r L , . . . E E satisfyiiig thp following conditions: llzL$= 1 ,

IISx, 2 1

t/32,

l\aklj (= 2”l,

and (x,,uk) = t , l

Since S is not a n injection, we can find 2 , E E such that 1 1 . ~ ~ 1 1 = 1 and IIXxt.,ij5 t i 3 . Moreover, there is (1, E E‘ with ia11j 5 1 and (xl, a,) = 1. Tf sl,..., 5 , E E and n , , . . ., cie E E’ are chosen, piit &In:== (z E E : (x,11,) =

... = (x,a,)

=0).

1. Operator Ideals on Banach Spaces

49

Then M nis finite codimensional, arid SJ$, cannot be an injection. So we may select T , + ~E M , such that jlx,+l[l = 1 and /~S.T,+,]] 5 ~ / 3 ~ +Now ' . there is u:+l E E:' with kill = 1 and ( x ~ +a:,,) ~ , = 1. If n

a,+, :=

-

o \ ak, (xk, an-l/

1

then n

]/an+ll\ 51

+- 12k-1= 2"

and (.zrh+,,a,&,,) = 1.

1

Moreover, ( xi, n,,,)

i
= 0 for

+1

and (x,,~,ak) = 0 for E


This proves the existence of the desired sequences. Let M e be the subspace generated by x l , x2, ..., and put

sn:-

n ak

0Sq.

1

If x belongs to the linear span of xl,x2, ..., then x number of summands differs froin zero. Since m

SJ,x

=

1(x,ak)Sxk

m

=

1:(.T.

(l,;;

x k , where only a finite

1

n

and S,,J,z = 3 (x,nk) 8xk,

1

1

we have m

00

llh"JLXll5 1 I(X,

ak)l

Il8zklI

5 & 11x11 2' 2k-1,:
1

T

I

b,Xl!

1

a TI d m

i:(S - 8 s ) J,zll

5

This proves that l\SJ6ii5

m

i(x, (/k)l

5 E l!xll 2 2k-1/3k= (?/3)nt /xl\.

1'8~kIl

n t l

n-1

E

and lim IJSJ,-- 8,Jeii

= 0.

11

1.9.2. Let S E g ( E , F ) . Then S is called a Kato operator (or strictl?y sirzgular) if S J $ is riot an injection for any infinite dimensional subspace ill of E . The class of all Kato operators is denoted by G.

1.9.3. Theoreiii. Au operator S E 2 ( E , F ) i s a Kato operntor if and only i f every infinite dimensional subspace ilI of E contains some infinite dimensional subspace Mo such that SJCI i s upproximable (compact). Proof. Let S E G(E,F ) . If i%Z is a n infinite diiircnsional subspace of E , thenLemma 1.9.1 can be applied to SJC. Consequently there exists a finite codiniensional subspace M , of 191 such that S J f f , is approxiniahlc and, therefore, compact. Obviously JfO must be infinite dimensional. Conversely, let S E B(E, F ) satisfy the condition foriiiulatrd above. Suppose that S J ; is an injection for some infinite dimensional subspace M of E . Then there exists :in infinite dimensional subspace M , of M such that SJCois approxiniahle (compact). 'Phis is a contradiction, since S J f f aninst be an injection; cf. 2.3.4. 1.9.4. We are now ready to establish the main result.

Theorem.

G i s an operator ideal.

50

Part 1. Theorv of 0Dcr:Ltor' Ideals

Proof. Obviously 1% E 6. (1) Let S,, S, E G ( E ,F ) . If 31 is a n infinite diiiiensional subspace of E , the:? by 1.9.3 we can successively find infinite dimensional subspaccs M , and M , such that A ~ , J and & ~ S,JE2 are approximable, M , & N , C- M . Therefore (8, S,) J & IS also appi-oxiniable. Applying 1.9.3 once again, we obtain X1 S , E G(E,3'). ( 2 ) Let T E e ( E , , E),S E G ( E ,P ) , and R E 2 ( F , F,). Furthermore suppose that (RST)J2f)ois an injection for some subspace M, of E,. Then T o:= 5''JfiOis also an injection. Consider the subspace iV:= M(!!',). It follows Croiii S E G ( E , F ) that J f must be finite dimensional. Since To is one-to-one, the same is true for M,. Thiq proves that KS!!' E G(E,, F,).

+

+

1.9.5. Finally, we give sonic non-trivial examples. P r o p o s i t i o n . G(Zl, 7,) = i?(Zl, Z2). Proof. Let S E &Zl, I,) and J E g ( M , Zl) such that S J is an injection. Then M is isomorphic to a Hilbert space. Conscquently every sequence (Jz,,)with Z, E U , has a weakly convergent subsequence (Jxn,)which is norm convergent by 2.5.3. Therefore the closed unit ball U , must be coinpact. This iinplies that df is finite tiiiuensional. Hence S E G(Zl, Z2).

1.10.

Pelczyiiski Operators

I n this section all proofs will be omitted, since they are dual to those of 1.9.

1.10.1. L e m m a . Let S E 2 ( E )F ) hrcvetheproperty that QgS i s not a surjection for a : y finite dimensional subspace N of F . Then, given E > 8 , there exists a n infinite codimensional subspace N , such thtrt Q,S is approxirnable and ]~Q,S~~ 5 E , where Q, denotes tht surjection from F onto Flh',. 1.10.2. Let S E 2(E, F ) . Then S is called a Pebxyriski operator (or strictly cosingdar) if Q5S is not a surjection for any infinite codiinensional subspace Z\r of F . The class of all Pelczyriski operators is denoted by 2.

1.10.3. T h e o r e m . An operator S E 2(E,P ) i s a Pelczynski qerator if and only if every infinite codimensional subspace iV of F i s contained in some infinite codimensional subspace N o such that QFN,Si s approximable (compact). 1.10.4. T h e o r e m . 2 i s a n operator ideal. 1.10.5. We now mention a n interesting example; A. PELCZY~ISKI [3]. P r o p o s i t i o n . The identity map I from co into 1, is a Pelcxyliski operator

1.11.

Order Relations between Operator Ideals

1.11.1. Let us reinark that the collection of all operator ideals is something like a coinplete lattice with respect to the natural ordering. 1.11.?. P r o p o s i t i o n . G c 2.

51

1. Operator Ideals on Raiinch Spaces

Proof. The inclusion 0, & R is well-known; cf. 4.2.5. An example of a compact operator which is not approximable will be given in 10.4.6.

1.11.3. P r o p o s i t i o n . R c !&I. Proof. Since every norm conipact subset is weakly compact, we obtain !it Noreover, !it =/= !&I follows from 2.8.2.

1.11.4. P r o p o s i t i o n .

?B.

R c 8.

Proof. Let S E R(E, F),and consider a weak zero sequence (xn).Suppose that (Sx,)is not norm convergent. Using the compactness of S we can find a subsequence

(x,,)as well as y

E

F and

F

/i.lI-liInSx,‘ = y

> 0 such that and

I!Sx,,Jl 2

F

>0

for i = 1, 2 ,

...

i

Since (Sx,)is a weak zero sequence, we have y = 0. On the other hand, it follows that l]yll 2 E . This contradiction proves S E 8(E,P).Finally, R 8 is implied by 2.8.2.

+

9.11.5. P r o p o s i t i o n . !&I c U. Proof. Let S E !&I(E,F ) , and consider x weakly suinmable sequence (xn).Then the m

series ~A,S.-c, w e weakly convergent in F for all sequences (I,) E I,.

Now the

m

1

Orlicz-Pettis theorem. implies the norm convergence of =/= U follows from 2.8.2.

X E U ( E ,F).Moreover, 2B

2 Sx,;[DAY, p. SO]. Hence 1

1.11.6. Proposition. 8 c U. P r o o f . Let S E %(E,F ) , and consider a weakly suillmable sequence (.rn). Then the partial sums form a weak Cauchy sequence a norm convergent sequence implied by 2.8.2.

(I2

(,2R x,1

which is mapped, by 1.6.3, into

S.Y,&. Consequently

)

S E U ( E ,E”). Pinrdly, 8 =+ U is

1.11.7. P r o p o s i t i o n . !it c X. Proof. The inclusion R & X follows from the fact that every compact subset of a Eanach space is separable; [KOT, p. 271. Obviously R f X.

1.11.8. The order relations are collected in the following diagrani, where the arrows point from the smaller operator ideals a t the larger ones:

T

6 All inclusions are strict. Moreover !@,&! 3, and X are incomparable; cf. 2.5.2.

52

Part 1. Theory of Operator Ideals

1.11.9. Proposition.

a c B and

c 2. Proof. The inclusions follow immediately from 1.9.3 and 1.10.3.

Moreover, by 1.9.5, every surjection Q from I , onto I, belongs to G. Clearly Q is not a Pelczyriski operator. We know from 1.10.5 that the identity ninp I from c, into I , belongs to 2.Obviously I is not a Kato operator. This proves that G and 2 are incomparable. Hence R =/= G and =# 2.

1.12.

Notes

The coiicept of a11 operator ideal on the class of Banach spaccs was introduced by the author about 1969. For the historical background the reader is referred to the classical papers of E. GOURSAT [l], D. HILRERT113, E. SCHMIDT [l], [el, F. RIESZ [l], S. KAEUTAXI[l], and K. YOSKIA [I]. In particular, the ideal of compact operators is treated in almost all textbooks on functional analysis. Further informations about strictly singular and cosingular operators can be found in [GOL, pp. 76-93] and [ROL, pp. 252-2633. See also T. 1c.4~0[l], A. PELCZY&SKI [2], and K.J. VLADIMIRSK~J [l]. The theory of unconditionally summing operators is due to A. PELCZYI~SKI [I]. Recommendations for further reading:

J. W. BRACE/R. R. KNEECE[l], D. VAW DULST[l], N. R. FORYUM [l], I. C. GOHBERG/ 9. S. MAREIJS/I.A. FELDNAN[l], S. GOLDRERG/E. 0. THORP [l], A. GROTHEBDIECK[2], IC. GUSTAFSON[l]. R. H. HERMAK[2], J . HOWARD[t], [2], r3], T. K
Banach Space Ideals

2.

For every operator ideal the class of all Banach spaces l2 with I E t possesses some significant properties. This circumstance gives rise to the introduction of socalled space ideals. It turns out that every space ideal A can be generated by a special operator ideal which consists of those operators admitting a factorizat'ion through a Banach space M E A. The most interesting example is the ideal of reflexive Banach spaces which corresponds to the ideal of weakly compact operators. Moreover, every such operator factors through a reflexive Banach space. Proper operator ideals are defined by the property that the only identity maps which they contain are those of finite dimensional Banach spaces. For instance, the ideals of finite, approximable, compact, strictly singular, and strictly cosingular operators are proper.

8.1.

Space Ideals

2.1 .I. Let us recall that L denotes the class of all Banach spaces.

A space ideal A is a subclass of L with the following properties: (Sb) The 1-dimensional Banach space % belongs to A. (SI,) It follows from E,, E , E A that El x E, E A. (SI,) Let Eo E. Then E A implies Eo E A.

<

Remark. Space ideals are always denoted by bold capital sanserif letters. Remark. The relation Eo

< E is defined in B.4.3.

8.1.2. Let 3 be an operator ideal. Then Space (a)is the class of all Banach spaces E such that I , E 3. 2.1.3. Theorem. Space (a)is a space ideal. Proof. Since I , E a, we have X Space (a). (1) If E l , E , :. Space (U). then I,,, I,, E a. Using the notation introduced in B.4.11 we have

I E ,E~~

+

J~IE~Q J J E~I Q , C

U

s

7% proves that 6,x E , E Space (3).

<

(2) Let Eo E. By B.4.4 there are J E B(E0, E ) and Q I,o = QI,J. Hence E E Space (a)implies SoE Space (a).

2.1.4. A Banach space M is said to be Cartesian if M set of all Banach spaces E with E M is denoted by

<

2.1.5. Theorein. F,t, is

II

x M FM.

2 ( E , Eo) such t h a t

< M . I n this case, the

space idenl / o r every Cartesian Hrcnach space .V

4lo).

54

Faart 1. Theory of Operator lrleals

2.1.6. Finally, let us mention some important examples. P r o p o s i t i o n . The Banach spaces C[O, I ] and co, L,[O, 11 and lp where 1 5 p 5 co, ure Cartesiun.

2.2.

A-Factorable Operators

2.2.1. Let A be a space ideal. An operator S E B(E, F ) is called A-factorable if there exists a factorization AS’ = Y A such that A E B(E,M ) , Y E 2 ( M , F ) , and N A. The class of all A-factorable operators is denoted by 011(A).

2.2.2. T h e o r e m . Op (A) is an operator ideal. Proof. Since S E A, we have I x E Op (A). (1) Let S, E 2 ( E , F ) be A-factorable. Then S, = Y z A , with Ai E 2(E,M!), Y, E B ( M , , F ) , and M 2E A. Using the notation introduced in B.4.11 we have

s, i-4 = (YIQ, + PzQzl

( J , A , 4- J2Az).

+-

Hence S1 S , factors through M , x M , E A. This proves that S , (2) The property (01,)is trivial.

2.2.3. Let M be a Cartesian Banach space. An operator S E able if it belongs to BM := Op (FM).

+ S, E Op (A).

B(E,F ) is called B-fnctor-

2.2.4. Obviously we have the following criterion. P r o p o s i t i o n . An operator S E 2(E,F ) is M-factorable if and onl?j if there exists a factol-ization S = Y A , where A E 2(E,M ) und Y E B(M, F ) .

2.2.5. We now show that U --t Space (U)is a map from the collection of all operator ideals onto the collection of all space ideals. T h e o r e m . Let A be a space ideal. Then

A

= Space [Op

(A)].

P r o o f . Put 8, := Op (A). Since E E A implies I, E a,, we have A & Space (8,). On the other hand, E E Space (U,) iiieans that I , E U,. Hence I , = QJ with J E 2 ( E , M ) , Q E 2(3f,E ) , and M E A. By B.4.4 we know that E < H . So 31 E A implies E E A. This proves that Space (Uo) A.

2.2.6. P r o p o s i t i o n . Let

be an operator ideal. Then

% 2 Op [Space (U)]. P r o o f . Let S E 2 ( E ,F )belong to Op [Space (U)].Then S = I’A, where A E f?(E,N ) , Y E C(iV,F ) , and M E Space (U).Since I , E U, we have S = Y I l l A E U ( E ,F ) . R e m a r k . The operator ideal A shows that identity does not hold in general. cf. 2.3.4. 9.3.7. The next result follows immediately from 2.2.5 and 2.2.6.

T h e o r e m . Let A be a space ideal. Tlben Op (A) is the smullest operator i d e n 1 3 iuith

A = Space (a).

2. Bannch Space Ideals

55

1.2.8. As a supplement to the preceding theorem we state the open

P r o b l e m . Given an arbitrary space ideal A. Does there exist a largest operator ideal with A = Space (a)?

2.3.

Finite Dimensional Banach Spaces

2.3.1. The class all finite dimensional Banach spaces is denoted by F. “3.2. Theorem. F is the smallest space ideal. Proof. Every finite dimensional Banach space E is isomorphic to a finite Cartesian power of the scalar field. Consequently E belongs to every space ideal. 2.3.3. An operator ideal U is called proper if Space (U) = F.

2.3.4. P r o p o s i t i o n . The operator ideals 8, 8 , and

are proper.

P r o o f . By a classical result of F. RIESZa Banach space is finite dimensional if and only if the closed unit ball is compact ; [DUE, p. 2451.

9.3.6. The next statement is trivial. P r o p o s i t i o n . The operator ideals G and Z are proper.

2.3.6. As a special case of 2.2.8 we formulate the P r o b l e m . Does there exist a largest proper operator ideal?

2.4.

aeflexive Banach Spaces

2.1.1. The class of all reflexive Banach spaces is denoted by W. 2.4.2. Theorem. W = Space (m). P r o o f . A Banach space is reflexive if and only if the closed unit ball is weakly chompact; [DUN, p. 4251. 2.4.3. The following result is very impressive. = Op (W). Theorem. I

P r o o f . Let S E m ( E ,F ) , and select positive numbers cr, such that CQ

c 1

.:

= IlS/l2-

Bp lly/lln := inf

{ g f hIixli

+ lly - Szjl: x E E }

for y E F

we define a norin ;!.\I,, on E’. Obviously l[yilsr5 llyll. Moreover, it follows from

+

iISlI-l IkII 5 ori IIsIl-’ (IlfJ~Il f Ily - 841) < gn llzll IIYII - 8-41 are equivalent. that on \l,S\l-l \\yl[5 I l ~ l [Therefore ~. 11.1) and Gn

66

Part I . Theory of Operator Ideals

Form the Banach space M consisting of all f

Hence Y E f ? ( M ,F ) and IIYil

5 1. Moreover, S

E F with

=

PA.

Finally, the reflexivity of M will be shown via the weak compactness of U,. For this purpose we consider a family ( f i ) , where f i E ULwfor i E 1. Furthermore, let U be a n ultrafilter on the index set I. Since llfill 5 IlfillM 5 1, by the weak F’compactness of UF., there is E PI‘ with ft = F‘-lim KFfi.We now choose x,,, u such tthat

ft

6, lbinll

+ llfi - sxinll 5 3

iifilln-

Since llxinll 5 2~;;’~there are 2: E E“ with

2; = E’-lim KExin. It

u

follows from

Consequently f,” = ~ ~ . ~ ~S”x:. - l i m Since S is weakly compact, according t o 1.5.4, n

we have S“x; E M ( K F ) and therefore fi E M ( K F ) .So there exists f o E f;( = KFfo.Moreover, fo = F’-lim f i implies llfolln 2 lim IlfiIln. Conscyuently 11

P with

11

This proves that f o E U,. Let F , denote the Banach space F equipped with the norm jl.lj,. Then M can be canonically imbedded into Z2(F,),iff is mapped onto the sequence ( f , f , .. .). Therefore, by (2.4.2,every functional g E M ’ admits a representation m

(f, n> = S ( f , b,) 1

for all f E M ,

2. Banach Space Ideals

where b,, b,, ... E F‘ and set 1, E U such that

M

2 llbRli”,< 03.

Choose a natural number m and an index

1 m

m

C l]bnl125 e2 m

57

+1

and

I(fo - f i , b,)]

5c

for all i E

I,.

1

Then rn

J(f0

-fi,

s)l 5 s I(f0 - fi, 1

+E

00

b,)l

I(f0 -- f i ,

6,)1

mI-1

mt1

m f l

for d l i E I,. This means that f, = M‘-lii f i , and bhe weak compactness of b’.~ 11 is proved.

2.5.

Banach Spaces with Schur Property

2.5.1. A Banach space belonging to V := Space (93) is said to have the Bchur prgert!l.

2.5.2. The next statement will be proved in 3.1.8 (remark). Proposition. 8

=+

Op (V).

2.5.3. The most important examples are given in the following Proposition. For every index set I the B a m h space Zl(I)has the Schur property. Proof. The assertion follows from the fact that in Z,(I) weak and norm convergence of sequences are the same; [KOT, p. 2831. Remark. Let ( Q , p ) be a measure space. Then L , ( Q , p ) belongs to V if every singleton has positive measure; [DUN, p. 2951. On the other hand, by Riemann’s lemma the functions f,(t) := sin Snnt form a weak zero sequence in 2;,[0, 11 which is not norm convergent. Therefore we have WA11 B v.

2.6.

Banach Spaces with Pelczynski Property

%I. A Banach space belonging to U: = Space property.

%Le. Conjecture. U

(U)is said t o

have the Pekzy5-ski

=+= Op(U).

2.6.3. The following characterization can be derived from 1.7.3. Proposition. A Bcriurc?~space pos3esses the Petczyhski property if and mdy if if ro)vnins no coipy of co a-9 n subspace. 2-6.4. Proposition. For every meawre spme (9,p ) the Banach spnce Ll(Q.{ I ) has the Petczyhski property.

Proof. Since L1(Q,p)is weakly sequentially complctc, there docs not exist any subspace isoinorphic to c,; [DUN, p. 2901.

.?8

Part 1. Theory of Operator Ideals

2.7.

Separable Banach Spaces

2.7.1. The class of all separable Banach spaces is denoted by X. 4.7.3. P r o p o s i t i o n . X -= Space (T).

2.7.3. P r o p o s i t i o n . X

2.8.

=

Op (X).

Order Relations between Space Ideals

2.8.1. Let us remark that the collection of all space ideals is something like a complete lattice with respect t o the nat,nral ordering. 2.8.2. The diagram of 1.11.8 can be carried over from operator ideals t o the corresponding space ideals:

The following examples show that all inclusions are strict. hloreover, V, W, and X are incomparable.

u, L,[O, 11 6 v, ~ , [ O , 11 a w. Zl(I)E V and Zl(I)6 X, if card ( I )> No.

(1)

Z,(I) E W and &(I)c$ X, if card ( I )> No.

(31

L,[O, 11 E

co E

2.9.

x, co v, co c$ w.

(2)

(4)

Notes

The concept of a IJ,znach space i d e J was introduced by I. STEPHANI[4]. Special examples of operators which factor through a given class of Banach spaces were investigetad by T. FIQIEL [Z], A. GROTHENDIECK [4],11’. B. JOHNSON [i], S. KWAPIE~~ [6], J. LINDENSTRAUSS/ A. PELCZY~SHI El], A, PIETSCH [ill. and others. HOWEVER, the most striking result of this chapter is taken from W. J. DAVIS/T.FIGIEL/W.B. Jom-sos/A. Pt:ECZPr(SKI [l]. We also refer to the classical work of I, SCHCR [2] and an e d y paper of A. P E L C Z T ~ S K ZflJ. Recommendations for further redding:

J. DAZOED [l], J. DIESTEL [3], G. KOTHEti], [ZJ, L. K ~ C H B I[l], X H. PORTZ [a].

Products and Quotients of Operator Ideals

3.

J n the following we investigate products and quotients of operator ideals. Both concepts are defined in a natural way. To illustrate the general definitions we prove that 8 om = SZ and 8 = SZ o m-l. For two operator ideals 'u and B the class of all Banach spaces E such that % ( E , P ) E d(E,F ) , where F is arbitrary, coincides with the space ideal generated by the quotient %-l o 8. This statement gives rise to many important characterizat ions of special types of Banach spaces. See also Chapter -18.

3.1.

Products of Operator Ideals

:!.1.1. Let U and ?B be operator ideals. An operator S E 2(E,P) belongs to the plroddt fl o ?l3 if there is a factorization S = XI' with X E %(M,E') and Y i b ( E , H ) . Here M is a suitable Banach space. Clearly, fl" denotes the rL-th power of the operator ideal 21. 3.1.2. T h e o r e m . fl o 23 i s an operutor idecil. Proof. Since (01,)and (01,) are obvious, we check (01,)only. Let 8, t 91o B ( E ,F ) . Then Si= X , Y , with X iE fl(iW,, 2') and Y , E b ( E , M , ) . Put S:=XI&, X2Q2, Y : = J,Y, J z Y 2 , and X:=J I , x iM,. Now S1 +AS, = X Y , X i U(Jf, F), and Y E b ( E , M ) imply S, $- S,i % o B ( E , P ) .

+

+

3.1.3. To consider the following example we need a well-known lernnia which map be found in [LIT, vol. I, p. 301 and [SAF,, p. 1111. Lemma. Every compact -Tubset of n Banach space is contained in the closed absolutely convex hull of a zero sequeacc. = SZ.

Proposition. 8 o

Proof. Let S E S ( F , G ) and T E m ( E , F ) . By 1.5.3 every bounded seqiience (zn\ has a subsequence (xrt,)such that (T'xnc)is weakly convergent to some y E P. Thus (STx,) tends t o Sy 111 the norm topology. Consequently Sl' E R(E,G ) . So we have

2?0?B911RR. Consider S E B ( E , F ) . Then, bp the preceding lemma, we can find a zero sequence ( y n )such that S( U,) is contained in its closed absolutely convex hull. Write yn = Any: wch that 1, lirii ?,. = 0 , and lim /ly",l = 0. Let V , be the closed absolutely I1

n

csonvexhull of (&). Then F , := i~{pVo :p > 0 } is a Banacah space with the norm !&:= inf {p > 0 : y E ?T',}. It follows from 8(UE) V o that Sindnces an operator E e ( E ,Po) which is cottipact, since lim /lyJo = 0. Moreover, the canonical map K I1

frotn Po into F is coinpact, as well. So S R E R 055 &?LJ om.

=

KS, implies is i P ( E , P ) . Consequently

3.1.4. The next example shows that the iiiultipllcation of operator ideals is not commutative.

60

Part 1. Theory of Operator Ideals

+ &.

Proposition. I 08

,

P r o o f . Let Q be sonic surjection from I , into SE2 !B 0%. But S e St.

I,. Then 2.4.2 and 2.5.3 imply

is called idenipotent if '# = U2.

3.1.5. An operator ideal 3.1.6. We have the trivial

P r o p o s i t i o n . For ecery space ideal A the operator ideal 011 ( A ) i s idempotent. 3.1.7. We now list some concrete examples.

m, and% are ideirtpoteizt. P r o o f . The idenipotence of 8,I and , Z follows from 8 = Op (F), I = Op (W),

P r o p o s i t i o n . The operator ideak 5, 6, Jt,

and 3 = Op (X), respectively. Moreover, the forinula Jt = Jtz has been proved in 3.1.3.

Let S E 6 ( E , P).Then, given m

Ce

/[S,lj5 (1

S, and

S' = 1

CXoose operators X , ilx,\l= IIYill = I]A!3,{l1'', 6)

-Y :=

> 0, there exist S,, S,, ... E a ( E ,P) such that

E

+

E)

jjSj(.

1

g ( E , M i ) and Y , 8(Mi,P) such that Si = Pixi and where M i is a suitable Banach space. P u t M := Z&Wj). m

J i X i and Y := 2 YtQ,. Then

1

1

lpq = IIYll =

(P y

.

IlSjll

Since X E 6 ( E ,M ) and Y E O ( M , F ) , i t follows that S = Y X g W ( E ,F). This proves that 6 & Q2.

3.1.8. The following lemma can be found in [ROL, pp. 264 and 2671. Lemma. For every Bamch space F ,

@(C[O, 11, P ) = m\(C[O, 11, P). We now give a counterexample. P r o p o s i t i o n . The operator ideal B ! .is not aenapoteiif. P r o o f . Let J be the canonical map from C[O, 11 into L,[0, 13. Since &[O, I] is reflexive, we have J E Ill). By the preceding lenima, J i !& Suppose that ?B= ?B2. Then J = Y A , where A E %(C[O, 11, F ) and Y E B(P.&[0, 11). Therefore A E !lB(C[O,11, F ) . Now 3.1.3 implies J = Y-4 E B o = a. This is a contradiction. R e m a r k . By 3.1.6 we have%

+ Op (V).

3.1.9. C o n j e c t u r e . The operator ideal lf is not ideinpotent.

61

3. Products and Quotients o€ Operator Ideals

Quotients of Operator Ideals

3.2.

3.2.1. Let and ‘13 be operator ideals. An operator S 2 ( E , F ) belongs to the lefthand quotient N-l o b if Y S c b(E,Po)for all Y € N(F, Po),where F , is a n arbitrary Banach space. The right-hand quotient 2l o b-l is defined in a n analogous way.

are without any meaning.

Warning. The single syrnbols N-1 and 3.2.2. The following basic statenient is trivial.

Theorem.

a-l o 8 and 21 o b-1are operator ideals.

3.2.3. We next iniprovc 3.1.3. P r o p o s i t i o n . 9) = R o Proof. Let S E g ( E , F ) belong to 5l o m-1. Consider a weak zero sequence (r,). co

and put X ( t , l ):=

E l l x n for (5,)

I , . Then the co-weak conlpactness of U1 and

1

X ( X ’ ) 2 c, imply X E m(ll,E ) . Consequently S X E 5l(Zl,E ) . Sow it follows easily ) is a norm zero sequence. This proves that A o 9l-l G 8. Tlw that ( 8 ~=~(IOXe,) converse inclusion is a conseqrience of 3.1.3.

3.2.4. A proof of the folloiving statement will be given in 28.5.8. P r o p o s i t i o n . A ~ operntor L S E e ( E , F ) belomp to b-I o R i f nnrl sequence (Sx,)with x, E 77, admits a weak Cauchy subsequence.

077&J

if eueiy

R e m a r k . An operator belonging to 9)-1 o fi should be called a lbsenthal operator. R e m a r k . Since the identity map of c, is contained in 8-‘ o a, it follows that

fB*W105l. 3.2.6. P r o p o s i t i o n . A n operator S E B(E, P ) belongs lo 21V109)i f and only if E’-lim x, = o and F”-lim b, = o iwiply lim (Sx,, b,) = 0. n

11

¶

Proof. Let S E E(E, F ) belong to m-1 o B. P u t By := ((y. &)). Then B E E(F, cg). Moreover, M(B”) & c, implies B t m(P,co) by 1.5.4. Consequently B8 E%(E, PO). Now from lim ~~,?3Sx,~l = 0 we obtain lim (Sx,, b,) = 0. n

7k

Conversely, let S B(E, F ) satisfy the condition formulated above. Furthermore suppose that there exists E E m ( F , F,) with BS 4 B(E,F,). Then, given E > 0, we ~ can find a weak zero sequence (x,) such that jlBSxJ 2 E > 0. Choose b,O E U F with (BSXn,bi) = //BSx,/\.By 4.4.7 we have B’ E ?&(Fh, F ‘ ) . Consequently there exists a subsequence (b:,) such that b,, := B’b:< is weakly convergent to some b E l”. Hence Jim (A!!x,,,b,, - b! = 0. This ituplies lim (,?3Sxnl,b:,) = 0 , which is a contradiction. 1

L

R e m a r k . An operator ldonginp to operator.

!&-I

08 should be called a Dwnford-Petti8

3.2.6. We mention without proof a similar statement; cf. [DIC, p. 1791. P r o p o s i t i o n . Anoperntor S implies El’-lim S’b, = 0. U

:Z ( E , 3’) belongs t o X - l o !ID i f a n d o d y i f F-lh b,= n

o

Part 1 . ‘rheorv of Operator Ideals

63

R e m a r k . An operator belonging to X - l o m should be called a Grothendid operutor. 3.2.7. The following theorem can be used to characterize special classes of Banach spaces. T h e o r e m . Let % and B be operator ideals. Then Space (2I-l c d) consists of all Bamch spaces E’ such that N ( E , F ) & b ( E , F ) for arbitrary F .

(a-l

a-l

P r o o f . Tf R E Space o d),then IE E c 6.Hence for every S E have S = SIE E %(E, F ) . The converse implication is likewise trivial.

a(E,P ) we

R e m a r k . The space ideal Space (U o B-l) can be described in an analogous way.

3.2.8. A Banach space is said t o possess the Dunford-Yettis property if it belongs to Space (a-l o 8). 3.2.9. The main examples are treated in [DUN, pp. 494 and 5081. P r o p o s i t i o n 1. Let (9,,u)be any measure ,9pace. Then L l ( 9 ,p) has the Dunford-

Pettis property. P r o p o s i t i o n 2. Let K be aiiy compact Hausdorff space. Then C ( K )has the DunfordPettG property.

3.3.

Notes

The straightforward concept of the product of two operator idesls goes back t o the author, while J. PUHL[i] found the idea of their left- and right-hand quotient; rf. also 31.S. HSIEH[l]. The most examples, given in this chapter, are restatements oi well-knomi facts. The so-called Dunford-Pettis property of Banach spaces was introduced by A. GROTHENDICCH [2]. See also [ L N , pp. 181-184J and [RQL, pp. 263-2661. Recommendations for further rending:

J. DIESTEL [3], D. MORRIS/N. SALINAS [i], W. OOSTENBRIKK [l], N. SALINAS [3], S. SIMONS [Z].

Operator Ideals with Special Properties

4.

This chapter is devoted to constructions of new operator ideals from given ones. Such a rule ne20

: U -7

a""",

(tefining an operator ideal g n e w for every operator ideal a, IS called a procedure. Every procedure carries over itself to space ideals in a canonical way. Considering the class of all stable objects for a given procedure we obtain operator ideals and space ideals having special properties. First the norm closure BrJos is investigated. Then we introduce the radical grad which is of some interest in the spectral theory. The dual operator ideal IUdual consists of all operators S such that S' belongs to N. If 31 = then the operator ideal is called coinpletely symmetric. For exaniple, the classical theorems of 3 . SCHAUDER (1930) and V. GANTMACHER (1940) state that Jt and!& respectively, are completely symmetric. Next the regular hull %reg of an operator ideal is considered. This procedure will be of great importance in the sequel. As A. GROTHENDIECK observed, it may happen that an operator itself does not belong to but becomes a member of if the target Banach space is enlarged. This phenomenon leads to the definition of the injective hull Wl1J.The dual concept of the surjective hull N'"' is obtained in an analogous way. Both constructions were first studied by I. STEPHANI, using Grothendieck's work on tensor products as a basis. Finally, we introduce the maximal hull Urnaxand the niinimal kernel Urn'".These procedures are very important in the context of quasi-normed operator ideals.

a,

a

4.1.

Procedures

4.1.1. A rule

mew: ?il +%new which defines a new operator ideal gnew for every operator ideal 91 i s called a procedurr.

4.1.2. We now list some special properties: qjnew(monotony). (M) If U & b, then (%-)new = anew for all % (idempotence). (I) -4 monotone and idempotent procedure is called a Ivull procedure if U & ?Inewand a kernel procedure if U 2 for all operator ideals, respectively.

anew

4-1.3. Every procedure can be carried over to space ideals by setting Anew..- Space ([Op (A)]"""). R e m a r k . We always have Op (Anew)

[Op (A)]"""

and Space (2P") 2 [Space

but identity does not hold in general.

(a)]""",

64

Pert 1 . Theory of Operator Ideals

4.2.

Closed Operator Ideals

4.2.1. Let U be an operator ideal. An operator S E 2 ( E , P) belougs to the closure UC~OS if there are S,, S,, ... E U ( E ,F ) with lim IIS - Sni/= 0. n

4.2.2. The following statement is evident. Theorem.

CUClos

is an operator ideal.

4.2.3. Moreover, we have the T h e o r e m . The rule

clos: u --f U C l O S is (I hull procedure. 4.2.4. An operator ideal U is called closed if U = CUcloS. 4.2.5. P r o p o s i t i o n . The operator ideals 8 ,$3, m,!B, 3, and X are closed. Proof. By definition we obtain 6 := 810s. The ideals fi and !Xt are treated in

[DUN, pp. 486 and 4831. Let S E !Bclos(E, F), and consider a weal; zero sequence (zn).P u t e := sup Ill:nli. Then, given E > 0, there exists So E @(E,F ) with 1 1 8 - S0li 5 F . We can now find a natural number no such that llSgnll5 E for n 2 no.Consequently llflznll 5 ilsx, - f i o 4 I

+ Il4+Ai

5 (9

+ 1)

F.

Hence S 6 S ( E , F ) . This proves that ‘I) = ( 1 2 ~ 1 ~ . Let S E

UClos(E, P), and

consider a weakly suminable sequence (zn).Put

Then, given E > 0, there exists So E U ( E ,F ) with jjX - Sol/5 natural number no such that

E.

We can select a

Consequently

Hence S E U ( E , F ) . This proves that

u=

3~1~.

... i X(E,F ) with l h I/S- ~ $ 1 1 = 0. n We now choose r,,, x,,, ... E E’ such that Snxnl,SnxnZ,... are dense in iW(Sn).Then, Let S E 3 P s ( E , F ) . Then there are S,, S,,

given 1: E E and F > 0, there exist S, and IIS,z - SnxnJ 5 E. Hence

IlSx

- SZnjll

ynt

satisfying IjS - SnIIllrll 5 F and

5 jjSz - Sllxll + IISnx - S n ~ , i5 l / 28.

4. Operator ldeals with Special Properties

6B

Therefore Sxll, Sq,, Sx,,,... arc dense in M ( S ) , and S is separable. This proves that X = Xclos. R e m a r k . Obviously (fjis the smallest closed operator ideal.

4.2.6. As a n easy consequence of 4.2.5 we obtain the P r o p o s i t i o n . The operator ideuls%-l o R,%-l o'll mid X-I o ZU are closed.

4.2.7. P r o p o s i t i o n . The opeintor ideols 6 nnd

z are closed.

Proof. Let S E W O s ( E , F),and suppose that S J $ is an injection. Tiien there r,xists So E G(E,F ) with /IS- Sol]< j(SJ'&). Now it follows from B.3.11 that S,JfI IS also an injection. Therefore M is finite diniensional. This iiiiplies S < G ( E ,F ) . fIence G = Gclos. The proof of Z = ZClos is completely analogous.

-1.2.8. P r o p o s i t i o n . Let 3 be a n operator ideal. Y ' h m Space

(a)= Space ( a c 1 0 5 ) .

Proof. Let E E Space(aC1os).Since I E E Iuclos, we can find S t a ( E , E ) such that illE- S// < 1. Consequently S = 1, - (I, - S)is invertible. Finally I E = E iinplies E E Space (a).Therefore we have Space (Wlos) Space (a).The converse incluaion is evident. R e m a r k . The procedure clos is the identity for space ideals.

-1.3.

Radical

4.3.1. Let be an operator ideal. An operator S E B(E, F ) belongs to the radical 2 P d if for every L E B(F, E ) there exist U E B(E, E ) and X c U ( E ,E ) such that

U(IE - LS) -1.3.2. T h e o r e m . a

I, - X .

1

n d

is an operator ideal.

Proof. The property (01,) is trivial. (1) Let S,, S, E Urad(E, F ) . Then, given L C B(F, E ) , thare are U , B(E, E ) and XI t %(E,E ) with U l ( I E- LS,) = 1, - X,. We now choose U2 E B(E,E ) and X , E a ( E , E ) such that L',(I, - U,LS,) = I , - X,.Then

U,U,[I, - L(S,

+ S,)] = U,[I1 - X , - U,LS,] = I, - x,

Since X , $. U 2 X , E a(#,E ) , we have S, 4 S,

-

U2Xl.

E Urad(E, P).

(2) Let T E B(E,,E ) , S E Urad(E,F ) , and R E @ ( F , F,). Given L E B(F,, E,), there exist U E B(E, E ) and X E U ( E , E ) with C(IE - Y'LRS) = I E - X . Define the operators U , := I E , + LRSUT and X,:= LRSXT. Clearly So t U(E,,E,) and

U,(IEo- LRSl') = I E ,

LRSl' f I,RSU(I, - 7"LBS)T = Ih, - LRST LRS(I, - X ) T = IEo- Ll2SST = I E , - x,. -

-+

Therefore RSl' 5

t

Pietarh. 0 n r r ; ~ t o r

Ur&d(E,,,Po).

66

P a r t 1. Theory of Operator [deals

4.3.3. T h e o r e m . The rule rad: 'u --f

Wad

i s a hull procedure.

Proof. The property (M) is trivial. If S E % ( E ,F ) and L E B(F, E ) , put U := 1, and X := LS. Then X E % ( E ,E ) and U(IE - L S ) = I , - X . Consequently S E U m d ( E , F). This proves that 'u & W a d . Let S E 2 ( E , F ) belong to (%&)ran. Then, given L 2 ( F , E ) , there are U1 E 2(E,E ) and X , E Umd(E, E ) such that U,(IE - LS) = I , - X,. We can now find U2 E 2(E,E ) and X, E U(E, E ) with U 2 ( I E- X,) = IE - &.. Consequently

U,U,(IE

-

LS) = U,(I,vj - Xi) = IE - X2.

This implies S E Urad. Therefore evident, we have (I).

(Wad)rad

& Wa. Since the converse inclusion is

1.3.4. P r o p o s i t i o n . For every operator ideal 2 l the radical

is closed.

Proof. Let S E 2(E,F ) belong to (Urad)clos. Given L E 2 ( F ,E ) , there is So E Umd(E, F ) such that l!Lll /IS- Soli< 1. We now choose Uo E B(E,E ) and X o c %(E,E ) with

Uo(IE - [IE - L(S - So)]-' LSO) = I E - Xo. Put

u := uo[IE - L(s- so)]-'.

Then

U[IE - LS] = Uo[IE - L(S - So)]-' [I, - L(S - So) - LSo] = Uo(IE - [ I , - L(S - So)]-' LSo) = IE

Consequently S E U m d ( E , F). This proves that

(%md)clos

- Xo.

5 Wad.

R e m a r k. Obviously %cloS & %rad.

4.3.5. P r o p o s i t i o n . Let U be a n operutor ideal. Then a r a d = (Uc1os)rad.

P r o o f . It follows from

U(*los

5 W a d that

(2Ps)rad

E (Urad)rad = U r a d .

R e m a r k . I n 26.7.3 we shall prove that p d = !it&. The operators belonging to this ideal, denoted by ?&are called Gohberg operators (or inessential).

4.3.6. We now improve 4.2.8.

I' r o p o s i t i o 11. Let

be a n operator ideal. Then

Space ( 8 )= Space ( a n d ) . Proof. Let E c Space (Wad). Since I E E Urad, we can find U E 2 ( E , E ) and X E 'u ( E ,E ) such that U ( I , - IJE) = I &- X . Therefore I E = X E 94 and E E Space (a).Hence we have Space (%fad) & Space (a).The converse inclusion is evident. R e m a r k . The procedure rad is the identity for space ideals.

4. Operator Ideals with Special Properties

67

4.3.7. With respect to Problein 2.2.8 i t arises the following C o n j e c t u r e . Let A be a space ideal. Then Op (A)"d is the largest operator ideal % with A = Space (a). 4.3.8. Finally, we show that the asymmetry in the definition of

%pa*

can be removed.

Leniina 1. Let S c U?Zd

U r a d ( l 8 , F ) . Then for every L E B(F, E ) there ex& U E 2(E, E ) X I , X 2 E %(E,E ) such that

U(IE - LS) = 1, - XI and Proof. We can find U E Since

R := I E - U

( I , - LS) U = I B - X,.

B(E,E ) and XI E %(E,E) with C(I, - LS) = IE

=X I

- X,.

- ULS E 2lm*(E, E ) ,

there are Uo E B(E,E ) and Xo E %(E,E ) such that Uo(IE - R)= I E - Xo. We obtain from UoU Xo = I E that

+

+ Xo(IE - LS) U = Uo(IE - XI)U + Xo(IE - LS) U = 18 - X , , X2 := Xo(IE - U + L S U ) + UoXiU E %(E,E). (I, - LS) U

where

=

UoU(IE - LS) U

L e m m a 2. Let S E U r a d ( E , F ) . T h e n for every L E B(F, E ) there exist V E B(P,F ) and Y,, Y , € % ( F ,F ) such that

V ( I F- S L ) = IF - Y, Proof. Apply Lemma 1 to SL E

4.4.

and

( I , - SL) V = IF - Yf.

Wd(F,

F).

Symmetric Operator Ideals

4.4.1. Let U be a n operator ideal. An operator S E &(E,P ) belongs to the dun1 operator ideal %dual if S' c %(F', El). 4.4.2. The following statement is obvious. Theorem.

%dual

is an operator ideal.

4.4.3. Having in mind the property (M) defined in 4.1.2 we formulate the Theorem. T h e rule

dual: % --f

%dual

is rr monotone procedure.

-k%. We now carry over the procedure dual to space ideals. Theorem. Let A be a space ideal. T h e n that E' E A.

Adual

consists of all Banach spaces E such

Proof. By definition E E Adual is equivalent to I E E [Op (A)]dual. This means that = Q J , where J E 2(E',iM),Q E E(M, El), and M E A. In other words, E' is

1,.

68

Part 1. Theory of Operator Ideals

isomorphic to a complemented subspace of each other.

rW.Therefore E’ E A and E

E A d d iniply

5 (Udml. In case % = %dml the operator ideal is said to be cmpletely symmetric. The analogous definitions will be used for space ideals.

4.4.5. An operator idsal U is called symmetric if

4.4.6. The next result is trivial. P r o p o s i t i o n . For every (completely) symmetric oprcttor ideal % the space idecrl Space (a)is (cornpleteZy)symmetric, as well.

4.4.7. We now consider mine examples. P r o p o s i t i o n . T h e operator ideals $, 6, 8, and ?.Bnre completely sFjnimetric. Proof. The formula 5 = $dm1 is evident. In order t o show Q = %dud we need a special technique which will be developed later; cf. 11.7.4. The complete symmetry of 8 and is equivalent to the classical theorems of J. SCRAUDER and V. GAXTMACHER, respectively; [DUN, p. 4851.

4.4.8. P r o p o s i t i o n . l’he operafor ideals 8, U, c x.

x-1

r(nd

X

are not symmetric. iKoreouer,

Proof. The first part of the assertion follows froin the fact that the identity map of 2, belongs to ?B,a, and X, but not to Bdud, Udual. and Xdual. Let S E Xdual(E, F ) . Then there exist bl, bp,... f F’ such that S’b,, S’b,, dense in M(S‘). Choose x,, x2, 5 E’ with IIS‘bJi 5 2 !(xi, S’b,)l and 1ix2]] 5 1.

._.

. .. are

Let 2 1 1 be the subspace spanned by Sx,,Sx,,. .. If Sx 6j M , then there is b E F‘ such b), = 0 for i = 1, 2, ... Since we can find bio with that (Sx,b) $: 0 and ( S X ~ 1 /IS%- X’bioll 5 - IIS’bli, it follows that

4

IIX’bioli 5 2 and therefore

I(z,~,S’biJ1

= 2 ~ ( S Xbil, , ~, b)\

2 2 /jS’bbo- S’b/l

This contradiction iniplies M ( S ) 5 M . Consequently S E X(E,P).

4.4.9. A Banach space E is called quasi-dual if its canonical image is complemented in E”. The class of all quasi-dual Banach spaces is denoted by D. P r o p o s i t i o n . D i s a space ideal. P r o o f . Since the properties (51,) and (51,)are evident, we check (SI,) only. Let E . Then IEo= QJ, where J E L?(E,,E ) and Q 5 C ( E , E,). If E E D, then there exists P E 2(E”,E ) such that I , = PKE. Now IE0= QPKEJ = QPJ“KE,implies E, E D. Therefore (S1,) is satisfied. 4.4.10. An operator S E 2 ( E , P)is said to be dualisable if it belongs to 3 := Op (D). P r o p o s i t i o n . T h e wperntor ideal3 is symmetric, but not completely symmetric.

E,

<

69

4. Operator Ideals with Special Properties

Proof. It follows from I E p= ( K E ) ’ K Ethat j every dual Bniiach space B‘ belongs to D. Consequently D d u a l = L and 3 d u a l = 2. This proves that 3 c W a l . On the other hand, since co D, we have D += L and 9 B.

a

+

4.4.11. Xext let us inention that % and Proposition.

( Xdusl) dm l

c X and

(%dua1)dml

(3dual)dua 3

are incomparoble in general.

3.

4.1.12. For Inter use we state the P r o p o s i t i o n . 1,

6 (Vdua1)dml.

I’roof. It is known that the Banach spaces 1;‘ and LJO, 11” arc’ isomorphic:; [SED, p. 4801. By 2.5.3 (reniarlc) we have L,[O, 11 6 V. This implirs I,,[(), 11’’ 6 V. IIence 1;‘ 4 V.

43.

Regular 0pera.tor Ideals

4.5.1. Let U be an operator ideal. An operator S hull U r e g if KFS E % ( E ,F”). 1.5.2. Theorem.

@eg

2 ( E , F ) belongs to the regular

is a n operatoy ideal.

Proof. Since the ideal properties (01,)and (01,)are trivial, we check (01,)only. Let T E E(E,, E ) , S E %rcg(E: F ) , and R E e ( F , P o ) .Then

= R“(K,X)1‘ E %. This proves that RST Consequeritly KFo(RST)

E

Po).

4.5.3. Theorem. T h e rule

veg: % 4- w e g is

c(

iiull procedure.

P r o o f . We only show t h e irieiiipotencc. Let S < E(E. F )helong to (%reg)reg. Then fcfiS‘E a r e ~ ( E .F”) arid KF,,KFSE a ( E ,3“’”). No\\ I T , ,== (KFti’Kplt implies

KFS

=

(Kp)’KF,~KFLJI t %(E,P”)

and therefore S

< Weg(E,F ) .

Hence

(%reg)reg

&

%reg.

The converse inclusion is

trivial.

4.5.4. Of course, we i factory description of

m y

carry over the proceciure r e g t o spar(. ideals. But no satisseenia to be known.

Areg

70

P a r t 1. Theory of Operator Ideals

4.5.5. An operator ideal

is called regular if

= %reg. The analogous definition is

used for space ideals.

4.5.6. P r o p o s i t i o n . Let U be a n operator ideal. Then %dual i s regular. P r o o f . Let S E 2 ( E , F ) belong to (Udua1)rrq. Then KFS E Nd~al(E,F") and S'(KF)' E U(F"', E'). NOWIp = (KF)'K p implies S' = S'(KF)' K p E a(&", E') and S E Wu&l(E,F ) . Hence (Uda1)reg & %dual. The converse inclusion is trivial. 4.5.7. P r o p o s i t i o n . An operator ideal i.s conzpletely symmetric i f and only i f it is regular and symmetric. Proof. By 4.5.6 every completely synimetric operator ideal is regular and, of course, symmetric. Conversely, let U be regular and symmetric. If S E Udual(E, F), then we have S' E %(P',E') & Udual(F', E ' ) . This implies S" E %(E", 3"') and therefore KFS = X ' K , E U(E,F"). Thus S E % r e g ( E , F ) = U ( E ,F ) . This means that %dual & 3. 4.5.8. We now consider some examples. m, I, 23, U, and X are regular. P r o p o s i t i o n . The operator ideals 8,6, $ P r o o f . The regularity of 8,8 ,a, and !& is implied l by their complete symmetry. Let S E W e g ( E , F ) . Then every weakly convergent sequence (zn)is mapped into a norm convergent sequence (KFSX,,).Since KF is a n injection, the sequence (Sx,) must be norm convergent in F . Thus S E B ( E ,F ) . This proves that B r e g & 8. The regularity of U follows in the same way. The assertion is trivial for X. 4.5.9. By 4.4.10 and 4.5.7 we have the following negative exainple. P r o p o s i t i o n . The operator ideal 3 i s not regular.

Rinj

71

4. Operator Ideals with Special Properties

4.6.3. To establish the next theorem we need a result which is also of its own interest,. L e m m a . Let F be a Banach space possessing the extension prwerty. Then F) = U ( E ,F).

%inj(E,

Proof. By hypothesis there exists B E B ( F i n j , F )such that BJF = I F . Therefore a. The conS E U i n j ( E , F ) implies S = B(JF',S)E a ( E , B'). This proves that a i n j verse inclusion is obvious.

4.6.4. T h e o r e m . The rule

inj: a --f

ainj

is a hull procedure. Proof. We only show the idenipotence. Let S E B(E, F ) belorig to (2li"j)i"j. Then JFS E N i n j ( E , P j ) , and the preceding lemiiia implies JFS E a ( E ,F i n j ) . Consequently U i n j . The converse inclusion is trivial. S E % i n j ( E , F). Thus we have ((Uinj)inj

4.6.5. We now carry over the procedure inj t o space ideals. Theorem. Let A be a space ideal. Then A i n j consists of all Bancnch space.s i5oiriorphic to u. subspace of some Banach spa,ce M E A. Proof. Clearly E E A i n j is equivalent to JE E Op (A). This means that there is a factorization J E = X J , where J E B(E, M ) , X E B(M, E i n j ) , and Ai! E A. Then J iiiust be an injection. Therefore E is isomorphic to a subspace of Ji!. Conversely, the last statement implies the existence of an injection J E B(E, M ) . Since E i n i has the extension properi,y, we can find X E e(M,E i n j ) such that J E = XJ. Hence Je E Op (A).

4.6.6. P r o p o s i t i o n . Let A be a space ideal. Then

[Op ( A ) ] i l l j

= Op

(Ainj)

.

Proof. According to 2.2.6? it follows from

Ainj

:= Space ([Op ( A ) ] i n j )

that

Op ( A i n j ) G [Op ( A ) p . Conversely, let, S E B(E, F ) belong to [Op ( A ) ] i n j . Then JFS = YL4, where =2 E B(E, M ) , Y E B(M, F i n j ) , and M E A. P u t M , := M ( A ) , and let A , E B(E,No) he the operat,or induced by A . Obviously Yf 6 M ( J F )for f E M ( A ) . Consequently Y(M,,) E M ( J F ) , and Yo := J;'YJ$o is well-defined. Finally, AS= YoAo and Op ( A i n j ) . E Ainj imply S E Op ( A i n j ) . This proves that [Op ( A ) ] i n j

4.6.7. P r o p o s i t i o n . Let

a be a n operator ideal. Then

[Space ( a ) ] i n j G Space ('U1.j). f'roof. By 2.2.6 we have Op [Space (a)]E U. Therefore [Space ( U ) ] i n j = Space [(Op [Space ( U ) ] ) i n j ]

Space [%inj].

R e m a r k . The operator ideal Z shows that identit,y does not hold in general.

4.6.8. An operator ideal k used for space ideals.

is called i,tajcctive if

= 3lixi.i. The analogous definition

72

P ~ i 1.t Theory of Operator Ideals

The injectivity of an operator ideal means that it does not depend on the of the target space P whether or not an operator S E e ( E , F ) belongs to (cr.

SIZP

4.6.9. T h e o r e m . An operator ideal i s injective i f and only i f for every injectiow 9 E B(F,,F ) a r ~ dfor every operator So E E(E, Po) it follow from J S , E %(E,F ) thccf S o E a@,Po). Proof. To check the necessity we consider an injective operator ideal 9z. Let So c B(E,F,) such that J S , E a(#,F ) . Since F r J has the extension property, there is B E e ( F , F F ) with JF, = RJ. Hcnce it follows from JpoS0= B(JS,) E %(E,Ft’) that So E SZI”J(E, F,) - U(E, F,). Conversely, let us suppose that the given condition is satisfied. If So E N ~ ~ J Po), (E, then JFoS, E %(E,I?:’). Since JF , is an injection, we obtain So E %(E, Po). Therefore U = Uln3, which proves the sufficiency.

4.6.10. Injectivity is a hereditary property. P r o p o s i t i o n . For every injective operator ideal U the spnce ideal Space (8)is injective, as well. Proof. It follows from 4.6.7 that [Space (N)pj

Space

(am]) = Space (a).

4.6.11. P r o p o s i t i o n . For every injective space ideal A the operator ideal Op (A) i s injective, as well. Proof. According to 4.6.6 we have [Op ( A ) ] ~ = J Op ( A ~ * J= ) Op (A).

1.6.12. Now some examples are considered. P r o p o s i t i o n . T h e operator ideals g,52, ?B%,a, , and X c m injective.

4.6.13. P r o p o s i t i o n . The operator ideals 8 and 3 nre not irLjective. P r o o f . For 6 theassertionwill be shownin 10.4.6. Theformula

W J= 2 is trivial.

R e m a r k . Since every Banach space F ~ * possesses J the approximation property, we have ($9”~ = 52; cf. C.3.3 and 10.1.3.

1.6.14. P r o p o s i t i o n . G is the lnrgest proper injective operator ideal. P r o o f . The injectivity of G is evident. Let, S E B(E, P ) belong to somc propcr irijective opcrator ideal !&. and suppose that XJE, is an injection. By 4.6.9, it follou s from SJ$I, c that 1T,M E a. Hence M must be finite dimensional. Therefore S is a Kato operator. This proves that % 2 G.

1.6.15. Finally, we pose the gencral P r o b l e m . Givexi any injective spacr ideal A. noes there exist a largest injective operator ideal % with A = Space (a)?

4. Operator Ideals with Special Properties

4.7.

73

Surjective Operator Ideals

1.7.1. Let % be a n operator ideal. An operator S :Q ( E ,F ) belongs to the surjecticc hull 8 9 ~ if 1 S(2C E %(Em., F ) . R e m a r k . Let F, denote the ideal of all Banach spaces with the lifting property Then 5,:= Op (F,) is the ideal of all operators admitting a factorization through a Banach space & ( I ) . Moreover, we have = % o (&)-l.

4.7.2. T h c o r e m . 9 1 b ~ ri s rin operator ideal.

4.7.3. We also mention the L e m m a . Let E bea Baytack space withtheliftingproperty.Then'll[sul(E,P)= n ( E , P ) .

4.7.4. T h e o r e m ?'he rule szcr: 'u ->

a"*

i.s a hull procedure.

4.7.5. We now carry over thc procedure suv to space ideals. T b e o r e m. Let A be a space ideal. Then Asup consGts of all Bmmcl~spaces isomorphic to a quotient of some B a n a h space M E A.

4.7.6. P r o p o s i t i o n . Let A be a space ideal. Then [Op ( A ) F = Op (Asu~). 4.7.7. P r o p o s i t i o n . Let 'u be a n operator ideal. The)& [Space ( % ) ] s ~ r& Space (USU~). R e m a r k . The operator ideal G shows that identity docs not hold in general.

4.7.8. An operator ideal 'u is called surjective if % = U S U S . The analogous definition is used for space ideals. The surjectivity of an operator ideal % means that it does not depend on the size of the source space li; whether or not an operator AS' E ( J ( E F , ) belongs to 'u.

4.7.9. Theorem. An operator ideal 3 i s surjective i f and o d y i f for every surjection and every operator So E %(E0,F ) i t folknos f70m SoQE %(E,F ) that 8, E 2I(Eo, F ) .

Q E i?(E,E,)

4.7.10. Surjectivity is a hereditary property. P r o p o s i t i o n . For every surjective operator ideal 8 f h z s p c e ideal Spacz (91) i s 'urjectice, as well.

4 . m . P r o p o s i t i o n . For ctiery surjective

spnce ideal A the operator ideal Q p (A)

is surjectiee, as well.

4.7.12. Next me consider some examples. P r o p o s i t i o n . The operator ideals &, R,?23, U, nnd 5 are surjective.

74

Part 1. Theory of Operator Ideals

4.7.13. P r o p o s i t i o n . The operator ideals 6,B,and 3 are not surjective. Proof. For 6 the assertion will be shown in 10.4.6. Since E s ~ Er V, we obtain = 2.The formula 3 s ~ . = 2 is also trivial.

Bsnr

R e m a r k . Since every Banach space Esw possesses the approximation property, = s; cf. C.3.7 and 10.1.4.

n-e have

4.7.14. P r o p o s i t i o n . 2 i s the largest proper surjective operator ideal. -1.7.15. Finally, we pose t,he general

P r o b l e m . Given any surjective space ideal A. Does there exist a largest surjective operator ideal U with A = Space (a)?

4.7.16. Injectivity and surjectivity are dual properties. Theorem.

(8dual)sur

= ( w n j ) d u a l a&

(3dual)inj

5 (Us,,

)dual.

Proof. ( I ) Let S E 2(E,P ) belong to (Udm1)sur. Then SQEE %dual and QhS' E U. Since Qk is an injection, it follows from 4.6.9 that S' E U l n j . Therefore S E ( 3 i n j ) d ~ a ' . This

proves that

(Udml)sur

(Ui*j)dual.

(2) Let S E B(E, ix) belong to (Uinj)dual. Then 8' E U i n j . Since (Esur)' possesses t h e extension porperty, using 4.6.3, from QkS' E U i n j we obtain QkS' E U. Consequently 8QE E %dual and S E (Udualfsur. Hence we have (Ulnj)dual E (%dual)sur. (3) Let S E 2 ( E , P ) belong to (Udua1)inj. Then JFS E U d m l and S'J; E U. Since Jk is a surjection, it follows from 4.7.9 that 8' E Usur. Therefore S E (Usur)dml. This (3sur)dml. prows that ((Udml)inj

4.7.17. The next example shows that the right-hand inclusion of the preceding theorem can be strict.

P r op o s i t i o n . (Bdua1)inj

=+

(!Wr)dual.

Proof. I n 4.7.13 it is stated that Bsur = 2. On the other hand, the identity map of 1, does not belong to ( B d u a 1 ) h j . Let 11s suppose the contrary. Then it follows from 4.6.3 that ( B d m 1 ) i n j (Z, E,) =?l?dual(lm, Zm). Therefore we would obtain I , f Vdual and 1, E (Vdua1)dual. But the Iast statement is false; cf. 4.4.12.

4.7.18. P r o p o s i t i o n 1. For ecery injective opercdor ideal the d u d operator ideal i s surjwtive. P r o o f . It follows from % = U l n j that

%dual

= (Ulu~)d1lal= (Udua1)sur.

P r o p o s i t i o n 2. For everij surjective operator ideal the dual operutor ideal is injactitx?.

4. Operator Ideals with Special Properties

Proof. It follows from 91 =

that

adual

= (Usw)dml 2

75

(91dual)inj.

4.7.19. As an immediate consequence of 4.1.18 we obtain the P r o p o s i t i o n . For every completely symmetric operator ideal injectivity and surjectivity are equivalent.

4.7.20. The next relation is trivial. P r o p o s i t i o n . (Uaj)sup = (Usur)lnj.

4.7.21. Finally, we mention the following example which has been found by I>. WEIS [I?* Proposition. G s ~ = r 21nj = rrZ-I o a. R e m a r k . It seems to be unknown whether there exists an operator ideal 91 such that %inj = G and 9 W = z.

4.8.

Minimal Operator Ideals

i.8.L. We start this section with a general statement. Theorem. Let b be an idempotent operator ideal. '/'hen the rule 2I + 91 o b kerizel procedure.

i.9

a

Proof. We only mention that (3o b) o b = 91 o B2= 2€o b. R e m a r k . The same result holds if we multiply from the left-hand side.

4.8.2. Let 91 be an operator ideal. An operator S E B(E, F ) belongs to the minimal kernel Umln if S = YSoA, where A E 6 ( E ,Eo), So E %(KO. Fo), and Y E (ti(Fo,2'). In other terms, U m i n := 6 o % o 6. 4.8.3. Obviously 3.2.2 yields the T h e o r e m . 2 I m h i s a n operator ideal.

4.8.4. By 3.1.7 and 4.8.1 we have the T h e o r e m . T h e rule

miri: 91 -> ir

?@in

a kernel procedure. R e m a r k . For every space ideal A we get

Amln

= F.

4.3.6. The procedure m2in is trivial for the most operator ideals considered till now. Significant examples will be treated in Chapter 18. P r o p o s i t i o n . Let 91 be

(L

closed opewitor ideal. T h e n W

Proof. The assertion follows froni 6 5 91. 4.8.6. An operator ideal U is called minimal if U = % m k .

i n

= 6.

76

Part 1. Theory of Operator Ideals

4.8.7. From 4.8.5 we immediately obtain the P r o p o s i t i o n . 6 is the only minimal closed opercitor

idtleol.

-1.8.8. The next statement is obvious. P r o p o s i t ion. !!'he operator ideal 8 is ~ n i ? ~ i ~ t i ~ l .

3.9.

Maximal Operator Ideals

4.9.1. 5;1'v start this section with a general StateJll~vlt. Theorem. Let b be cin dernpote~iloperator idml. Then the rule 41 -> % o b-3is a huU procedure. P r o o f . We only shon the idempotence. Let S E %(E,E") belong to (% o 4-l)o %-I. Because of b2= 4,every operator X E 4 ( E 0 ,E ) admits a factorization X = XIXo, where Xo b(E,,El) and X , E B(E,. E ) . Then we have SX1 E o 4-' and S X = (SX,) X , E 3. Therefore o b-1)o b-1& (er o 4-1.The converse inclusion is trivial.

(u

R e m a r k . The saim result holds for the left-hand quoticnt.

4.9.2. Let a hc an operator ideal. An operator 1.5' E E(E, F ) belongs to the maximal huU a m if BXX < a(&,Po)for all X Q(Eo,E ) and B Q ( F ; Po).In other terns. u- := 0-10 'ti 0 6 - 1 .

4.9.3. Obviously 3.2.2 yields the Theorem.

Zma*

is an operatoy ideal.

4.9.4. By 3.1.7 and 4.9.1 we have the T h e o r e m . The rule

m a x : 91 --f a m a x i s a hut1 procedure.

4.9.5. The procedure naax is trivial for the most operator ideals considered till now. Significant examples will be treated in Chapter 17 and 19. P r o p o s i t i o n . Let amax =

a be n cZosed operator

ideal. T h e n

2.

P r o o f . The assertion follows from 6 & 3. 4.9.6. Ar? operator ideal is called nirixininl if rU = Q P'. 4.9.7. From 4.9.5 we iniiiiediately obtain the P r o p o s i t i o n . 2 i s the only rnnximal closed opcivlor ideul. 4.9.8.

At first glance the nest result seems to be somelion, surprising.

P r o p o s i t i o n . !Z'?Le opemior ideal 8 is maximal.

4. Operator Ideals with Special Properties

77

Proof. Let S t 2 ( E , 4') btlonp to 5""".Furtherrl~oresuppose that S is not finite. Using the Hahri-Banach theoreiii we can find X , E i!(Zf, E ) and B, E O ( F , If) such that B,SX, = e,I, and l]Xn]i= ~ ~ =B 1., Hcre ~ ~I , denotrxs the identity inap of Zg, and en is a suitable positive nuiiiber. Form the Hilbert space 11 := Z2(Z;); cf. C.4.1. If

S

:=

- 1

- X,Q,

1

snd B :=

"I

- J,B,,

1 n

n

then X E B(H, E ) and B E Q ( F , H ) . Therefore B S S E 8. This is a contradiction, since Q,BSXJ, = ? r 2 p J n for n = 1, 2 , . ..

4.9.9. We non state the open P r o b l e m . Let A he a space ideal. What about R e m a r k . Obviously

Fmx =

4.9.10. P r o p o s i t i o n . Let (Uw

F.

Wmax

(U Dr an

Amax?

= L, and X m a v = L.

operator ideal. Then

& amax.

Proof. Let S 5 ( U r e g ( E , P). Then, given X E @(Eo,E ) and R E Q ( F , F,), from BSX = BnKFSXit follows that BSX E %(Eo,F0). Hence S E % m d x ( E , F ) .

4.9.11. As an immediate consequence we have the P r o p o s i t i o n . Every mximrrl operator ideal i s regular.

4.9.12. Finally, we deal with the connection between the procedures max and min. P r o p o s i t i o n . Let Iu be an operator ideul. The?z (%min)max = (Umax

a&

((Umax)min = (Umin.

Proof. Let S E 2 ( E , F ) belong to a m a x . We know froiii 3.1.7 that X E Q(E,, E ) and B E B ( F , Po)can be written in th(1 foriii X = S,Xo arid B = BOB,,respectively, X , E @(El,E ) , B, E Q ( F , F , ) , and BoE @(F,, Fo).Now S t where X o E Q(E,,El), implies B,XX, E (U and B S X = Bo(BISX,)X o E (Umln. Thus S i ((Um1n)max. This proves that %ma\ & ((Um1n)mau. The converse inclusion is trivial. Let S E O(E, F ) belong to ((Umax)ml*. Then there are A E Q ( E , ,Yo), So E %""(&,,FO), and Y E @ ( P oF , ) such that S = YSOA.By 3.1.7 we have A = AoA, and Y = YIYo, where A , E W E , El), A , E @(El,Eo), Y o E @(go, F,),and Y , E @(Fl,F ) . Now it that S 2 W n . Consequently follows from S = Y,(YoSoA,)A , and YoS,,AoE (W-)min %mi*. The converse inclusion is trivial.

4.10.

Notes

The concept of a dosed operator ideal is straightforward. The radical of an operator ideal seems t o be defined in this monograp!i for the first time. I n thc special case of compact operators this coiistruction goes back to R. TOOD [l] and I. C . GOHBERQ/A. s. M ~ ~ a r r s /9. I . FELD-

78

Part 1. Theory of Operator Ideals

MAN [l]. See also the lecture notes [CAR, pp. 95-1021. Symmetric and regular as well as minimal and maximal operator ideals were studied by A. PIETSCH [13] and H.-U. SCHWARZ [l], [2]. The concepts of injectivity and snrjectivity are taken from the theory of tensor pro[4]. For further informations we refer t o the work of I. STEPHANI ducts; cf. A. GROTHENDIECK PI, ~31.

Recommendations for further reading:

J. DIESTEL/B.FAIRES[l], V. R. GANTMACHER [l], D. KLEINECKE[l], J. SCHATJDER [l], L. WEIS 111, R. J. U’HITLEY[l], B. Yoon [l].

5.

Closed Operator Ideals on Cla.ssica,lBanaeh Spaces

I n this chapter we investigate the lattice of all closed ideals in the operator algebra of a fixed classical Banach space. The first result of this kind goes back t o J. W. CALKINin 1940. He proved that for l2 there is only one non-trivial closed ideal. This fundamental theorem was extended to the sequence spaces l p , where 1 5 p < 00, and co by I. C. GOHBERG, A. S. MARKUS, and I. A. FELDMAN. Until now no other Banach spaces with this property are known. The situation is completely different in function spaces. Of course, Calkin’s theorem holds for LJO, 11. On the other hand, there are a t least countably many closed ideals in the operator algebra of Lp[O,11, where 1 < p < co and p =# 2. I n the case of C[O, 11 we even construct a n uncountable chain of closed ideals. The problem seems to be unsolved for L,[O, 11. Another generalization of Calkin’s theorem is due to B. GRAMSCHand E. LUFT. Both observed independently that, for a n arbitrary Hilbert space, the lattice of closed ideals is well-ordered.

5.1.

Operators in Sequence Spaces

5.1.1. At this point let E be either l p (1 5 p < co) or co and let F be either 1, (1 (I < 00) or co. The natural projection Pnis defined by pn(t1,

..

En, Sn+l,

-

.) := ( E l ,

* * *,

En,

0, - * * I

and ek denotes the k-th unit sequence. L e m m a 1. Let x l , z2,... E E such that 0 < a-l 5 l ] X k / / 6 < 00 nnd lxil A 1 9 1=o for i 7 k. Then there are X E B(E, E ) and A E B(E,E ) with IlXll f and llAil 5 a, x e k = X k , and A X k = ek for k = 1, 2,

...

Proof. P u t

respectively. Hence IIx’ll 5 E .

80 T

Pal t 1. Theory of Operator Idcals i

h e r e exists up E E' with ( x k , ak) = 1 and Ilakli = ~ ~ k. Define the operator A by that !ail A jakl = o for i

+

~

Moreover, ~ ~ ~ - we 1 may . suppose

W

A := 2'

@

ek

e

1

Using the same technique as above we obtain jlAlj

Ia.

L e m m a 2. Let S E B(E, b'). Suppose that lim 8xk = o wheneoer ( x k ) -is 6

= o crnd lim PmSxk = o for sequence such that lim PTnxk h k wpproxim a 61e.

nb

= 1, 2,

...

c7

bounded

Then S i s

Proof. Assume that X is not approximable. Then we have g := inf jjS

-

PnS,j > 0.

n

Choose x, E E with

[!Sxn- PnSx,/l 2 e/l" and llxnll = 1 for n

= 1, 2,

. ..

By a diagonalization process we select a subsequence (xnh)such that (Pmxn,) and (Pmi3xnk) are Cauchy sequences for m = 1,2, ... Using the method of 1.6.3 we can prove that (Sx,,) is norm convergent. Put y := lim Sx,,.It follows from k

IISxn, - P m S x n , / l 2 that IIy - Pmyll 2 el2 for m

2 @ / 2 for

- Pn,Sx,,jl = 1, 5.

%k

2 wz

. . . This is a contradiction.

We are now prepared t o prove

x

L e m m a 3. Let S E B(E, F ) . If S .is not approxiwiable, then there are q r a t o r s € e ( E , E ) and B E B(F, F ) such that BSXek = ek for k = 1,2, ...

Proof. Without loss of generality we may suppose that ljS\l= 1. According to Lemma 2 there are a sequence (x,) and a number E > 0 such that lim Pmxn= o and

im PmSxn= o for m

= 1,

2, ... as well as llxnll 5 1and /iSxnll2 E

11

Put

&k

n

> 0 for n = 1,2,...

:= &/Zk+l and y, := Sx,. If n, := 1, then there exists m, with

Ikn1

- P m , ~ n , / I5 4 2

and

I i ~ n ,-

pmlynll/ 5 4 2 .

Let n, < -..< nk+ and m 1 < ..-< m k - 1 be chosen. Then we can find nk > n k - , such that

d

~ ~ p i n h - l ~ nEk/2 ~ ~ ~

and

5

~ ~ ~ m 2 - l E k~ / 2n* ~ ~ ~

Now take mk > mk-, with !/Enk - Pmkxn,.!I

2 Fk/2

and

Ikne - pm,!/n,.ll

5 Ek/2*

So we obtain sequences (x,~)and (yn,). Set

x i := x,,

and xi := Pmkxn,- Pmk-lxnh,

y i := Yn,

and

YE :=Pm,Yn,

- Pm,-,yn,

>

5 . Closed Operator Ideals on Classical Banach Spaces

81

where Pm,:= 0. Then arid

-+

I14 - 4II 5 lkrck - Z.'mir,cr~i V'rnB-,x,Ln;i 5 ~k

1 IIY ~ ~2112 I I Y ~-~ ~PrnLyna,~ + iIPrn,-,ynkIIS & k . Moreover, since E 5 li.zi/l 5 1 and E 5 Ily:l] 5 1, we have

3~/4 I I J $/ - /Ix;

-~$1

5 1ldJ15 l l ~ f k l l+ /iQ - ~ i i 2i

1

t

E

and

344 5 11y:i

-

1 0

hy, -

?/:.I1 5 ly:ll 5 lY;ll t I]!/; - y:1, 5 1 t E . A lyil = o for ?L =j= k. Therefore, by Leniina 1 , there are

Clearly A = o and Iy:] operators X,, A, E 2(E,E ) and Yo ,Bo E 2(F , F ) such that

Xoek =- .I-:, A , x ~= e, , and as

Y#?k = g:,

Boy!

== e k

wcll as

I!Xo]l5 1

+

E,

llAoJl5 4 / 3 ~and

/ ] Y o5 /] 1

+

E,

l/Bol'5 4/36.

Since

2

m

11z: - z i 1 1 E~/ 2

and

2 ~ l yi yilj I &/2,

1

1

by setting m

X I :=

61

ek

8 (2;- xi)

and

Y , :=

1

ek

(3(y? - y:j

1

we obtain operators X, E B(E,F ) and Y, t 2(E,P ) with &XI\]5 612 and l]Y,/i5 $2. If X := X , X, and Y := Yo Y,, then S P=~xj and Y p k = y.; Moreover, IlB,Y,\( 5 213 implies the existence of B := ( I $ BOYlj-' B,. Kext it follows from BOY,= I, that BY = I,. Therefore

+

+

BSXq

= BSxk = By: = Bye/, = en.

5.1.2. We now easily obtain the

Ip). Theorem. If 1 5 q < I ,< cn3, then 2(Zp,2), = S(&, Proof. Suppose that there exists S 2(Ip,Z,,) which is not approximable. According to 5.1.1 (Lemma 3) we can find X E 2(Zp,l p ) and B E 2(7,, I,,) with BSXen = e,. Then it follows that

But this is impossible. R e m a r k . We also have e(c,, 2),

5.2.

= S(co$ Zqj for 1

5 Q < oc.

Simple Banach Spaces

52.1. A Banach space I2 is called simple if the algebra B(E, E ) contains one and only one non-trivial closed ideal. 6 Pictach, Operator

82

Part 1. Theory of Operator Ideals

5.2.2. We now give the only examples which are known until now.

T h e o r e in. The Bamch spaces Zp ( 1 5 p

< co)a

d c,, are simple.

Proof. Let E denote either Zp or c,. Suppose that the closed ideal U ( E ,E ) contains a n operator S which is not approximable. By 5.1.1 (Leinma 3) there are operators X , A € B(E, E ) with AXXe, = ek. This means that IE = ASX € %(A!,E ) . Consequently U(E, E ) = P(E, E ) . Therefore B(E, E ) is the only non-trivial closed ideal in 2 ( E , E ) .

5.2.3. Finally, we formulate a n open question. C o n j e c t u r e 1. Every simple Banach space is isomorphic either to l p (1 or to c,.

5 p < 00)

Con j ec t u r e 3. Every simple Banach space is separable.

5.3.

Non-Simple Banach Spaces

5.3.1. We begin with a n elementary result. The Cartesian product El corresponding operators are defined in B.4.11.

x E, and the

P r o p o s i t i o n . Let E := El x E,. Every ideal %(E,E ) tk uniquely d e t e r m i d by the components

U(Ei7 Ek) := {Sik € B(Ei, Ek): JkSjkQi € % ( E ,E ) )

7

where i = 1,2, and k

=

1,2.

We then write

5.3.2. Now a n interesting example is given. T h e o r e m . Let 1 closed ideals :

5 p < q < CQ. Then 2(lpx lq, lp x I,) conta,ins the fdlmuing

Purther non-trivial closed ideals can mdy occur between the ideals connected by i l i ~ broken arrow.

5. Closed Operator Ideals on Classical Banach Spaces

83

Proof. Let %(I, x I,, ,Z x I,) be a non-trivial closed ideal. Then %(Ip, ),Z and I,) are closed ideals in i?(Zp, I,) and B(Z,, Z,) respectively. Moreover, it follows I,) = B(Z,, I,) = @(I,, Z,). from 5.1.2 that %(Z,

%(Z,

We claim that %(I,, 1,) = B(Zp, I,) implies %(Ip, la) = B(Zp, I,). To prove this assertion we denote the identity map of I, by Ipp.Then JpIppQpE %(I, x I,, 1, x I,), where J p and Qp are defined as in B.4.11. Hence, for S,, E B(Zp, I,), we have JqSpqQp

= (JqSpqQp)

(JpIppQp)

c

X

b, zp X

zq)-

I,). Analogously we can see that %(Z,,

This shows that S,, E %(Z, plies %(Ip, I,) = S(Zp,I,).

x I,, Zp x )2, different

We now observe that every non-trivial closed ideal %(Z, from

(

QVp,

w p

I,), > I,),

@G,,

I,, I,)

)

w,>

or

("'".~,), w p >

41,

W q ,

5))

QVq,

I,)

I,) = g(l,, 2,) im-

is of the form

QV,,

(

w p ,

1.

@U,> 1,) 44, @(I,, 4)

ZpL

The two obvious cases are U(Z,, I,) = i?(Zp, I,) and %(Zp, I,) = Q(I,,

Z,).

R e m a r k . We obtain another non-trivial closed ideal %(Zp x I,, Zp x I,) if a(&,Z,) is defined to be the closure of the set of those operators S,, E %(Ip, Z 4 ) which factor through the identity map I,, from 1, into 1,; cf. V. D. MILMAN[l].

5.3.3. The preceding theorem leads to the open P r o b l e m . Let 1 5 p < q < 00. Does the algebra B(Z, nitely many different closed ideals?

x Z, Zp x I,) contain infi-

5.3.4. Let M be a Cartesian Banach space. Then we put

QM :=

rp,where vM

denotes the ideal of M-factorable operators; cf. 2.2.3.

<

5.3.5. Lemma. Let M and N be Cartesiaiz Banach spaces. Suppose that M E and N. Q M ( E , E ) QN(E,E ) . Then M P r o o f . By hypothesis we have IM = Q J , where J c B(M, E ) and Q E B(E,M ) . Then P := JQ E v M ( EE , ) & (fjM(E, E). Therefore P E Q N ( E ,E). Now it follows M ) . By 4.2.8 we get IM g N ( M ,M ) . This means that that I M = QPJ E @iJv(M, ii N.

<

<

<

<

5.3.6. If M N and N M , we write M N . I n this way an equivalence relation is defined on the class of all Banach spaces. N

R e m a r k . Under some additional assumption M isomorphic; [SED, p. 3801.

N

N implies that M and N are

5.3.7. Let Car ( E )denote the collection of all Cartesian Banach spaces M such that M < E. Obviously M E Car ( E ) and M

-

N imply N E Car ( E ) .

84

Part 1. Theory of Operator Ideals

5.3.8. We now observe that there is a one-to-one correspondencc between the equivalence classes of Car ( E )and a certain subset of closed ideals belonging t o g ( B , E ) ; cf. 5.3.5. P r o p o s i t i o n . Let M , N E Car(& T h e n M equivalent.

-

W and 6 . , f ( Z ,E )

= @j,r(E,E ) ore

5.3.9. $pplying the preceding proposition we now obtain the following result.

Theorem. There are infinitely many closed ideals in the algebra f?(L,[O,11, LJO, 11) for 1 < p < 00 und p =!= 2. Proof. Sinre the case 1 < p < 2 can be treated by duality, we may s;pposc that 2 < p < cy). Then, according to H. I).ROSEKTHAL [ 2 ] , we introduce the Banach space X , := (5:x E I , and wx 1 2 } , where w = ( n ( 2 - P ) ' 2 P ) . The norm is given by

II4l := ma= (lI4lp l l ~ ~ l l -2 ) It can be shown that X ,

Car (LJO, 13); cf. [LIT, vol. L, 1). 1701.

The next construction is due to G. SCHECHTMAN [l]. Identifying X , with a subspace of LJ0. 11 we define XF' to be the closed linear span in L,([O, 11") of all products f l (2 * * . 0fn(t1, . . *, t,) := f l ( t 1 ) - * * f"(t,'). where f l ,

...)f V b:, X,. Then the Banach spaces X F ) , . . XF"', ... .)

are niutually non-equivalent. Since X F ) t Car (LJO, l]), the assertion follows froin 5.3.8. Z&)

R e m a r k . Other Banach spaces belonging to Car &[O, ; cf. J. T,IXDEKSTRA~JSS/A. PELCZP~SKI [ 11.

11) are l,, I, x I,, and

5.3.10. P r o b l e m . Let 1 p; < 00 and p + 2. Does the algebra f$(L,[O,11, L,[O, 11) contain uncountably rriany different closed ideals? 5.3.11. T h e o r e m . There are uncountably nian!y closed ic1enl.s in e(C[O, 11, C[O, 11). P r o o f . I n the following let a be a countably infinite ordinal. Then I'(LY) denotes the set of all ordinals 6 such that { 5 x. Consider the order topology in which the intervals ( 5 E ] ' ( a ) :[ < &,I and t T(ix):5 > to]form a subbase of open sets. Observe that I'(a) becomes a. compact Hausdorff space. Moreover, C(I'(a)) is R Banach space belonging to Car (C[O, I]). We know froin C. BESSAGA/A. PELCZYrliSKI 113 that C(l'(a)) C ( r ( P ) )implies > ,9, where Q is the first infinite ordinal. Therefore the Banach spaces

(e

N

C ( T ( w ) )C(T(OY)), , . . .) C(l'(0"

",), ...

are mutually non-equivalent. Since there are uncountably many ordinals of the first type, the assertion follows froin 5.3.8.

5. Closed ODerator Ideals on Classical Banach Spaces

85

5.3.12. Finally, we state a n open

P r o b l e m . How many closed ideals arc contained in the algebra 2(Ll[0,1], L1[O,l])?

5.4.

Closed Operator Ideals on Hilbert Spaces

5.4.1. An operator S E e ( E ,P) is called an N,-Hilbert operator if S = Y A with A E 2 ( E , H ) and Y E e ( H , F ) , where H is a Hilbert space such that dim ( H ) < N,

The class of all N,-Hilbert operators is denoted by 8.. R e m a r k . Obviously $jo = 5.I n 6.6.1 we will define $j :=

u &. Y.

5.4.2. By the same technique as used in 2.2.2 we obtain the

Theorem. $, is an operator ideal. 5.4.3. Let H, be the class of all Banach spaces which are isomorphic to some Hilbert space H with dim ( H ) < N,. 5.4.4. Theorem. H, = Space (&).

Proof. If E E Space (&). Then E is isomorphic to a (complemented) subspace M of some Hilbert space H with dim ( H ) < Na.Now i t follows from dim ( M ) 5 dim ( H ) that E E H,. Consequently we have Space (if&) E Ha.The converse inclusion is trivial. 5.4.5. P r o p o s i t i o n . If

LY

< B, then H, c H,. Moreover, sjb c 8,and $:lo' c $;l"s

Proof. By 5.4.4 and 4.2.8 the assumption SjS = $jaor H, = H,, which is obviously false.

&:lo'

= $j;loS would imply

5.4.6. We now prove the main result of this section.

Theorem. On the class of all Hilbert spaces there are only the closed operator ideals and 2.

8,C'OS

P r o o f . Let U be any closed operator ideal. If Space (a) contains all Hilbert spaces, then = 2.Otherwise put

8, := inf (dim ( H ) :H

6 Space (a)).

We now consider an operator So E U(L,,(Q,p), L,(Q, p ) ) such that Sof = sf, where 8 E L,(R, p). Let Q E := ( m E R : is(w)l 2 E ) for e > 0. The restriction of p to Q, is denoted by p,. Furthermore, we define the injection J , E S(L,(Q,,p , ) , L&2, p)) and the surjection &, E 2(L2(Q,p ) , L,(Qc,p E ) )in a canonical way. Then S , := Q,S,JJ, is = S,S;l E a. Consequently dim (Lz(Qe, p E ) )< N,. invertible, and we have IL,(n,,pF) By setting L, := SoqJ,Q,we now obtain a n operator L, E Bnsuch that ll8, - L J 5 F . Hence So E $Elos. Let S E % ( H , K ) . Using the notation of D.3.1 we see that SoE a(L,(Q,p), 2;,(R, p)). s o 8, E ~ ~ ' " s ( L , (ps)Z, L,(Q, , p ) ) and therefore S E $Sj",los((H, K ) . This proves that ?I 2 #?. The convers? inclusion is implied by 8. %.

86

P a r t 1. Theory of Operator Ideals

5.5.

Notes

The theory presented in this chapter has been initiated by the classical paper of J. W. CALKIK [l]. The main result, stating the simplicity of the sequence spaces I , with 1 5 p < 00 and e,, was proved by I. C. GOHBERG/A. S. MARKUS/I.A. FELDMAN [i], See also R. H. HERMAN 111. The given examples of non-simple Banach spares are based on the work of C. BESSA GB/ A. PELCZYBSKI [l], J. LINDENSTRAUSS/A. PELCZY~SKI [l]. H. P. ROSENTHAL [ 2 ] , [3], and G . SCHECHTMAN [l]. The reader is also referred to V. D. MILMAN113. Closed operator ideals o n a fixed Hilbert space of arbitrary dimension were investigated by B. GnAniScH [I] and E. LUFT [l]. Recommendations for flirther reading:

E. LACEY!R. J. WHITLEY[i], H. R. PITT[l], H. PORTA [i], [2], [3], H. P. ROSENTHAL [l], P. VOLIWANN[i].

Part II:

Theory of Quasi-Normed Operator Ideals

Quasi-Normed Operator Ideals on Banach Spaces

6.

'The tiieoiy of noriiicd operator ideals has been founded by A. GROTIIENUIEOK and I<. $CHATTEN in the carly fifties. Basic examplcs are the ideals of nuclear, integral. and ahsolutt~lysuniniing operators. In the following we deal with quasi-nornied operator ideals. This generalization is motivated by the fact t h a t some important operator ideals which do not possess a natural norm should also be covered. On c.vt:ry oI>c*riltorideal there exists a lot, of cliffcrcnt quasi-norms. However, t hc ,.iiice" quasi-norms are selected by the conipletenebs of the corresponding topology. Prom this point of view for every operator ideal t,hrre is, up to equivalence, at most one rc.aaonablc quasi-norm. Because of this fact it swnis convenient t,o includr the cvmpletcness into the definition of quasi-normed operator ideals.

6.1.

Quasi-Normed Operator Ideals

6.1.1. Let U be an operator ideal. A map A from U into W+ is called a quasi-norm if the following conditions arc satisfied :

(QOI,) A(1,) = 1, where X denotes the 1-diitiensional Banach space. (QOI,) There exists a constant x 2 1 such that B(S1

+ 8,)5 x[A(Sl) + A(S,)]

for A"], 8,E U(E,F ) .

(QOI,) If 7' € f?(EO,E ) , S E g ( E , F ) , and R E Z ( F , Po),then

A(RS7')5 jlBl] A(S)IITII. R e m a r k . Quasi-norms on operator ideals will he denoted by bold capital Roman letters. 6.1.2. Propositioii. Let 8 be an operator irEal with a qumi-norm A . Then all corttpcrnents %(E, F ) are linear l o p o l o c ~ Hausdorff l spaces.

P r o o f . Obvioiisly we have A(U)= II) A($) for S E a(#,F ) and il X . Thus A is a quasi-norm in the sense of LKi)T, p. 1621and we niny generate the corresponding t 0pology. Rcniark. We refer to this topology if we use concel)ts like A-lim, A-Cauchy wquence, A-bounded scit . etc.

6.1.3. A quasi-norwed opercctor ideal ['u, A ] is an operat,or ideal U with a quasi-norm -1such that all linear topological Hausdorff s p w s 9J(E,F ) are complete. 6.1.4. P r o p o s i t i o n . Let [VJ. A] be u quasi-normed operaior ideal. Il'k~nIlSIl 2 A ( S ) for ccll S U.

90

Part 2. Theory of Quasi-Piormed Operator Ideals

Proof. Given S E U(E,3'). Let z t U , and 6 E Up.. Then it follows from = (6 @) 1)S(1@) z) that

(Sx,b) I x

I(fk-915 IF 0 I l l WJ) 1110511 5 -4(X). Hence llSll 5 A(&). 6.1.5. P r o p o s i t i o n . Let [%, A] be a qzcnsi-norttux? o p r o f o r d e a l . T h e n we hncc M a o 0yo) = llaol///yoilfor all a, E E' and yo E F . Proof. We know from B.1.2 and 6.1.4 that llaoil IIYd = /la00Yo11

5 A(a0 @ Yo).

On the other hand. the diagrain a0

E

\Ryo

1

@3 1

a,

+F 1 0 Yo

c

I

s-

+%-

I x

implies A(ao0yo) 5 111 Q yo;:A ( I x )

I/ao@

111 = ~ ~ l]y0\i. u o ~ ~

6.1.6. Next we state a fundamental result.

[a, A] and [d,B] be quasi-nornzed operator ideals. If e 2 0 swch that B(S) 2 eA(S) for all S E U.

T h e o r e m . Let there is a constant

9z

d,then

Proof. Supposethat such a constant does not exist. Thenwe can find S, E U(E,, F,,) such that and B(S,,) 1n for n = 1,3,...

A(&,) 5 (2x)-"

Put E := Z2(En)and F := Z2(Fn)as defined in C.4.1. Since I

k

A f h i l

\

m

.JnSnQn)5 C xiA(Sh,,)2 ( i ? ~ ) - ~ . i=l

k

00

the partial sums 2 JnSnQ,forin a n A-Cauchy sequence. Consequent,lyS := 2' J,S,Q, 1

belongs to 8. Now it follows froin S E 8 that n 2 B(S,) = B(&,SJ,) is a contradiction.

1

5 B(S), which

6.1.7. Let U be an operator ideal. The quasi-norms A, and A2 defined on 41 are called equivalent if there exists a constant e 2 0 such that Al(S)5 eA2(S) and A2(S) 5 eAl(S) for all S E U. 6.1.8. 4 s a n iniinediate consequence of 6.1.6 it follows the iniportant

T h e o r e m . For every quasi-?tornbed operator iden1 [?I, A] the quasi-norm A i s wzique up to equivalence.

6. Quasi-Normed Operator Ideals on Banach Spaces

91

6.1.9. Unfortunately, a quasi-norm A need not be continuous in its own topology. We only know that S = A-lim S , implies n

A(&) 5 x lim inf A(&) n

and lim sup A(Sn) 5 x A ( S ) . n

An easy counterexample is the following. Let

x

> 1 and put

A(S) := !I811 for S E (6 and A(S) := 5t llSll for S

48 .

Then A is a quasi-norm on 2 with

+ S2)5 x[A(S,) + A@',)]

B(8,

for S,, S, E %(E,3').

However, if E is infinite dimensional and S E @(E,E ) , it follows that

-4(S A(S) = llS/l and lim E'O

+ &IE)= x IISll.

This quasi-norm catastrophe is very unpleasant, but not critical. Indeed, we see from 6.2.5 that every quasi-normed operator ideal admits an equivalent quasi-norm which is automatically continuous.

6.2.

p-Normed Operator Ideals

6.2.1. A quasi-norin A on the operator ideal % is said to be a p-norm (0 < p 5 1) if the p-triangle inequality holds :

A(Sl

+

S,)P

5 A(Sl)P

+ A(S,P

for S,, S,E

a(#,F ) .

If p = 1, then A is simply called a norm. R e m a r k . The constant x := 2llp-l can be used to satisfy condition (QOI,). R e m a r k . Obviously every p-norm A is continuous in its own topology.

6.2.2. A p-normed operator ideal [a,A] is a n operator ideal 91 with a p-norm A such that all linear topological Hausdorff spaces % ( E ,F ) are complete. If I, = 1, then [a,A] is said t o be a normed operator ideal. R e m a r k . Every closed operator ideal 91 is a normed operator ideal respect to the operator norm.

[a,11.111 with

6.2.3. We now formulate an important criterion which will be permanently used in the sequel. T h e o r e m . Let % be a subclass of 2 with a n 2*-mlued functim A such that the are .ratisfierJ (0 < p 5 1):

f o ~ ~ o ~ uconditions irq

u

(0) I,y- € and A ( I x ) = 1. co (1) I t Jollowsfrom S,, S,, ... E %(E,F ) and 2 A(S,)P CQ

and A ( S p

5 2' -4(S,)p.

1

< c*3 that8 :=

m

SnE %(E,P) 1

1

T E 2(E,, E ) , S E U ( E ,F ) , and R E 2 ( F , F,) imp&J RST E %(E,, F,) and d ( R S T )S IlBIl A(S) IITII. (3)

T1~e-n[a,A] i-sa p-normed operator ideal.

92

Part 2. Theory of Quaai-Normed Operator ‘Ideals

Proof. The only point is to observe that (1) summarizes the p-triangle inequality and the completeness, as well. 6.2.4. We now consider a quasi-normed operator ideal [a,A], where SL denotes the constant in the quasi-triangle inequality (QOI,). Define the exponent p by 2l’P-l = z. Since SL 2 1 , we have 0 < p 5 1.

Lemma. Let Si E U(E,F ) and ki E (0:1,

...). Suppose

that

A(Si)P

5 2-Od and

II

2-kl = 1. Otherwise,

Proof. Without loss of generality we may sssunie that choose k,+l,

..., k,

m

F,

1

{ O , 1, ...) with 2 2-k6 = 1and put Si := 0 for i = n

+ 1, ...,

?)I.

1

We prove the lemnia by induction. The case inax {kl, ..., k,} = 0 is trivial. S u p p o s ~ ~ that the assertion is true for max (k17..., k,) = h. Xow let niax {k,,..., k,} = h 1.

+

n

2’ 2-kl = 1, the set I = (i:ki = h + 1) contains a n even number of Writing I = { i , ,...,&,) we obtain

Since

indices.

1

for u = 1, ..., 1. Consequently the operator

n

2 Si

can also bc represented in the

1

sn

n,

form

2 Ti such

that -4(Ti)p5 2-kj and

1

2 - k ~= 1 , but max (hl,

...,h,,) = h.

1

Therefore, by hypothesis, it follows that A P r o p o s i t i o n . If Si E %(E,F ) , then

n

1 A(AS,),5 1/2, and choose k , E ( 0 ,1, ...} such that I n n 2-k*5 1. Now it follows froiri 2-kp1 2 A(S;)P 5 2-kf.Since 2’ 2 - k 4 5 1/2, we have P r o o f . Suppose that

1

thp preceding Ienima that A

(: 1

1

S , 5 1. This provcs the assertion.

6.2.5. Finally, we show that every quasi-nonned operator ideal is indeed a p-nornied opcratm ideal.

Theorem. Let vale& p-norm A,.

[a,A] be a

quasi-no,med operalor idenl. Then there exists a n equi-

6. Quasi-Normed Operator ldeala on Iianach Spaces

Proof. Compute the number p from

ic

99

= Z l l P - l , and put

n

Obviously A,(& 5 A(&).Since S = 2’ Si implies 1

we have I/Slj5 Ap(S).Hence A p ( I x ) = 1. The p-triangle inequality follows irnmediately froin thc definition of Ap, and condition (QOI,) is fulfilled, as well. Using 6.2.4 we get A(S) 2 21’pAp(S).This proves. together with A p ( S )5 A(S), the equivalence Qf A and A,.

6.3.

Nuclear Operators

6.3.1. An operator S E e ( E , E’) is called 7~uclenrif m

s = :cg?yi, 0,

1

with a,, a2,... t E‘ and y,, y2, ... E F such that 05

x IbJI ll?/lll <

cYz

+

1

We put M

N(S) := 1nf 2’ 11UJ I yJ, 1

where the infiniuni

IS taken

over all so-called nuclear represe?btution s described above

The class of all nuclear operators is denoted by 92. m

R e m a r k . The series ,?: CI,

8 y,

converges in the norm topology of e ( E , E ) .

1

Therefore the operator S is approxiuiable.

6.3.2. T h e o r e m .

[a, S ] is a normed operator ideal.

Proof. We use criterion 6.2.3.

(0) It follows froin Ix

= t

1 that I y E 92 and S ( I

5

1. Moreover, let

CL

Ix

= &yt, J q1 1 m

be any nuclear representation. Then 1 = 2’ therefore 1 N(Ix). 1

b)

Consequently 1 5

/n,q,l and 1

94

Part 2. Theory of Quasi-Normed Operator Ideals m

(1) Let S17 IS,, representations

... E %(E,P)such that 2 N(Si) <

00.

Given E

> 0, choose nuclear

1

m

Sn

X a n i 0Yni i=l

=

with a,

2 l b n i l l llynili S i=l

(1

+

E)

K(Sn).

Then m

M

S :=

2 anf@ y,i n.i=l

Sn=

n=l

and a,

,Z'

Ibnill

n.i=l

ll~nill2 (1

m

+

E)

z'R(fJn) n= 1 m

imply S E %(E,P)and N(S) (= (1 4E ) 2 Pi(&). n=l

(2) Let X

R(E,F ) and E > 0. Consider a nuclear representation m

S = ,Z ai 0Yi 1

such that 8

,Z llaill llyill I (1 1

+

E)

N(4

-

If T E E(Eo,E ) and R E E(F, Po),then M

RST

= ,Z T ' u ~ @

Zyi

1

and m

2' IIT'aill JPYill 1

I (1

+

E)

IIRIl N(S) IITll-

This proves that RST E %(Eo,Po)and N(RST) 5 (1

+

E)

(jRIlN(S) IlTll.

6.3.3. We now state an import,ant characterization. Theorem. An qerator S E E(E,P) is nuclear i f and only i f there exists a commutative diagram

E

S

6. Quasi-Normed Operator Ideals on Banach Spaces

95

such that So E B(lm, 11) is a diagonal operator of the form So(E,)= (oiti)with (a5)E E l . Furthermore, A E B(E)1,) nnd Y E B(ll, F ) . In this cuse,

N(S) = inf llyll llSollIIAli, where the infimum i s taken over alE possible jactorizations. Proof. Let S E %(E,F ) and E

> 0. Consider a nuclear representation

m

r 9 = 2 u i @ yi 1

such that

21 Ibill IIYtll 5 (1 + E ) “8) 00

ai

Put a! := -,

1~: := L -!

IIYiII ’

llaall

m

*

and ai := llaill IIyill. Next, by A z := ((2,a!)) and

Y(yi) := 2 qiyp we define operators A E B(E,I,)

and Y E f$,,

F ) . Then /[All= 1

m

1

and llYl1 = 1. Obviously IISoil = 2 ai. Consequently, there exists a factorization 1

S

=

YSoA such that jjYII llSoll[IAll 2 (1

+

E)

N(S).

Conversely, every diagonal operator So(ti)= (oJi) with (ai) E I, belongs to ?JZ(Zm, Zl), since M

So = 2 olei @ ei . 1 m

Moreover, ?i(So)5 2 loi/= IISoll. Therefore S = YS& is nuclear and we have W) I IIYII llS0ll IIAIL

6.3.4. I n the proof of the following lemma we consider a n important example of a non-trivial nuclear representation. Here I denotes the identity map. Lemma. N(1: 1;” --f

EL) = 1.

P r o o f . Let dn be the set of all n-tuples e = ( q )with

E~

= -j= 1.

Then

96

Part 2. Theory of Quasi-Normed Operator Ideals

Then

N(S,) 5

SUP 1.rig:n

I c ~ N(I: / 2'; -+ ZL) 5 SUP

10il.

I

Analogously we get

N(S, - 8,)

sup \ci\ for

nL

> n.

n d l m

Therefore (S,) is a n N-Cauchy sequence. Since S is the only possible limit, it follows that S E ?X&, Z-). Moreover,

N(X) = lim N(8,)

4 sup / O i l .

R

1

Conversely, we have sup IUiJ = IJSllg N(8). a

6.5.6. We now consider another important example. P r o p o s i t i o n . An operator S E 2(Zl,Zl) i s nuclear if and cmly if there is an infinite ( o i k ) such that

mcltriz

In thi.9 case, a3

Y(S) = ,r sup 1=1

Id,k/.

k

Proof. Suppose that S admits the described representation. By setting 8

2 o,~&

(2,a,>:=

for r

=

(&)

E I,

k - I

m

we define furictioiials a, E I , with /ju,(ioo= sup joIk{. Since S = k that S E %(Zl, Zl) and m

W S )I s Ib&ICJ lk,ill = 1

CJ

csup I

N

1

such that

bzkl

*

-1 k

Conversely, let S E R(2,, Zl). Given S=xai@yj

a , @ ei,it follows 1

E

> 0,consider a nuclear rcpresentation

6. Quasi-Normed Operator Ideals on Baiach Spaces

6.4.

97

Integral Operators

(i.i.1. An operator S

E B(E, F ) is called integral if there is a constant u 2 0 such that

/trace ( S L ) J5 c IILjl for all L E g ( F , E ) .

We put

I(S) := inf c. The class of all integral operators is denoted by 3. 6.4.2. T h e o r e m .

[a, I] is a normed

operator idecil.

Proof. We use criterion 6.2.3. (0) It follows from I, = 1 @ 1 that , 5 1[Ll1 for L c [trace(Ixh)1 = / ( L ( l )I)/ Hence 1% E 3 and I(1,)

5 1. Obviously 1 5 I(1Ap). b?

(1) Let S,, S,, ... F, 3 ( E ,F ) such that for every finite operator m

L=

g ( X ,3').

bj '3xi

1

we obtain

m

m

2' I(&) < co, and put AS:= 2 8,. 1

1

Then

98

Part 2. Theory of Quasi-Normed Operator Ideals

6.4.3. L e m m a . I(I: 1; -+ 1;) = 1. Proof. Using the representation

I = 2-"X e @ e cv"

described in 6.3.4 we obtain

I ( I : 1;

--f

5 2-"

1:)

I(e @ e ) = 1 . 8"

The converse estimate is trivial. 6.4.4. Finally we consider a n important example. P r o p o s i t i o n . The embedding map I froml, into I , i s integral. Moreover, we have I(I:1, +lm) = 1. Proof. P u t &,(51,

7

.

-,En, En+,,

.-.) := (51,.- *,

6n)

and

Jn(L ..., L):= (51,

...I

En, 0, ...).

Then 6.4.3 implies ltraee (JnQnL)/2 l/L\/ for all L

g(Zm,Z1).

Since trace ( I L )= lim trace (Jn&,L), we obtain 98

]trace ( I L ) /5 I/L\I for all L E &(la,l l ) . This proves that I E 3(Zl,I,)

6.5.

and I(I: I, -+1,)

i1.

Absolutely Summing Operators

6.5.1. An operator S E B(E,F ) is called absolutely summing if there is a constant cs 2 0 such that rn

for every fipite family of elements z,,

..., z, E E .

We put

P(S) := inf

G.

The class of all absolutely suinining operators is denoted by 'p. 6.5.2. Theorem.

[q,PI i s a normed operator ideal.

P r o o f . We use criterion 6.2.3. (0) Obviously 1% E 'p and P ( I x ) = 1.

99

6. Quasi-Xormed Operator Ideals on Banach Spaces

(1) Let X,,S,, ... E @(E, F ) such that xl, . . ., x, E E , then

m

m

2 P(S,) <

00,

and put S := 2' X,,.Jf 1

1

m

Consequently, we have 8 E @(E, F ) and P(S) 5 2 P(Xn). 1

(2) Let T E S(E0, E ) , S E @(E,F ) , and R E i?(F,Po).Then m

m

2 IIRST~~ll P IlRlI LY IlSTxBIl 1

for z:,

..., x:

1

E Eo. Hence RST E ?@(Eo, Po)and P(RST) 5 [jRljP(S) lITl[.

6.5.3. L e m m a . P(I: 1: -+ I f )

5 cK.

P r o o f . Khintchine's inequality (E.5.3) means that I/z/j25 cK2+

z,.(I

e)l

for all z E I : .

&"

Let zl,

...,x, E ZI;. Since llellm = 1, we obtain m

m

2 l l ~ i I l 25 c,T~-' 8C Y., i=l '' i = l

)(xi, e)l I CK sup

ZI(zi, a)l: Ilall, {im= l

=

<

ll.

This proves the assertion.

6.5.4. As a n easy consequence we have the P r o p o s i t i o n . The embedding map I from 1, into 1, tk absolutely summing. Moreover, 3 2,) 2 c,.

P(1: I ,

R e m a r k . Recently it was proved by S. J. SZAREK [l] that P(1: I , +Z2) the real case.

6.6.

= fiin

Hilbert Operators

6.6.1. An operator S E &(E,F ) is calleda Hilbert operator if S = Y A with A E and I'E & ( H ,F ) , where H is a Hilbert space. We put := inf IIYll IlAlI,

where the infimum is taken over all possible factorizations. The class of all Hilbert operators is denoted by

8.

2(E,H )

100

Part 2. Theory of Quasi-Norined Operator Ideals

6.6.2. Theorem. [Sj,HI is a normed operator ideal. P r o o f . We use criterion 6.2.3. (0) Obviously I, E $j and H(IAc) = 1, since X is a Hilhert space. 00

. .. E Sj(j(E, F ) with H(S,) < co. Given E > 0, we choose factori1 zations S,= Y , A , such that A, E e ( E ,H,) and Y , E e ( H , , F ) satisfy the conditions (1) Let S,, S,,

llA,/I2 5 (1

+

E)

H(S,) and llYnl125 (1

+

E)

H(S,\.

Form the Hilhert space W := Z,(fI,); cf. C.4.1. Put cu

00

A := 2 J,A,

Y := 2 Yn&,.

and

1

1

Then m

W

IlAI12 5 2' lIAflIIZ and 1

lIY1l25 2 IIYnl?. 1

Finally, it follows froin S = Y A that S E Sj(E, F ) . Moreover, we have

H(S) 5 IIYII IIAII 5 (1

+

m E)

2 H(Sn)* 1

(2) This property is trivial. 6.7.

Order Relations between Quasi-Normed Operator Ideals

6.7.1. Let [a,A] and [%, B] be quasi-normed operator ideals. Then [a,A] [b,B] means that 3 5 !B and -4(S) 2 B(S) for all S E a. The collectioii of all quasinormed operator ideals is something like a lattice with respect to the natural ordering. 6.7.2. First we state an improvement of 6.3.2. T h e o r e m . [%, N] is the smallest normed operator ideal.

> 0, we consider a

Proof. Let S E %(E,F ) . Given E

nuclear representation

m

a , 8y t

S = 1

such that 00

c IIaill ll?%ll 5 (1 f 1

n

Put S,:= C 0 , @ y t . Let

8)

W).

[a,A] be any norined operator ideal. Then

1

m

A(S, - 8,) 5 ,Z' A(a, 3 yJ

m

=

2 Ilaill lly,ll

for

m

>~

2 .

n+l

n+l

This shows that (8,) is an A-Cauchy sequence. Since S is the only possible limit, we have S C U(E, P).Moreover, 11

A(S) = lim A(&',) 5 lirn n

IluJ]liyill 1

5 (1

+

E)

N(S) .

6 . Quasi-Normed Operator Ideals on Banach Spaccs

101

R e m a r k . It follows from 6.2.5 and 18.5.1 that there docs not exist a smallest quasi-nornied operator ideal.

6.7.3. P r o p o s i t i o n . [%, X ] c [3, I]. Proof. The inclusion is implied by 6.7.2. On the other hand, the integral embedding map I from I , into I , cannot be nuclear, since I 4 R.

R e m a r k . Further informations on the relationship between [%, N] and [a, I] may be found in 5.6.4 and 8.7.6.

6.7.4. P r o p o s i t i o n . [3, I] c [q, PI. Proof. Let S E 3 ( E , F ) . If xl, ..., x, E E , then choose b,,

..., b,,* E F'

such that

m

(A!.rt, h,) = ilSxiil and ilbiij = 1. Put L := 2 bi @ x i . We now have 1

1ILl = sup (I(LY,a>l:llvll I 1, llail 5 11

Consequently

I

1

This proves that S q ( E , E') and P(S) 5 I(#). The embedding map 1 from I , into I, is absolutely summing, but not integral.

6.7.6. For coinpleteness we inentioil the following result which follows from 17.3.7 and 17.3.9. Proposition.

6.8.

[q.P] c [$,HI.

Finite Nuclear Norm

63.1. Let 8 E g ( E , F ) m d put n

PL'O(8)

:= i l l f

1/(7rli

ijyt'j,

1

ithere the infimum is taken over all finite representations n

s = 1a , 3 yi. 1

6.8.2. First \be state a n easy result. Theorem. KO is the Inmest norm on the operdor ideal 5.

102

Part 2. Theory of Quasi-Normed Operator Ideals

6.8.3. It can be shown that E and NO do not coincide for all finite operators. Consequently the abstract completion of 8 with respect to the norm "0 is not a normetl operator ideal. However, we have the iniportant P r o p o s i t i o n . Let E or P be finite dimemional. Then No@) = X(S) = I(S)for alf S

E B(E,F ) . Proof. By 6.7.2 and 6.8.2 we have I(S) 5 N(S) 5 No(&).

Now the converse inequality will be checked. Applying the Hahn-Banach theorem we can find a functional L on B(E, F ) such that (8,L ) = W ( S )and I(T, L)j 5 No(T) for all T E B(E, F ) . If E is finite dimensional, then there exists L E B(F, E ) with (a0y, L ) = (Ly,u) whenever a E E' and y E F. Hence ( S ,L ) = trace (SL). Moreover, it follows from I(Ly, u)I 5 No(u 0y) 5 llu/\IIyll that IlLll 5 1. Therefore

No(#)= (8,L ) = trace (SL) 5 I(&)IIL'l 5 I(S). Now suppose that F is finite dimensional. Given such that

> 0, we choose yl,..., y,

E

5 L7F

n

UF

s u (Y*+ SUFI, 1

+

8

where 6 := -. Then [lLll 5 (1 E ) sup IILy,lj for all L E B(F, E ) . By setting I+& a ((Lyi),Ao):= trace (SL) we define a functional A , on the linear subset M := ( ( L y i ) :L E B(F, E ) ]of Z",(E). Moreover,

I((k/i)> A0)I

=

/trace (S-41

s I(&)llLll 5 (1 4-

E)

IfS)SUP lILYz11-

By the Hahn-Banach theorem there exists an extension A such that l((zn), A)I

5 (1

+

E)

I(S) SUP IlzeII for all (J,) E z",(E)-

This functional admits a representation 71

((xJ, A ) = 2 '( x d ,ai) with

..., u,

CI~,

E E',

1

n

and we have

2'

11aJ 5

(1

+

E)

I@). Now, if Lo := b @ x, it follows that

1 n

(83, a> = trace (SL,) = ((Lo?/t),A ) =

z

E E and b F'. This means that S

2' a, @ g?.Consequently

(z,

4 (Yz,b )

1

n

for all .E

=

1

n

No(S) 5 2 ' Ilat'l IIyeII 5 (1 1

+

I(#).

So we have seen that W(S) 5 I(S).

6.5.4. We now prove an easy consequence of 6.8.3. L e m m a 1. Let X E g(Eo,E ) nnd S Z(E,3).Then S"(SS)

s I(S)11x1'.

6. Quasi-Normed Operator Idcnls on Bnnach Spaces

103

Proof. Take the factorization X = JX,, where J is the injection from M ( X ) into E. Then it follows from l&Coll = IlXIj that

s I(S)(IX(1.

NO(SX) I NO(SJ) llXoll = I(SJ)!lXol1

Using the same method we can also check the dual L e m m a 2. Let S E 3 ( E , F ) and B E $(F, F,). Then

NO(BS) 5 IIBil I(S). 6.8.5. Finally, we state a uscful inequality.

P r o p o s i t i o n . Let

.6.9.

S E $(E, E ) . Then [trace (S)[ 5 KO@).

Notes

The concept of a quasi-normed operator ideal on the class of Banach spaces was introduced by the author about 1969. However, the main ideas go back t o the theory of tensor products. Indispensable references are the memoir [GRO] and the “RR6sumir de la theorie mbtrique des produits tensoriels topologiques” by the same author. See also I. AMEMIYA/K. SHIQA[l]. Suclear operators in Banach spaces were independently defined by A. GROTHENDIECK [l] and A. F. RUSTON[l], [ 2 ] . The notions of an integral and absolutely summing operator as well as a Hilbert operator are also due to A. GROTHENDIECK. A few results are taken from A. PELCZYNSKI/W. SZLENK[l] and A. PIETSCH r3], [ 3 ] . Recommendations for further reading:

[LOTI. 3. ROSENBEBGER [l].

Products and Quotients of Quasi-Normed Operator Ideals

7.

The purpose of this chapter is to show that products and quotients of quasi-normed operator ideals may he equipped with a quasi-norm, as well. Important examples will be treated in later chapters.

7.1.

Produets of Quasi-Normed Operator Ideals

i.l.l. Let [a.A] and [d, B] be quasi-nonned operator ideals. For every operator N c ( E ,G) belonging to the product 'u o b we put

A o B(S) := inf A(X) B( Y ) , where the infimum is taken over all factorizations S = 2-1' with A- E %(F. C ) a d Y € B(E, F ) . Clearly, ,4" denotes the n-th power of the quasi-norm A. R e m z r k . The product

7.1.2. Theoreni.

[ao $3,

[ao B, A o B] will frequently be written as [U, A] o [%, HI. A o B] is a quasi-normed operator ideal.

P r o o f . Using the method of 3.1.2 the quasi-norm properties of A o B can easily be checked.

To show the eonipleteness, according to 6.2.5, we may suppose that [a,A] and are p-nornied and 2-normed, respectively. Then, using 6.3.2, we can prove that [U o b, A o B] is an r-normed operator ideal, where l i r := lip l/q. Let S, E o B(E,F ) such that

[a,B]

+

bo

A o B(S,)'

< M.

1

Given E > 0, we can find factorizations X, = X,Yn such that the following conditions are satisfied:

+ c) A o B(S,)]'/p, + A o B(S,)]'/Q.

X , E U(M,, -Fj and A(X,) 5 [(l Y , E B(E,M,) and B( Y r L5 ) [( 1 co

Put S :=

3 X,Q,,

N

E)

W

Y :=2 ' J,Y,, and M := Z2(Wn).Then

B( ITn? 5 (1 + c

m

2 A o B(S,)'

) ~

1

1

irnply X E % ( M .F ) and Y E B(E,M ) . Moreover, we have A(X)P 5 (I +. c)r

m

A o B(AY,)~ 1

7. Products and Quotients of Quasi-Nolined Operator Ideals

105

and m

B ( Y ) Q5 (1

+ E)'ZA o B(S,)'. 1

m

Since S :=2 S, has the factorization S = X Y it follows that S E

o

b(E,P)and

1

m

A o B(S)' 5 (1

+ 2' A o B(S,)'. t)'

1

Sow 6.2.3 yields the assertion.

7.1.3. Kext wr give an example which will be treated in 24.6.2.

[m, li.l]]o [3,I] = [a, N]. 7.1.4. Theorem. Let [a,ll.il] and [b,ii.il] be [?l 0 b, li.l;l = [a,11.113 [B, 11.113. Proposition.

nor?iied operakor ideab. TiLen we have

0

Proof. We consider an operator S t B(E, F ) belonging to o b. Then there exists jmtorization S = XI' with X E % ( M , P ) and Y t B(E, M ) , where M is a suitable Banach space. Obviously, we may suppose that IlY;!5 IISil. Sow

3

llflio := max (llfil, IlXfIl) for f E M defines an equivalent norm l].llo on M . Let M , denote the Banach space obtained from M by this change of the norm. Piirtherniore, let XoE31(Mo,P)and YoEB(E,M,) be the operators given by the same rules as X and Y , respectively. It follows from IlXOfli 5 l l f i l o that liXo/]5 1. Moreover,

/IYozllo= niax (IIYsli, IlSxll) 5 liSI/ ll.~ll for all x 5 E implies llYoll 5 l]Sl/.This proves that IlSIl = inf /jX//IIYl;, where the infimumis taken over all admissible factorizations. R e m a r k . We see froin 7.1.2 and the above theorem that the product of closed operator ideals is closed, as well.

7.1.6. A quasi-norined operator ideal

[a,A] is called idempotent if [a,A] = [W. A']].

7.l.G. From 3.1.7 and 7.1.4 we obtain the P r o p o s i t i o n . Y'he normed operator ideals [6,iI.ll1, [52, il.111, [%k (!re idempotent.

7.1.7. P r o p o s i t i o n . T h e norined operator ideal [Sj,HI is idempotent. 7.1.8. Finally, let us mention a fact which follows from later results. P r o p o s i t i o n . The noi,mcl operator ideals poteizt.

[a, S ] . [3. 21, n7d [?$3. PI are nof i d e w

106

Part, 2. Theory of Quasi-Normed Operator Ideals

7.2.

Quotients of Quasi-Normed Operator Ideals

7.2.1. Let

[a,A]

and [b,B] be quasi-normed operator ideals. 3For every operator

X E e ( E , F ) belonging to the left-hand quotient 9l-l o b me put A-l o B(S) := SUP (B(Y8): Y E %(F, Po), A ( Y ) 5 1). Here Fo ranges over all Banach spaces. On the right-hand quotient a o b-1 the expression A o W(S)is definedanalogously. R e m a r k . The quotients [a o 8-1, A o B-11 and [U-l o b, A-l o B] will frequent,ly be written as [a,A] o [b,B]-1 and [a,A]-l o [b,B], respectively. 7.2.2. Theorem. iderJs.

[%-I

o 8, :Z-1 o B] and

[ao b-l, A o B-l] are quasi-normed operator

Proof. The main point is to establish the existence of A-l o B. Therefore let us suppose that the supremum is not finite for some operator S E U-l o b(E,F ) . Then we can find Y, E a ( F , F,) such that

A(Y,) 5 ( 2 ~ ) -and ~ B(Y,S) 2 n for

n=

1, 2,

...

Put, F,, := I2(FVb). Since

k

m

the partial sums 2' J,Y, form a n A-Cauchy sequence. Consequently Y := 2 J,Y, 1

belongs to tradiction.

a, and we obtain

1

.n

B(Y,S)

= B(&,YX)

5 B(YX), which is a con-

Finally, it is easy to check the ideal properties and the completeness, as well.

7.3.

Notes

See the bibliogra,phical notes of Chapter 3.

8.

Quasi-Normed Operator Ideals with Special Properties

I n the following we investigate procedures on the collection of all quasi-normed operator ideals. I n this case the rule

new: [a,A] + [ a n e ~ Anew] , transforms every quasi-normed operator ideal [a,A] into a quasi-normed operator ideal [ a n e w , Ane~v]. I n comparision with Chapter 4 we have of course to take into account some modifications due to the presence of quasi-norms. Most of them, however, are straightforward. Of considerable importance are the procedures min and maz. In particular, we inention the formulas

and J.L. The concept of ultrastability, introduced by D.DACUNHA-CASTELLE a useful tool to prove the niaxiniality of some special quasinormed operator ideals; cf. Chapter 14 and 19-

KRIVINE in 1971, is

8.1.

Procedures

8.1.1. A rule

new:

[a,A] -+

[!Blew,

Anew]

which defines a new quasi-nornied operator ideal normed operator ideal [a,A] is called a procedure. R e m a r k . The quasi-normed operator ideal as [a,Apw.

[anew,

[anew,

Anew] for every quasi-

Anew] will frequently be written

8.1.2. We now list some special properties: (M) If [a,A] S [a,B], then [U, A]new G [S, B P w (monotony). (I) ([a,A]new)new= [U, A]ne!%.for all [a,A] (idenipotence). A monotone and idempotent procedure is a hull procedure if [U, A] 2 [%, A p w m d a kernel procedure if [a.A] 2 [a,A P w for all quasi-normed operator ideals. Impectively .

5.2.

Symmetric Quasi-Normed Operator Ideals

8.2.1. Let

[a,A] be a quasi-normed operator ideal. Then we set

Adm'(S) := A@') for all X and

[?@ml,

Udml(E,F ) ,

Adual] is called the dual quasi-normed operator ideal.

108

Part 2 . Theory of Quasi-Normed operator Ideals

8.2.2. Obviously we have the

Theorem.

[ U d l ~ a l , A d ~ ~ ils] a

quasi-normed operator d e a l .

8.2.3. Since the property (M) is satisfied, we can formulate the T h e o r e m . T h e rule

[a,A]

dual:

--7

[%dual, Adual]

is CL monotone procedure. 8.2.4. il quasi-noriiied operator ideal [a,A] is called symmetric if [a,A] & [a,A]dual. I n case [a,A] = [a,A]dual the quasi-normed operator ideal is said to be coinpletel?y s?jmmetric.

8.3.5. We now consider some examples. P r o p o s i t i o n . The nortned operator ideals [S I] and [ S j , H] are cornpletel~s~jmt~tetrir. Proof. Sccording t,o 19.1.1 and 19.3.8 the assertion is a special case of 19.1.4.

8.3.6. P r o p o s i t i o n . The normed operator ideal pletely symmetric.

[a, N] i s

syvitiietric, but not con1-

P r o o f . The inclusion [a, N] [a,X]dual is evident. On the other hand, T. FIGIEL/ W. B. JOHNSON [ I] have constructed an operator S E ?&dual which fails to be nuclear.

8.2.7. P r o p o s i t i o n . The normed operator ideal [$,P] i s iiot symmetric. Proof. By 6.5.4, the embedding inap I from 1, into I , is absolutely summing. the other hand I 6 ?@dual(ll, 12).

8.3.

Regular Quasi-NormedOperator Ideals

8.3.1. Let

[a,A] be a quasi-normed operator ideal. Then we set

Areg(S)

and

011

[ W g , drep]

:= A(KFS) for all

X t Areg(E, F ) ,

is called the regular hull of

8.3.2. Theorem.

[%reg, Areg]

[a,A].

is a quasi-normed operator idetrl.

P r o o f . Using the method of 4.5.2 we can easily show that The coinpletencss of U..g(E, F ) is evidcrit, a s u-ell.

AIeg

is a quasi-norni.

8.3.3. Corresponding to 4.5.3 we have the Theorein. il'he :tile reg:

[a.8 1 -3

[%reg, Alrg]

i s n hull procedzw. 8.3.4. A quzsi-norrned operator ideal [U, A] is called regzclar if

[a,A] = [Zr, h ] r e g . .

8. Quasi-Normed Operator Ideals with Special Properties

I09

8.3.5, P r o p o s i t i o n . Let [U, A] be a quasi-nolnzed operator ideal. il'tien 18, A]*ua1 i 4 regutur. P r o o f . Use the method of 4.5.6.

8.3.6. The next result is analogous to 4.5.7. P r o p o s i t i o n . A quasi-normed operator idenl if it is regular and symmetric.

completely sywmzetric i f and only

is

5.3.7. We now consider some examples. P r o p o s i t i o n . T h e normed operator ideals [3,I],

[v,PI, and 18,HI ure 7egular.

P r o o f . The regularity of [3, I] and [$j, HI follows from 8.2.5 and 8.3.6. The assertion is trivial for [P, PI; cf. 8.4.9.

5.3.8. By 8.2.6 and 8.3.6 we have the P r o p o s i t i o n . T h e nornzed operator idenl

8.4.

[a, S] i.s no! regular.

Injective Quasi-Xormed Operat,orIdeals

8.1.1. Let

[a,A] be a quasi-normed operator ideal. Then we set

A I ~ J ( S:= ) A(J,S)

for all S E %~"J(F:,F ) ,

and [!@"J, Ainj] is called the injective hull of

[a,A].

8.9.2. T h e o r e m . [ I U ~ Z ~8J1, n ~ 1is CG quasi-norwed operutor ideal Proof. Use the method of 4.6.2.

8.4.3. Corresponding to 1.6.4 we have the T h e o r e m . T h e rule

inj: [a,A] --f [ ~ I I A ~ JI ~, J ]

is a hull procedure.

5.4.4. To prove the next rcsult we need a L e m m a . Let So E 2(E,F,) and S E 2 ( E ,J'). Suppose that Yo has the metric extension property. If ilSoxll 5 ~/S.I$for all x E E , then tl~ereexists an operator B E 2(F, F,) sue?& that So = BS and l\Bl!5 1. Proof. By setting B o y := yo, where y = Sx and yo = S,a, we obtain a welldefined linear map from M ( S ) into F , with ~ ~ 5B 1., Consequently ~ ~ there is a n extension B E B(F, F,) such that So = BS and IIBIl 5 1. P r o p o s i t i o n . Let [a,A] be a quasi-normed operator ideal. Lei So t e ( E ,F,) and 8 E %(E, F ) . I TfllSoxll 5 &!3xll for all x E E , then SoE Um](E, F,) and ,4ln3(SO)5 A(S).

Proof. By the preceding lemma, we can find an operator B E e ( F , PinJ) such that JF,So= BS and IjBIj 5 1. Hence JF,S, E U(E, FYI) arid A(Jp0S,)5 A@'). This roves the assertion.

110

Part 2. Theory of Quasi-Normed Operat,or Ideals

8.4.5. P r o p o s i t i o n . An operator S E B(E, F ) belorup to Rinj if and only if there exists an absolutely summable sequence of fzcnctionak n,, a2, . .. E E' such that m

lIS2ll 5

i(z, ai>l for all

2

E E.

1

In

this case, m

Ninj(S)

2 ]la&,

= inf

1

where the infirmrn is talcen over all admissible sequences ( a d ) .

Proof. Let S E ! R ~ J ( E F ), . Then, given E

> 0, we can find a nuclear representation

m

a j 18 yp

JFS = 1

such that

X1 llaill IlyPil 5 (1 + M

E)

N'W).

Obviously we may suppose that

lly&ll= 1. Then

00

2 llaill 5 (1 + E ) Nbj(S). 1

Conversely, let 00

llSsl/ 5 2 I(%, ai>l for all z E E . 1

00

By setting A z := ((2,ai)) we define a n operator A E B(E, Z,) with N(S) 5 Z Ila& 1

Moreover, IlSzIl 5 IIAzll for all z E E. Now, by 8.4.4, it follows that S E Binj(E, F ) m

and W j ( S )

5

llaill. 1

8.4.6. As an immediate consequence of the preceding result we obtain the L e m m a . Pu'hj(1:l; + 1;) 5 cK. Proof. Khintchine's inequality means that ~2 '2\(z, e)l - ~for all z E 1;.

1 1 ~ 1 15~ ~

6"

Thereforc

Nmj(1: 2:

--f

Z:)

5c

= ~ - ~lleilm X = cK. 8"

8.4.7. We state without proof the following criterion which can be derived from 17.3.2 and 19.2.7.

8. Quasi-NormedOperator Ideals with Special Properties

111

P r o p o s i t i o n . An operator S E B(E,F ) belongs to 3 i n j i f and only if there exists a finite regular Borel measure p on the weakly compact set UE,such that IjSzil 5

1l(x,

a)I dp(a) f o r aZl

x E E.

u,

I n this case,

Iinj(S)

= inf

p( U E , ) ,

d i e r e the infimum i s taken over all admissible Borel measures.

R e m a r k . We have N i n j ( S ) = Iinj(S) for all S E P 8.4.8. A quasi-normed operator ideal

i ;

cf. 10.3.1.

[a,A] is called injective if [g,A] = [N, A]i*ij.

S.1.9. P r o p o s i t i o n . T h e normed operator ideats ['p, P] and

[a,HI are injective.

Proof. The injectivity of [Q, PI is evident. Then there exists a factoriza.tion J F S= Y A , where A E @(E,H ) ,

Let S 6 $jinj(E,F).

Y E B(H, Finj),and jjYlj IlAlj 5 (1 + E ) H h j ( S ) . P u t Ho := Y-l[M(JF)],and denote

the injection from Ho into H by J . Let Q be the orthogonal surjection from H onto H,. Setting A , := QA and Yo := JF'YJ we obtain operators A , E B(E, H,) and Y o E B(H,, F ) with liAoll 5 l/A/\and jlYoll 5 I)YII.Since S = Y,A,, it follows that S E @(E,F ) and H(S) 5 IIY,ll /IA,lI 5 (1 c) H n j ( S ) . This proves the inclusion [$,H p [B, HI.

+

8.4.10. P r o p o s i t i o n . T h e nopmed operator ideals

[a,N] and [a, I] are not injective.

P r o o f . Consider a diagonal operator S(&) = (an(,). Then S E ! 3 1 2 , 12) implies (a,) E I,. On the other hand, it follows from 8.4.6 that S E ainj(Z2, 1,) whenever (a,) E 1,. Therefore %(&, 12) $: W j ( E , , Z2) and S(12,Z2) =+ 3inj(Z2, 1,).

8.5.

Surjective Quasi-Normed Operator Ideals

8.5.1, Let

[a, A] be a quasi-normed operator ideal. Then we set

Asur(S) := -k(SQE)

and

[Usur, B s u r ]

for all 8 E USUr(E, F),

is called the surjective hull of [U, A].

8.52. Theorem.

[Usur, A s ~ r ]is

a quasi-normed operator ideal.

8.5.3. Corresponding to 4.7.4we have the Theorem. T h e rule SUI':

[a,A]

--f

[rUsur > - Asur

1

is ( I hull procedure. s.5.P. The proof of t h e following result is based on the

L e m m a . Let SoE B(Eo, F ) and S E 2(E,F ) . SuppGse that E, has the metric lifting property. If S,(UEo)& S ( U E )and E > 0, then there exists c m operator A B(E',, E ) such that So = S A and /!All _I 1. E.

+

112

Part 2 . Theory of Quasi-Normed Oprrator ldrals

P r o p o s i t i o n . Let

[a,A] be

(I

quasi-normed operator ideal. Let So <- 2(Eo,P ) and ASu1(SO)5 A(&).

S E %(E,F ) . If Bo(UE0)5 A S ( L ~then ~ ) , So ?@l(EO,F ) und

8.5.5. P r o p o s i t i o n . An operntor S E Z ( E , F ) bdongs to T !W if and only if there exhts a n absolutely suwiniable sequence of elements yl, yz. .. . E 3' such that

In this case. 00

Ns~r(8)= inf 2 llyJ. 1

w h e ~ ethe infimum i s taken over all admissible sequences (1~~). 8.5.6. A quasi-normed operator ideal

[a,A] is called surjective if [a,A] = [H, A p .

8.5.7. P r o p o s i t i o n . The normed operator ideal [$j, HI i s surjective. Proof. By 8.5.11, the assertion follows from 8.2.5 and 8.4.9.

8.5.8. P r o p o s i t i o n . T'he normed operator ideals surjective.

[a, N], [S, I],

and [!@,1'1 are not

8.5.9. Injectivity a i d surjectivity are dual properties.

Theorem.

([a,A]dual)su

=

([a,A]lnj)dul

and

([a,A]t'ual)in3

&

([a,A]SUl)dml.

8.5.10. P r o p o s i t i o n 1. For every injective quasi-normed operator ideal the dual quasinormed operator ideal i s surjective. P r o p o s i t i o n 2 . For every surjective quasi-normed operator ideal the dual quasinormed operator ideal i s injective.

8.5.11. P r o p o s i t i o n . For every completely symmetric quasi-nornzed operator ideal injwtivity and surjectivity are eyuiinlent properties. 5.5.12. P r o p o s i t i o n .

8.6.

([a,A]m~)sur

=

([a,A]sur)lnJ.

Minimal Quasi-Normed Operator Ideals

[a,8 1 be a quasi-normed operator ideal. Then [ a m i n , -4min] := [8,i1.111 o [a,A] 0 [8,(/.I[] is called the minimal kernel of [a,Al. 8.6.1. Let

8.6.2. By 7.1.3 we have the

Theorem.

[amin, Amin]

i s a quasi-nornied operutor ideal.

8. Quasi-Normed Operator Ideals with Special Properties

8.6.3. The next result is related to 4.8.4. Theorem. T h e rule m i r i : [U,A]

--f [ U m m

>A m i n I

is CI kernel procedwe. A.6.4. To investigate the following exanrple we need an elementary Leniuia. Let ( o f )i I,. l'hen, given M

E

> 0, there exit& (ei)

N

P r o o f . We only treat the non-trivial case, where

Choose a natural number m with

and put

m

na

co .such that

113

114

Part 3. Theory of Quasi-Normed Operator Ideals

P u t Soo(fi) := (pi20iti)and R ( t j ):= (piti). Then we have the diagram R 4

ly

+fB

SO

It follows from R'A E 6 ( E , Z,) Noreover,

ZI), and Y E E 6(Zl,F ) that S E %mi"(E,F).

So,E %(Zm, 00

,z eT2 bil 5 (1 + &I2 N(S) 5 R(S). Hence [a,N] E [a,XImin G [z, Ilmin.

IIYBlI N W O O ) llE'4 5 IIYll Mll iniplies P i n ( S )

1

Suppose that X E g ( E , Eo), So E 3 ( E o ,Yo), and B E g(Fo,F ) . Then there are factorizations X = XoQ: and B = J$Bo, where N := N ( X ) and M := M ( B ) .Since llXoll = //XI/and llBoil = IlBlI, by 6.8.3, we have

IV(BS0X) 5 W

O S O ~ O = )

I(BosOx0)5

1 1 4 1W O )

11x11.

(* 1

Xow let X E 6 ( E , ,To), So E 3 ( E o ,Fo),and B E @(Po,F ) . Then we can find operators X n E g(E,Eo)and B, E g ( F o ,F ) such that

X = Il.ll-lim X , and B

= ll.]l-limB,.

16

n

Using (*) we obtain

+

N(BnSoXn- B,80Xm) 2 K ( ( B n - B,) Sox',) K(BmSo(Xn- Xm))

5 I P n - B m I I I(&,) IIXnll

+ IlBmIl I(&) Ilxn

-

Xmll.

Hence (BnSoXn)must be an N-Cauchy sequence. Since BSoX = 1I.li-lim BnSoX, n is the only possible limit, it follows that BXoX E %(E,F ) . Moreover,

This provts that

[3,I]m1nC= [a,N].

8.6.5. A quasi-uormed operator ideal [U, A] is called wzininzal if [U. A] = [U, AJmin. 8.6.6. P r o p o s i t i o n . The nornied operator ideal [%, pu'] is minimal. Proof. The assertion follows from 8.6.4.

8.6.7. P r o p o s i t i o n . T h e normed operator ideals [S, I], [!$ PI, I and , [@, €11 are noi nzinimal. P r o o f . Every nrinimal operator ideal consists of approxirnnble operators only. Therefore the assertion is implied by 6.4.4, 6.5.4, and I , 8.

Y. Quasi-Normecl Operator Ideals with Special Properties

8.7.

11.5

Maximal Quasi-Normed Operator Ideals

[a,A] be a quasi-norined operator ideal. Then [ a m a x , Amax] := [8,l].Jl]-' 0 [U, A] 0 [a, ll.lJ]-l is called the maximal huZZ of [a,A4]. 8.7.1. Lct

8.7.2. By 7.2.2 we have the T h e o r e in.

[%ma-\, Am"']

i s a quasi-normed operatoi ideal.

8.7.3. The next result corresponds to 4.9.4. Theorem. The rule

mux: [a,A] -+ [ a m a x ,

-4ma\]

i s a hull procedure.

[a,

8.7.4. Theorem. Let A] be a p-novmed operator ideal. Then S E B(E,F ) belongs to i f and only i f there exists a constant (T 2 0 such that A(BSX) I G /IRll /jXll for all X E g(Eo,E ) and B E g ( F , P o ) , where Eo and Fo are arbitrary Bnnach spaces. I n this case, Amah(S)

:= inf

G.

Proof. If S E %max(E, F ) , then &n&x(S) := sup AlBSX), where the supremum is taken over all X E Q(Eo,E ) and B E Q ( F , Fo) such that liX'lI 2 1 and IIBil 5 1. Hence

A(BSX)5

Amax(S)

IlBll IjXlj for all

X E g ( E o ,E )

and B E g ( F , P o ) .

Conversely, let S E E(E, F ) satisfy the above condition. If X E @(E,, E ) and B E @ ( F ,F o ) , then there are X,, E 8(Eo,E ) and B, E g ( F , Fo) with X = l/.ll-lh X, n and B = ~ ~ . ~ ~Bra. - l iIt m follows from n

A(BnSXn- B,XX,)p

5 A((Bn- B,)

S9,)P

- @[llBn- BdiplI&llp

1A(B,S(X,z -

+ lIB,l?

I\&

X,))'

- Xd'l

that (B,SXn) is an A-Cauchy sequence. Since BSX = ll.~l-hn B,XX, is thc only 1z possible liniit, ~ 7 have e BSX E 'U(Eo,F o ) .Moreover,

A(BSX) = lim A(B,#X;,) =( (r lim llBnIl llXnll = G IIBI!IIXII. IL

Hence S E

'Umax(E,

F ) aiid

11

Ama\(S)

5 (T.

R e m a r k . The following theorem shows that in the above criterion the Banach spaces Eo and Po may supposed to he finite dimensional.

8.7.5. Thvnrciti. Let [a,-41 be a p-normed operator ideal. Then S 5 2 ( E , F ) belongs to a m a ' ( E , E') if t r n d only if tkere exists a constant 5 2 0 such that A(Q;&'Jff) 5 5 for nlZ 31

Dim ( E ) and

N E Cod (3').

116

Pa16 2. Theory of Quasi-No1ined Operator Idcals

I n this case. .imax(Aqj

= inf

(T.

P r o o f . Let S c % - x ( E , finite.

Z'). Then S(QTSJTf)5

Am~~~(S since ),

QZ- and J g arc

Converdely, let X E g(E,, E') and R < g(F,Po). Then theie are factorizations = JCX, and B = BoQT, such that //Xoll= IIXli and ]IB,ji = IIBII,where M := M ( S j and N := A7(B).If the ahore condition is satisfied, then

S

P)and Ama'(S) 5 6,by 8.7.4.

Therefore S f %max(E,

Remark. Obviously [Umax, Amax] is unicply dt+winined if t h e p-norm A is only linown on the collection of all elementary operators.

[a, plT]max = [3. I].

8.7.6. P r o p o s i t i o n .

Proof. Let S E !.l@ax(E, F ) a n d L E g ( F , 3;). T h ~ ntherc is a factorization L = J:&,&$ snch that IILoi\I= \ID/,where 111 := N ( L j and N := N ( L ) . R y 6.8.3 and 6.8.5 we have Itrace ( S L ) /= /trace ( & ~ . S J ~ L ,5) /K o ( QiS J sLo ) =

Hence S

s(Q:,sJE,r;,) 5 N(Q;sJ;I)

1IL,II 5 P

a y s )

ilL\I.

E 3 ( E , F ) and I(S) 5 Nmax(S).

Conversely, let S

N(&:+!!Jif)

3 ( E , P),41 E Diin ( E ) ,ancl N =

c

Cod ( F ) .According to 6.8.3

I(&:.SJ$f)5 J(S).

Vsing 8.7.5 we obtain S E %max(E, F ) and 8.i.7. A qmsi noriried operator ideal

Pax(&')

5 I@).

[a,A] is called nzcrxinzal if

[U, A]

j21,

:

.i]m~11.

8.7.8. We now consider soine examples.

Pi o p o s i t i o n . The norwed operntor ideals [3.I], ib,PI. nnd

Pi oof. The maxiiuality of [3,11 follows from [3, I] Let S E ?@max(E,F ) and x l , ....a , E E. S o w \Srt, b,) = I/Sx,/iand llbLl; 1. By setting

UP

-=

[a,Xjma'.

choose b,,

n

X ( t , ) :-=

ttzL and By := ((y, b , ) ) 1

we define operators XI E

g(Z2,E ) and R f g(F,.);Z

[B, HI m e nzaxrmal.

Jlorcovcr,

..., 6,

t

F'

P W ~t h ~

t

8. Qumi-Normed Operator Ideals with Spccial Properties

117

Xow i t follows that n

7I

Hence S E p ( E , 3') and P(S) 5 P n ~ a x ( S ) .

The inaximality of [SJ, HI is a consequence of 19.3.8.

[a, S ] i s not ?nuximul. Proof. The assertion follows from [a, K] =+ [S, I] = [a, 'N]mm.

4.5.9. P r o p o s i t i o n . T h e norined operator ideal

S.i.10. P r o p o s i t i o n . Let

[a,A]

be a quasi-normed operator ideal. Then

[U, _4]reg & [U,A ] m x . Proof. Use the method of 4.9.10. b.7.11. As an immediate consequence we have the

P r o p o s i t i o n . Every iwzximnl quasi-norrned operator ideal is regular.

8.7.12. P r o p o s i t i o n . Let

([a,A]dual)max

=

[a,A] be a p-normed

operator ideal. T h e n

([U,A p = ) d u a l .

P r o o f . Let S E 2(E,F ) belong to (Umax)dml. If X E g(Eo, E ) and B E g ( F , PO). then A(X'S'B') 5 Amx(S') l(X']]IlB'11.

It follows that AduaI(BSX)

Hence S E

5 ( A m - ) d m l ( S ) lIBl]IlXll.

(Udml)max(E,

P)and (Adual )max ( 8 )5 (Am=)dual(S).

Conversely, let S E 2 ( E , F ) belong to ( U d ~ 1 ) m a x . Suppose that Y s ( P &F ' ) and 14 E 3(E', EL), where the Banach spaces E, and F , are finite din~ensional. Then there is B E $ ( F , Po) such that Y = B'. Obviously llBll = llY11. Given E > 0, by E.3.2, we can find X E $(Eo, E) with AS'B' = X'S'B' and llXll 5 (1 $- E ) IIAll. Since AS'Y = (BSX)',we have

A(AS'Y) = A d l * I ( B S X ) 2

(Bdal)ma(S)

5 (I + F ) S o w 8.7.4 (remark) yields S E

(Umax)dml

(A-1

liB/l ilXll

1 . (8)iiAiI lil'il. Inax

and (-4max)dml(S) 5

(Adml)max(S).

R e m a r k . It is very likely that the above formula also holds for arbitrary quasiiioriued operator ideals.

*

8.7.13. P r o p o s i t i o n . Let [U, A] be a quasi-normed operator ideal. T h e n

([U,A]inj)-

=

([a,A]max)inj.

118

Part 2 . Theory of Quasi-Normed Operator Ideds

Proof. Let S E e ( E , F ) belong to (Urnax)lnj.Suppose that X E B(E,, E ) and R E B(F, F,). Then there csists B, E B(F1*j,F F ) with JFoB= BoJFand JIB,]]5 (1 c) IlBl].So we have the diagram

+

s

P-E

JF

Since JFSE Iumax(E, F i n j ) , iC, follows that

A(JFeBXX)

=

B(B,J,XX)

5 ;lmax(JpS) $?lOii jjSlj.

We obtain

A i n j ( B S X ) 5 (1

+

E)

(_i"ax)inj(S) ]lBll IiXil.

Hence S E ((Uinj)max and (Ainj)max(S) 5 (Amax)inj(S). Conversely, 1t.t S E 2(E,F ) belong to (Uinj)max. Suppose that X E @(E,, E ) and B, E Q(Pinj, F,). By 18.3.2 and 19.3.9 there exists a factorization B,JF = I'B, where B E B(F,, ) ,Z Y E @(Zm, F,), and jJYlj llB!l 5 (1 + E ) [iBoll.So we have the diagram

_

_

BSX

_

_

_

_

_

f

F-,I

Y

Since I , possesses the metric extension property, it follows that

A(B0JFSX)

=

A ( Y B S X ) 5 / / Y /A/ ~ ~ J ( B (= ~ X( - )4 i n ~ ) m x(8) ljYl/ ijBlj j\Xll.

Hence 8 E (Umax)ln~and (Amax)ln~(8)(= (A1nJ)max (8)*

8.7.14. I n a similar way we obtain the

[a,A] be cc quasi-normed opemtor idecrl. The??

P r o p o s i t i o n . Let

([a,A ] s m ) m x

=

([U. A]m-)s~r

.

5.7.15. Using the method of 4.9.12 we may establish the P r o p o s i t i o n . Let [a, A] he n quasi-normed operutor ideal. Y ' h e ) ~

([a, A]mln)m*x

=

[U, A]max

and

([U,A]-)mln

=

[U, Almin.

8. Quasi-Normed Operator Ideals with Special Properties

119

8.7.16. Let A be a quasi-norm on the operator ideal 3. Then A is said to be wenkly lower semi-continuoue if _4(S)5 lim inf A(8,) Y

for every directed family of operators X, E %(E, F ) converging to S E % ( E ,F ) in the weal; operator topology.

8.7.17. As shown in 6.1.9 an arbitrary quasi-norm A need not be lower semi-contii w o u s in its own topology. However, we have the P r o p o s i t i o n . Let [a,A ] be n maximal p-narmed operator ideal. Thez A i s weakly loiuer semi-continuous. Proof. Let X c g(E,, E ) and 23 t 3(F,P o ) ,where E , a i d Fo are finite dimensional. Clearly B(E,, F,) is also finite dimensional. Suppose that the directed family of opertitors X, E % ( E , F ) tends to S E %(E,F ) in the weak operator topology. Then R S X = A-liin BS,X. Consequently

A(BSX) = lim A(BX,X) 5 lim inf A(S,) liBll IlXii. Y

Y

The assertion now follows from 8.7.4 (remark).

5.7.18. It can be deduced from Enflo’s counterexanlple that the nuclear norm Pi is not weakly lower semi-continuous. A much simpler example is given in the following P r o p o s i t i o n . T h e equation

A(S) := llSjl

+ inf (118-

L E z ( E , F ) ) for S E B(E,F )

dffines a norm A on 2 which fails to be weakly lower semi-continuous,

8.8.

Ultrastable Quasi-Normed Operator Ideals

8.8.1. I n the following let (E,) be a fanlily of Banach spaces. Furthermore suppose that a n ultrafilter u is given on the index set I. The Banach space of all bovnded families (xi),where x, E Ei for i E I . is denoted by l w ( E t ,I ) . Moreover, put cu(E,. I ) := (x,)E l w ( E l ,I ) :lim llxLii = 0). U

We now form the quotient space ( E t ) u:= l w ( E t ,I)/c,(E,, I ) . If .XI = ( z z ) U denotes the equivalence class corresponding to (xz),then the norm of z can be computed by

The Banach space ( E z ) uobtained in this way is called tlic ultraproduct of the Banach spaces E , with respect t o the ultrafilte lt.

120

Part 2. Theory of Quasi-Normcd Operator Ideals

8.8.2. Let (a,)E Zm(Ei,I ) . By setting

(x,a) := lim (xz,C I , ) for x = (xi)u U we define a functional a == ( u ~ )on , ~(Ei)uwith \]ail= lim Ilajll. U

R e m a r k . This means that the ultraproduct (E:), is metrically isomorphic to subspace of [(EJ,,]’.

R

8.S.3. Let (E,) and ( F , ) be families of Banach spaces. Suppose that (8,)is a bounded faniily of operators X,c B(Ei,P i ) .By setting

(&)u

(5th

:== ( & 4 l l

we define a n operator (St)llfroin (Bz)tlinto (F,)uwhich is called the ultraproduct of the operators S , with respect to the ultrafilter 11. Moreover, Il(Sj)ull = 1iF liS,ll.

8.8.4. Suppose that S E O(E, F ) . Let I be the set of all indices i = ( M , N ) with Jf E Dim ( E ) and N E Cod ( F ) . Choose a n ultrafilter U containing all subsets {iE I:M 2 M,,N 5 X 0 ] , where io= (&lo, N o ) is fixed. Finally, put E , := N . P,:= FIN, and Si:= QgSJg. We now show how the operator S can be reconstructed from its elementary parts. Leninia. There are operators J E B(E,(E&) and Q € B((Ft)lt,J”’) such that 1, IIQII 2 1, and K 4 = Q(X,), J .

iIJd 2

Proof. The operator J is defined by

Jx Obviously

x if x E M , o if x 4 X.

:= (xi)ll with x i :=

jiJll 2

1. Moreover, let

whwe (yi)E Zw(Pi,I) such that

Q ( ( Y ~ ) := ~ ) F’-lim KFyi, U

Q:yj = yp.

First we check that the right-hand expression does not depend on the special choice = o and y” := F’-lim KFyi. Then of ( y i ) . For this purpose assume that I1

i(KFyi,b)j 5 IiQZ-Yil] JIbilwhenever N is contained in the null space of b E P’.Therefore I(Y”, b)I = 1$ (KFyi, b) 2 l i p IIGyiIi IIbII = 0 -

I

I

Hence y” = 0. This proves that Q is well-defined. If the faniily (yj) is chosen such that sup llyiil 5 (1 E ) II(yp)uil, then we get

+

I

IIQ((yP)u)iI5

Therefore

IIyiII S (1

+

E)

II(yP)uII-

5 1. Finally, observe that

&(LY,)~ Jrc = &(Sirci)u

=

P’-lim KFSJgxi = K,Sx U

for all x E E

I

S. Quasi-Normed operator Ideals with Special Properties

8.8.5. -4 quasi-norined operator ideal (&)I1

(ml)and

E )U((Ei),l,

121

[a,A] is called ultrastable if A((S,)u) 5 linl 11 -4(Si)

for every A-bounded fainily of operators Si E a ( E i ,Fi) and cvery ultrafilter U. R e m a r k . Let [a,A] be a quasi-normed operator ideal. Then we can define t1.c. ultrastable hull [Ndt, A ~ l t ]as the intersection of all ultrastable quasi-normed operatror ideals containing [a.A]. Until now no direct characterization of this hull is known. 8.8.6. We now prow the main result of this section.

[a,A] be nn ultrastable quctsi-norvied [a,A]max = [a,A]reg.

T hpor em. Let

operator ideal. T h e n

Proof. Suppose that S E a m a x ( E , F ) . As in 8.8.4 we put S, := Q;SJ$. Then A(#,) 6 A ~ Y ( QSince ).

it follows that K,S

A(K, S )

= Q(S,)u J

Ei:= M , F , := H / X , and

t % ( E ,F')and

A(Q(S,), J ) 5 -i((St)u) 5 l i p A@,) 5

-@ax(S).

Hence S E % r e g ( E , F) and A l e g ( S ) 5 Amax(#). So [a, A]max 2 inclusion has been checked in 8.7.10.

[a,A p g . The converse

5.8.7. As an immediate consequence we have the

P r o p o s i t ion. Every ultrastable reqular quasi-nornied operator ideal is muximl. R e m a r k . Let us mention that, conversely, every niaxinial p-normed operator ideal is ultrastable and regular; K. D. KURSTEN[ 11.

8.8.8. P r o p o s i t i o n . For every ultrastable quasi-norm& operator ideal the injective hull i s ultrastable, as well. P r o o f . Let [a,A] be a n ultrastable quasi-normed operator ideal. Supposs t h n t (S,) is an -41n~-boundedfamily of operators 8, E UlnJ(Et,F , ) . Since

l l ( ~ t 4 u=l ~'/(JF,Stz,)ui/

for all ( 4 u E

(4h

by 8.4.4, i t follows tjhat (S2)aE 211n~((Ez)U, (F&) and

5 lini A4(JFIS,) = lim A ~ I * J ( S ~ ) . A~~J((5 S ~A((JF,Sg)U) )~) U

11

5.8.9. Csing 8.5.4 the next result can he checked in a similar u7ay. P r o p o s i t ion. For eiisry ultrastable qumi-normed operator ideal the surjective hd1 well.

i s dtrastable, a s

8.8.10. As recently proved by S. HEINRICH [3] we also have the P r o p o s i t i o n . For every ultrastable quasi-nwmed operator ideal the d m 1 quasi?Lorma?oyeratoi ideal i s wltrastable, as wel:.

122

P a r t 2 . Theory of Quasi-Normed Operator Idealr

8.8.11. Finally, we give a n example. P r o p o s i t i o n . The normed operator ideal [@, PI is ultrastable. Proof. Let (8,)be a P-bounded faiiiily of operators S , E @(E,,F t ) . Then we put ( P ( 8 , ):i E I].Now, given ( x ~ .~..,) ( ~x ,, , ~E )(~E , ) u ,we have

a .= sup

for all i

I . Hence

According to 8.8.2, the functional corresponding to (a& belongs to the closed unit is absolutely summing and ball of [(Ei)u]’.Consequently the ultraproduct (8i)n P((Si)u)5 sup {P(S,):i E r).

8.9.

Notes

As stated before the idens of injectivity and surjectivity first appeared in I. STEPHANI[Z], [3]. The corresponding notions in the context of tensor products go back t o A. GROTHENDIECK [4].The most important example, namely the injective hnll of the normed ideal of nuclear operators, was investigated b y A. PIETSCH [6]. Symmetrie, regular, minimal, and maximal quasi-normed operator ideals were studied by A. PIETSCH [13] and H.-U. SCHWARZ [l], [4]. I n a famous paper D. DICUKIIA-CASTELLE/J. L. KRIVIKE [l] observed t h a t the concept of ultrastability is an important tool t o deal with quasi-normed operator ideals. See also [SEMI,. exp. Csj. Recommendations for further reading:

[LOT]. [SEMI,. exp. 7, 81.

S. HEINRICH [3], J. R. HOLUB[2], J. L. KRIVINE[l], I<. D. KURSTEN[i], A. PIETSCH [20],

S.SrMoxsjT. J. LEIH[l], J. STERN[i].

Adjoint Operator Ideals

9.

Annlogously to G. KOTHE'Sconstruction in the theory of ,cquence ?paces i t is possible to define the concept of a n adjoiut operator ideal. In the folIowing we only dtal with the case of a normed operator ideal [a,A]. Then thr adjoint [a*,A*] is a normed operator ideal, as well. We state some elementary results a1:d give t L e 1, isic examples. I n particular, perfect and self-adjoint nornicd operatoi ideals a x invLhgated. Prom t h g historical point of view the notion of a n adjoint norincd operator ideal was introduced in order t o describe the dun1 Banach space of the conipoiicnts [U(E,F ) , A]. However, such a representation theorem does not hold in gexral. So, in a first step, we treat the finite diixensional case only. A more satisfactory result is proved in the following chapter.

9.1.

Adjoint Nsrmed Operator Ideals

9.1.1. Let [a,A] be a normrd operator ideal. An operator S E C ( E . F ) beloag:.. to the adjoint operator ideal a* if there exists a constant G 2 0 such that ltrace (SXLoB)j2

B

!!XI]A(Lo)IIRij

for all B E g ( F , F o ) , Lo E %(Fo,Eo), and X E g ( E o ,E ) , where Eo and Po are arbitrary Banach spaces. We put

A*(S) := inf

B.

R e m a r k . The above situation is represented in the follomifig non-commutative diagram :

E

s

Eo-

+F

FO

Lo R e m a r k . The next theorern shows that in the above definition the Banach spaces Eo and Fo may supposed to be finite dimensional. R e m a r k . The adjoint normed operator ideal [%*, A*] will frequently be denoted by [U, A]*.

9.1.2. Theorem. Let [U, A] be u nornied operator ideal. 'l'heii S F 2 ( E . F ) belongs t o U" i f and o d y if there exists a constant CT 2 0 m c h that Itrace (SJSLQ;,)]5 GA(L) for all

M E Dim ( E ) ,N E Cod ( P ) ,crnd L

< P ( F / N ,M ) .

124

Part 2. Throw of Quasi-Sormed Operator Idea13

I n this CClSb,

A*(S) -= i d cr. Proof. Let B E g ( F , P o ) ,Lo :a ( F o ,Eo),and X factorization

E(Eo,E ) . Consider t h o canonical

where M := M(X) a n d W :==N ( H ) . Then l/Xoll= //Xi:and /]Boil = //Bl/. Put h := XoLoBo.If the above inequality holds, thtm we get

jtrace (SXLoB)II :/trace (SJ",Q;,)I

GA(L)5

G

/iX/lA(L,) ilBl/.

Consequently S % * ( E , F ) a n d A*@) 5 G . This proves the sufficiency of the condition. The necessity is trivial, sinre the operators J;[ and Q,: are finite by hypothesis.

R e m a r k . It is important t o ixention that the adjoint riorrned operator ideal is uniquely deterrninrd if the norm B is only k w w n on the collection of all elementary operators.

[a*,A*]

9.1.3. Theorem.

[a*,A*] is a normed operator ideal.

P r o o f . We use criterion 6.2.3. (0) Obviously I, E ? and ? Al ** ( f f ) = 1; cf. 6.4.2. ro

W

(1)

L c A t

S,, Xz,... f %*(I?, F ) such that J'.4*(A',,)< 1

1 00

Therefore S E U*(E, F )and A*(X) S

A*(&). 1

and p L t t S :=

8,. If 1

Eo),and X c S(Eo,E ) , then R 5 $ ( F , F,), Lo %(Po, 1

CO,

135

9. Adioint ODerator Ideals

9.1.4. The next result is evident. P r o p o s i t i o n . Let [a,A] wxl [b,B] be nornied operator idecr1.s. Then,[a,A] E [!B.B] irrp%.r [a,A]* 2 [b,B]*.

9.1.6. The following statement is also trivial. P r o p o s i t i o n . Let

[a,A] be a normed

operator ideal. Theti [N, A] & [N, A]**.

9.1.6. We now prove a formula which is related to that given in 8.7.12. P r o p o s i t i o n . Let [%,A]beunoimedoperntorideal. Then ([a, A]dml)* == Proof. Let S 2 ( K , 2’) belong to S E g(E,, E ) , then w e 1uvc

(a*)dml. If

B E g ( F , Po),Lo c

([a,A]*)dml.

2Pim1(F0,

Eo),and

/trace (SXL,B)/ = /trace (S’B’LhX‘)I

5 A*(S’)IiB’Il A(Lh) ~ ~ X ’ ~ ~ 5 (-k*)dm1(8)/Ixlj_4d”’(Lo)!/Bl,. Therefore S E (%dual)* and

(Ad1ial)*(S)

5 (A*)dual(S).

Conversely, let S c 2(E,F ) belong to (2Fal)*. ConsiderA E g(E’,Eh), KOE B(E&F;), and Y E g(F6, F’), where E, and F , are finite dimensional. Then there are operators B E g ( F , Po)arid Lo E 2(Fo,B,) such that Y = B’ and KO= L;. Obviously IjBll = IIYlj and Adml(Lo) = -4(K,). Given E > 0, by E.3.2, we can find X E 8 ( E o ,E ) with AS‘B‘ = X’S’B’ and llXlj 5 ( I E ) IlAli. Since

+

trace (8’YK,A)

=

tracc(S’B’LhX’) = trace (SXL,B),

we have

jtrace (S’YKoA)/5 (-4dm’)*(S) IIXli iidml(Lo) /lBl/

+

5 (1 E ) (AdUa’)*(S)i!YlI AWO) I l 4 * Therefore S‘ E a* and A*(S’)5 (1 4-E ) (Aoml)*(S). This proves that 5‘ E (%*)dm’ arid (A*)dml(S) 5 (,ldm*)*(S). I). I .7. Finally, we consider the basic examples.

Proposition.

[e,I!.il]* = [3, I] and 19,I]* = 12,11.11].

Proof. Let S E 3 ( E , F ) . If B E g ( F , F,), Lo C 2(Po,Eo), and Y C g(Eo,E ) , then SL& E $(F, E ) . Consequcntly

5 I(S) llxll 1 1 4 1 1IlBIlItrace (SXLoB)I5 I(&)Ii~LdIl l’herefore S E 2*(E, 3)and JIS/I* 5 I(&. This shows that [3,I]

i2,11.11]*.

Conversely, let S E B*(E, F ) . If L E g ( F , E ) , then we consider the cai?ocical factorization 1; = JfILo&$,where X := M ( L ) and N := W ( L ) .This yields jtrace ( S L ) ]= Itrace (SJ$Lo,Q%)I5 IlSIj* IILoll = IIS/j* &L,;. Therefore S E a ( E , F ) and I(S)5 ilSli*. So we have [i?, l/.ll]*& [3, I], and the first equation is proved.

126

P a t 2. Thcory of Quasi-Kornied Operator itlrals

Finally, by 6.1.4 and 9.1.5, it follows that

[I?, 11.113

c= [e,ll.lll**

= [S.,11"

G [I?, ii-lil.

9.1.8. We conclude this section with an interesting formula. Lemma. Let

{a,A] he a normed

operator ideal. Then

A*(I: I: + 1;) d(I:1; + 1;)

= 12..

Proof. We denote the identity i m p from 1: onto 1; by I for brevity. Obviously

n = trace (11-1)5 A*(I) A ( I - l ) . I n order to establish the converse estimate we choose L E I?(l;, Zt) such that A*(I) = trace(1L) and d(L)5 1. Form the collection 9 of all pairs y = (e, n), where e E 8* and iz is any permutation. Then c := card (F) = 2%!. Let U , and 11, be the operator in l: and 16, respectively, defined by (El, * *

-7

5%)

--f

(81L(1,>

..* > &Jn(,LJ *

It follows that

with some constant A. We have A n = trace (MI-1) = c-l

trace (IU;1LF7y)

9" = c - 1 2 trace ( Vi'ILV,) 9" = trace

(IL)= A * ( I )

and

This yields

A*(1) A(1-l)

9.2.

= AnA(1-I-l)I n.

Representation of Functionals

9.2.1. The foIlowing representation theoren? was the starting-point of the ideas developed in this chapter. A more general result will be stated in 10.3.5. Theorem. Let [a,A] be a norm& opelator ideal. Suppose that E and F are finite diw~ensioizal.If T E U*(F, E ) , then

(8,T ) := trace (87') defines a functional T o n %(E,B'). T h e correspondence obtained in t?uis way is [B*(P,E ) , A*] and [%(E,F ) , A]'.

c(

metric isomorphism between

127

9. Adjoint Operator Ideals

Proof. Observe that a ( E , F ) = C ( E , F ) and %t*(E,E ) = i?(F, E ) . Obviously

/(a,T)I = /trace ( S I h T I F )5/ A(S) A % ( T ) . Therefore A’(T) 5 A4*(T),where A’(T) denotes the norin of the corresponding functional T . Conversely, let, T be some functional on a ( E ,F ) . Sinec E is reflexive, we can find a n operator T E a*@‘, E ) such that ( a @ y, T ) = ( T y , a ) for all a E E‘ and y P. Using the fact that every operator S E B(E, F ) adinits a finite representation n

S = ’2 ai @ yi weget (8, T ) = trace ( S T ) .Furthermore, if A E g ( E .Eo),Lo E U(E0,Po), 1

and Y

c $(Fo,F ) , then /trace (TYLoA)j= I(YLoA, T)I 5 d‘(T) A(YL0A)5 A‘(T) lIyll a(&)IIAI!.

Hence A*(T) 5 A’(T). 9.2.2. P r o p o s i t i o n . Let [a,A] be a normed operator ideal. Suppose that E and F w e finite dimensionul. TheTL A(&) = A**(S) for all S E B(E,F ) . Proof. Applying the preceding theorem twice and using the reflexivity cf

[a@,F ) , A ] we observe that A and A** coincide on B(E, F ) = %(E,F ) = %**(E, P ) . 9.2.3. For later use we give some supplements of 9.2.1. P r o p o s i t i o n 1. Let E be finite dimensio~ml.If T E % ( F ,E ) , then

(8,T ) := trace (ST) defines a fzLnctioncc1 T on B(E, P ) . T?ce correspondence obtained in this [ W F ,E ) , NI and [W,F ) , 11.111’.

t(;uy

i s a melric i.somorphism between

Proof. Observe that % ( F , E ) = B(F, E ) . Obviously l(S, T ) / = jtrace (ST)I 5 I(T) 1 1 8 1 1 5 X ( T ) ilSll.

Therefore IITII’ (= X(T), where IITII’denotes the norm of the corresponding functional. Conversely, let T be any functional on B(&, F ) . Then we can find T E % ( F , E ) such that (S, T ) = trace ( S T )€or all S E B(E, F ) . Moreover, it follows from 6.8.3 ar,d /trace ( S T ) /= I ( &

T)l 2 IITll’ IlSlI

that N ( T ) = I(T) 5 I;Tll‘. P r o p o s i t i o n 2 . Let E be finite climenswnul. If S E 2(E,P’), thtn

(S, T ) := trace (ST”) defines a functional S on % ( F ,E ) . T h e correspondence obtained in this way is u metric i2(E, F”), 11.11] and [!&(F,E),XI’.

‘isomoi pl,?st:i

izturcn

128

P a r t 2. Theory of Quasi-IVormed Operator Ideals

P r o o f . Given E

> 0, by 6.8.3, we can write

n

such that n

L' libill li.c,/I 5 ( 1 i-&) N(T)1

Then I(S,

I:

I

, 5 (1 + E ) IISll S ( T ) . T j l = 2' X X ~bj)

Il~S'l. where R'(S) denotes the norm of the corresponding functioiial.

Therefore N'(S)

Conversely, let S be any functional on % ( F , E ) . Then we can find S E B(E, F") such that (S, b @ x) = (Sx,b) whenever x E E and b E F'. Consequently we have (8,!I!')= trace (ST")for all T E %(F, E ) . Moreover, I(S5, b>l = I@', b

(3)I. S

X'(S)

Il4I lib11

implies IISll 5 "(8). R e m a r k . Coinbining Proposition 1 and 2 we observe that [B(E,F"),11.11] and [!2(E, P ) , 11.111" are metrically isomorphic. With respect to this identification the transforins , S into KFS. evaluation map K O ( E , F

9.3.

Perfect Normed Operator Ideals

9.3.1. We start this scction with an important Theorem. Let [U, A] be a normed qperator ideal. Il'hen

[U, A]**

=

[U, i l l m a g .

Proof. Let S t a**(E,F ) . If M E Dim ( E ) and AT E Cod ( F ) , then we know from 9.2.2 that

A(&;SJE,)

=

A * * ( & 3 J f f )5 A**(S) .

Hence S E Um*x(E, F ) and

Amax(S)

5 A**(& by 8.7.5.

Conversely, let S < Umax(E;, F ) . Suppose that M L E L?(F/N,iM).Then

E Dini ( E ) . N (5 Cod

Itrace (SJffLQ:)I = I t r a c e ( L r , , N Q ~ S J ~ ~ I ) [

5 A*(L) llIF,Nil A(QFNSJ5) Jdi'l 5 Amax(S) A * ( L ) . ~

According to 9.1.2, we get S E ?[**(E',F )and A**(S) 5 9.3.2. A normed operator ideal

[a,A] is called perfect

if

Amax(S).

[a,A] = [%. _\I**.

(P).and

9. Adjoint Operator Ideals

129

9.3.3. By 9.3.1, me have the

P r o p o s i t i o n . A normed operator ideal i s perfect if and only i f it it$ maxi?rial.

[a,A] be n normed operator ideal. Then [a,A]* i s perfect. Proof. Using 9.1.4,froin [a,A] & [a,A]** we get [a,A]* 2 [a,A]***. 0 1 1 the other hand, 9.1.5 implies [a,A]* & [a,A]***. !).3.4. P r o p o s i t i o n . Let

9.3.5. Finally, we give the following example.

I] is the svnnllest perfect normcd operator idetrl. P r o p o s i t i o n . [3,

9.4.

Self-Adjoint Normed Operator Ideals

9.4.1. A nornied operator ideal

[a,A] is called self-ctdjoint if [S, A] = [a,A]*.

R e m a r k . The most important example of a self-adjoint nornied operator ideal will be treated in 19.2.14.

9. t.2. The next statement follows irnniediately from 9.1.6. P r o p o s i t i o n . Let [a,A] be a self-adjoint nornied operotor ideul. l’hcn the normed operator ideal [a,A]dual i s self-adjoint, as well.

9.4.3. Using a n average method developed by R. SCHATTEN the following result can be proved; cf. A. PIETSCH [133. T h e o r e m . There exists a self-adjoint urul synitnetric normed opelator ideul. R e m a r k . Unfortunately R. SCHATTEN’S construction is quite a technical one. So; we should like to know a inore conifortable description which gives a simple chzrncterization of the operators belonging to this ideal. R e m a r k . Very rpcently J. PGHL[ 2 ] was able to construct infinitely rnany nornied operator ideals of this kind.

9.6.

Notes

Tile concept of a n adjoint (associate, conjugate, duaf) object has a long history. For sequence sp;:ces it was introduced by G. I~OTHE/O.TORPLITZ [l]. Some kind of an adjoint normed operator ideal on Hilbert spaces was studied in the monographs (GOH, pp. 125-1361 and [SAT,, pp. 73-78]. The final definition of a n adjoint riormed operator ideal appeared in A. PIBTSCH [13], where also the basic examples were treated. Supposing t h a t all Banach [l] has spaces under consideration have the metric npproximat,ion property H.-U. SCHWARZ simplified this const,ruction. Rwornmendations for forther rending:

[LOT]. D. W. DEAN[l], CH. GAPAILLARD/PHAM TXE La1 [l], D. J. H. GARLINQ[Z], H. G . GA~WIK [I], Y. GORDON/D. R.LEWIS/J.R.RETHERFORI) [l], 121, A. PERSSON/A. PIETSCH [l].

10.

Banach Spaces with Approximation Property

This chapter deals with the (metric) approximation property of Banach space?. JYCJ establish this property for the classical Banach spaces and give the fanious counterexample of P. ESFLOin the elegant version due to A. 31.DAVIE. The niain puryost is to show that the theory of operator ideals becomes mncli simpler if tlir underlying Banach spaces and their duals have the metric approximation property.

10.1.

Approximation Property

10.1.1. A Banach space E possesses the approxfmation property if for every compact subsct K and every t > 0 there exists a n operator L E S ( E , E ) such that

11%

- Lsll

5 t uhenever x

E K.

The class of all Banach spaces with this property is denoted by G. 10.1.2. We state the trivial Theorem. G is u space ideal.

10.1.3. The significance of the approximation property comes from the following result. P r o p o s i t i o n . If F ha8 t?re approximatioiz property, then 6 ( E ,P ) = R(E, F ) for ull Banach spaces E. Proof. Let flE.!Q(E',F). Since S ( U E )is contained in a coinpact subset of F, there exists L E 3(P,3')such that 118s - LSxjI 5 E for all s E UE. This means that IjS - LSlj 5 E . Hence S E Q ( E , P). R e m a r k . The above condition is characteristic for Banach spaces belonging to G. In other terms, G = Space (R-l o 6).This is proved in [LIT, vol. I, p. 321 and [SAF,, p. 1131.

10.1.4. Using the complete synimetry of Q and

we obtain the dual

P r o p o s i t i o n . If E' has the approximation property, then Q(E,F ) = B ( E , F ) for all Banach spoces F.

10.2.

Metric Approximation Property

€0.2.1. A Banach space El possesses the metric approximation pioperty if for every compact subset K and every E > 0 there exists an operator L E g ( E , E ) such that lILl1 5 1 and

/jx - L.qj 5 E whenever

zE

K.

131

10. Bnnach Spaces with Approximation Property

10.2.2. Leillnla. A Banach space E has the metric approximation property i f , given x1, ..., x, E E and E > 0 , there exists a n operator L E g ( E , E ) such that IjLil 5 1 and jlxc - Lr,jJ

for

E

i = 1, ..., n.

Proof. Let K be a compact subset of E. Then we can find xl, ...>5, E E with n

K

U {xi + E UE}. 1

By hypothesis, there is L E g ( E , E ) such that llLil 5 1 and llxi - Lxill 5 i = 1, . . ., ‘12. Let x E K and choose x, with jlx - xkll 5 E . Then jln - Lxlj 5

1 1 5

- xkil

+ i]xk - Lxkll f IjLx,

E

for

- Lx/l 5 3 ~ .

10.2.3. The following result will be proved in 19.2.4 and 19.3.5. T h e o r e m 1. Let (Q, p ) be any measure space and let 1 has the metric approximation property.

p 5

00.

Then L,(Q, p )

T h e o r e m 2. Let K be any compact Hausdorff space. Then C ( K ) has tlzc metric approximation property.

10.2.4. L e m m a . Suppose thatE has the metricapproximationproperty. Let M Dim ( E ) and let E > 0. Then there exists an operator A S ( E ,E ) such that IlAll 5 1 E and Ax=xfmallxEM. 6 Proof. P u t n := din1 ( M ) and choose 6 such that 0 < 6 < 1and n -I E . 1-6Then we can find L E g ( E , E ) with IjLlj 5 1 and 11% - Lxll (= 6 whenever x E U M . Consequently

+

llLxll 2 IlxIl

-

ijx - Lx1/ 2 (1 - 6) ljzlj for all x E M

This means that the restriction of L on M is one-to-one. Therefore dim ( L ( M ) )= n. By B.4.8, there are x l , ..., r, E Jl and a,, ..., a, E E’ such that IiLxill = 1, lla,Jl = 1, and @xi, ak) = cik. P u t

P := 1TE

+

n ak

‘3(xk - Lx,).

1

Then IIn,

-

Lql/ _I 6 ilxkli and (1 - 6) IIxkii 5 llLxkll = 1 iniyly

Since xl, .._. x, is a basis of A1 acd T’Lx, = xl,we have 1’Lx = x for all x E M . Consequently A := PL satisfies the desired conditions. 10.2.5. We now obtain the fundcniental Lemma. Suppose that 3’ has the metric approximation propert!j. Let S $ B(E, F ) a n d let E > 0. Then there exists a n operator B € s ( P , E’) such t h t IIRII S 1 E and BS = S.

+

132

Part 2. Theory of Quasi-Normed Operator Idcalb

10.2.6. Finally we state a dual version of the preceding result. Lcinrna. Suppose that E' has the metric approximation property. Let S 6 g ( E , F ) and let t > 0. Then there exists a n operator X E g ( E , E ) such that llXlj 5 1 E and

+

xx = s.

P r o o f . Choose 6 > 0 with (1 + 6)2 5 1 + E . Applying 10.2.5 to S' we can find A E g ( E ' , E') such that ~~A~~ 5 1 + 6 and AS' = S'. By E.3.2 there exists X E g ( E ,E ) with lIX1i (1 i6) IlAli and X'S' = S'. Consequently I/X//5 (1 4 21 E and SX = A'.

+

Consequences of the Metric Approximation Property

10.3.

10.3.1. We begin with an improvement of 6.8.3. P r o p o s i t i o n . Let E' or F possess the metric approximation property. Then Y(S) = N(8) = I(S)for aZE S 6 g ( E , F ) . Proof. Suppose that E' has the metric approximation propcrty. Given E > 0, by 10.2.6, there exists X E g ( E , E ) such that [\Xi/5 1 E and SX = S. Using 6.8.4 (Lemma 1)we obtain

+

W ( S ) = NO(SX) 5 I(S)I1Lq

s (1 +

E)

I(S1.

Therefore Fio(S)5 I@). This proves that No(S) = S ( S ) = I@').

Tile other case can be treated analogously.

10.3.2. T h e o r e m . Let E possess the metric approximation property. Then every operntor S E %(E,E ) has a well-defined trace which can be computed by Kl

trace (8)= 2 (ai,x i ) , 1 CQ

where S =

aj @

xi i s any nuclear representation. Moreover,

1

jtrace (AS) 1 5 N(S)

.

Proof. By 6.8.5 and 10.3.1 we have Itrace (S)] =( &(AS) for all S E g ( E , E ) . Therefore the functional, S -+trace (S),admits a unique extension on %(B,E ) . Furthermore, a7

if

S = 2 ai@ x i is any nuclear

representation, then

1

/ n

\

a

;

trace (27)= lim trace n

R e m a r k . Obviously the same statement is true if E' possesses the metric approximation property. Indeed, this is a stronger assumption, since E has the metric approximation property if E' does; cf. [DIU, p. 2441 and [GRO, chap. I, p. 1801.

10.3.3. Theoreni. Let [a,A] be (I quasi-normed operator ideal. Suppose that E' and P have the metric approximation property. Then (Umln(E, F ) consists of those operators S E U(E,F ) such that there exists a sequence (8,) of operators S, € g ( E , 3') with S = A-lim S,,. 91

10. Banach Spaces with Approximation Property

133

Proof. Let S f BUmin(E, F ) . Obviously A(S) 5 Amxn(S). We now consider a factorization X = YSoA, where A E Q(E,Eo), So E N(Eo,Fo), and Y E 6 ( F o ,F ) . Then we can find A , < $ ( E , E,) and Y , E g ( F o ,F ) such that

A P u t S,

=

1l.lI-lini A,

and

n

:= YnXoAn.It

Y = l~.ll-limY , . n

follows from S = Amln-lim S, that X = A-lim S,. This n

n

shows, without any assumption on E and F , the necessity of the above condition. We next consider an operator S $(E, F). By 10.2.5 and 10.2.6 there are X E $(E, E ) and B E g ( F , F) such that IlXIl 5 1 + C, IlBIl 5 1 F , and BSX = S. Consequently _4m'n(S) 5 (1 + F ) B(S). ~ This proves that A(&)= Amln(S) for all S g ( E ,F ) .

+

Finally, suppose that S

==

A-lim S,, where Sn E g ( E , F). It follows from n

A(& - 8,) = Amln(S,, - S,,,) that (8,) converges in possible limit, we have S E %mm(E, F ) .

amin.

Since S is the only

R e m a r k . The above statement means that %min(E, F ) is the A-closure of $ ( E , F ) in %(E,3').

10.3.4. T h e o r e m . Let [%, A] be a quasi-normed operator ideal. Suppose that B' and P have the metric: upproximation property. Then ama"(E,F ) consists of those operators S E C(E,F ) such that there exists a n A-bounded directed family (S,) of operators R, E $(E, F ) converging to S it& the weak operator topology. Proof. Let S satisfy the above condition. If M E Dim ( E ) and N E Cod ( F ) ,then = 8-lim Q$S,J$. This implies, by 6.1.9, that

2 ( X ,F I N ) is finite dimensional. Hence we have QZ#JZf A(Q$SJ$) 5 x lim A(Q!$S',J%). Y

Therefore S E U m a x ( E , F ) and Amax(S) 2 x sup A@,). Y

Conversely, let S E F). Since F possesses the iiietric approximation property, we may choose a directed family of operators B, E g(F.8')such that 'lBvl\2 1 and lim Bvy = y fcr all y E F . Given E > 0, by 10.2.6, there are X, E g ( E ,E ) Y

with //Xvll5 1

+ E and B,,SX, = SJ. P u t 8, := R,SX,.

Sx = lirn B,Sx

If x E E , we have

= Iim B,XX,x.

Y

Therefore the directed family (8,)tends t o X in the weak operator topology. Noreover, -I(s,) 5 ilBui'Amax(S)]/X,li5 (1 + E ) *4max(8).

10.3.5. Theorem. Let [%. A] be a n2isirmi normed operator idccil. Xuppso fhrrt E' rtnd P have the metric approximation property. If T t U * ( F ,K " ) , then

(8,T) := trace (A'T) defines a functionctl T on %(E, F )

134

Part 2. Theory of Qmii-Soimed Operator Ideds

The correspondence obtained in this way is a metric isomorphism between [%*(F, E"), A*] and [%(E,F),A]'. P r o o f . Suppose that T < %*(F, E"). First 1r.t AS' g ( E , F ) . Uiveii P > 0. hg 10.2.4 arid 10.2.5, we can find X E g ( E , F:) and B f g ( F . F ) siioh th'it ]]Xi:5 1 f 8. lIBll : 1 F, and B S S S. S o v

+-

1

itrace (XX"TL)i 5 A ( S ) //X"jj A*(/') llLl] for all L E g ( F , F ) , yields I(SXV) 2 A(&) [ / X A*(T).Using 6.8.4 it follows that

+

R'(S"T)= 3(RA'S7T) 5 jjBl/ A(S)IIXII Ai*(!Z1) 5 (1

F)')

A(S)A*(T).

Hence N(S"T) 5 A ( 8 ) A*(T). Since [%, A] I\ miqinml, for every S 6 %(E,F ) there are S,,E g ( E , F ) with S = A-lim S,. Then S(S;JZ- SLT) 5 A@,, - 8,) A*(T) n

implies that)'!$'SA( s o we get SnT

is an 3-Cxuchy sequciici~.i h i t S 1' k the only imssible liniit. R(F,F ) . By 10.3.2. thr traw of S T is well-tl~fiiietl.Moreover.

itrace (S*T)i5 ?i(-(sz/')

5 A(S) A * ( T ) .

Therefore A'(T) 5 A*@), where A'(?') denotes tt-w norm of the corresponding functional. Let now T he a functional on U ( E ,F ) . Then we citn find an operator T E g(17, E") such that ( a @ y, T ) = (Ty, a ) for all a E E' and y E F . Hence ( S , T ) = trace (ST) whenever S E g ( E , F ) . Next we prove that T E %*(F, E"). For this purpose 1Pt Lo E U(E;, To), and Y 5 g(Fo,If'), where Eo and F , are finite A E 8 ( E f fEg), , dimensional. Given F > 0 , by E.3.3, we can find X 5 g(K:. E ' ) such that AT Y = X'T I.' and IlXIl 5 ( 1 F ) IIAlj. Moreover, there exists A, :g ( K , E,) with X = Ah. Then we have A T Y = A:TY and lIA,jl 5 (I t) IiAjI. Henccb

+

+

trace (ATYL,) = trace (AiZIJ-Lo)= trace ( 1 7 L o K ~ o A ~= ? ' (YL&peA,. ) 'p) and Itrace (TYL,A)/ 5 A(PL,KFaA,)A'(T) 5 (1 -k

t)

A'(Y)

This proves that T E %*(E', El') and A*(Z1) 5 A'(T). Siricv finally obtain (AS', T ) = trace (S'T) for all S E %(E. F ) .

[a,A]

is niinimal, n e

10.3.6. We now show that the adjoint norrued operator ideal can be defined in a much simpler way if the underlying Banach spaces have the metric approxiination property. Theorein. Let [a, A] Ire rl n o r i d operator ideal. Szippow t h t E rl?zcl F possess the metric approximation property. Then S E g ( E , F ) beloup to %* if nnd 01?71/zf there exists (1 constavzt (T 2 0 such that

aA(L) for all 1; 6 g(F,E ) .

/trace (SL)i

I n W&& case, A*(& = inf

0.

10. Bsnach Spaces with Approximation Property

135

P r o o f . Let S satisfy thc above condition. Given B E g ( F , P o ) .Lo E U(F,, B0),and X E S(Eo,E),we piit I; := XLoB. Then Itrace (SXLoB)[= Itrace ( S L ) /5 oA(L) 5

(T

jIXl1 4(L0) 1;BjI.

Hence S E 'U*(E,F ) and A*(S) 5 (T. Conversely, let X E %*(I#, 3'). If L E g ( F , E) and E > 0, w-t can successively find X E g ( E , E ) and B E g ( E , E ) such that jlXll 5 1 E , jjBI1 5 1 E , X L = L, and BSXL = SXL. Then SL : XXL = BSXL. Therefore

+

+

+

E ) ~

-

0) Ok

ltrace (SL)I = jtrace (SXLB)j 5 A*(&')llX11 A(L) IiBI: 5 (1

A*@) A ( L ) .

This shows that /trace (SL)j 5 A*@) A ( L )for all I; E g ( F , 12).

10.4.

Counterexamples

10.4.1. We start this section with an elementary Then Lemma. Let 8 E [0,1] and ( E 9.

+ (1 - e) exp (-eg) 5 exp Proof. It is enough to treat the case < 1 0. Using O(1 - O)k f ( I e exp ((1- 6) ()

((2).

5 I . from

and

we obtain {2

(-3

1"4

0exp((1-0)r)+(l-e)exp(-e85)~1+++---+++.. 2! 3! 4. - exp (5) - C 5 exp ( { 2 ) .

10.4.2. Let bmbe the set of all vectors e = ( E ~ ., .., E , ~ )with cI = $ 1 . Pix a number 0 E [O, 11. Then we define a probability ,urnon d m by setting p n * ( ( e ):= ) ea(l - O ) @ for e E brn,where a := card fj:ci = +1] and B := card ( j : ci = - I } . Xoreovcxr, let f j ( e ) :=

1 - 0 for - 0 for

ci = $1, ci =

-1.

We now prove a generalization of E.5.2. Lemma. Let z = (Cl,

...,),[

E 1; and 0 < p < 00. Then

where co i s some positive constant.

136

Part 2. Theory of Qiinsi-Normed Operator Ideals

P r o o f . Observe that the randoin variables f l , case, by 10.4.1, WB have

..., f m

Sow choose a nat,ural number k snch that k

< p 5 k. Without loss of generality

-

1

are indcpendent. I n the real

m

we may suppose that

2 [i

= k. Then,

using

1

i'k

- 5 exp k! -

(C), k ! 5 k k ,

k1k2 < (1 ip)". , and

= llzllz,

we obtain the desired inequality

< - 2 exp (1)kl'zkl'z 5 cO(l c p)l'z 1 1 ~ 1 , ~ . The complex case

CBn

be treated by decomposition into real and imaginary parts.

(li) q 10.4.3. Lcmina. Let zh = (Cih)', ..., t',) numbers ill,.. ., Asr E (1 - A , -0) such that

l

~

1 ;i.,y1

for h

L

5 c( 1

1, .. ., r , where c

log T ) l i Z sup

n

for

JL

=

I.

..., r. Then.

ljZhj10

h 2s

same positive constmit.

Proof. Suppose that sup /I,zh'i2= 1. Put p := log r . By 10.4.2 we have h

ifrere

me

10. Banaeh Spaces with Approximation Property

for h = 1,

137

. ..,r . Hence

Consequently there exists e, E bmsuch that

rhen R, := fl(e,),

. .., i.,

:=fm(eo) satisfy the desircd condition.

10.4.9. 1,cmma 1. The 3 . 2*-th roots of unity can be decomposed into disjoint subsets (aaJ :j

such that

/I. ,"+l

1,

:

..., 29z+1)and {Bnj : j

I

&I:;] - 2 1'&, 5 cl(n + 1)1'2

j= 1

) =2"1

for h = 1, ..., 3

= 1,

..., 2")

2n12

an,where c1 i s Some poSitiue constant.

-

Proof. P u t 0 := 213, m := 3 2n, and r :I= 3 2". Furtherimre, let

znh := (i.;,,

..., c",)

for h = 1, ..., r ,

<,,, ..., in, are the m-th roots of unity. Theii j / ~ , ~ , ~ / \ = ~ ?nli2.By 10.4.3 we can

where find ?.l,

..., 2,

E {1/3, -2/3) such that

If A , := ( j : %I = 1/31 and B, := ( j : he written as follows,

r [ hnl

y

IFA.

-2 i€Bo

1

c;i

5 - C1(% 1

with some positive constant

=

-2/3],

then the above inequalities may

+ 1)1/2 2 n l 2 ,

2 cl.

By setting h = m we obtain

1 /card (A,) - 2 caid (Bo)l 5 - cl(n 2

+ 1)1/22n12.

Put A := 9,A D and B := B, n D,where D is soine subset of A , or 23,. Qioviouslp !+ is possiblc t3 choose D such that cerd

1 ( A ) = 2 card ( B ) and card (D)_I - cl(v, 6

So the desired decomposition

i s obtained.

+

Yiz.

138

Part 2 . Theory of Quasi-Normed Operator Ideals

In the following we state a supplement of the preceding result. Lenirna 2. There are enni= +1 such that

for

fL

2

-

1, ..., 3 3, and k = 1,

.... 3 . 2n+1,where c2 is some positive constant.

Proof. Put 8 := 112, ?n:= .?n-1, and r := 9 . 2"+l. Furthermore, let %hk

:= ('dp,k+11I

. * * ~ ~ ~ $ f + l r n )

for 72 = 1, ..., 3 . 2% and k = 1, ..., 3 . 2*+1. Then llznak1lz= ,nlie. By 10.4.3 wc can find F~~ = &l such that

where

is some positive constant.

L0.4.5. We are now ready to give the main construction of the counterexample.

P r o p o s i t i o n . There exists N E !Jl(Zl.

Zl) such that N 2 = 0 and trace ( N ) = 1.

Proof. According to 10.4.4 (Lemma 1 andLemma 2 ) we fix thenumbem a,j, arid E , ~ . Forin the matrices

bpi,

where h and k denote the row index and the column index, respectively. The fact 3.2'

that

where

Th = 0 for every 3 . 2"-th root of unity i

I, is the 3 P-th unit matrix.

+ 1 implies the following forinulas:

-

10. Banarli Spaces with Approximation Pmperty

1:;Lj

We now deEine soiiie infinit(%irmtriocs: A, 0 0

...

0 0 0

...

To0 0

...

...

S, 0 0 ...

...

...

Obviously the following iuultiplication table holds :

1:: A

L*

L*

R

L*

0

0

B

L*

L

-A

1

d

0

0

Xoreover, we have

STA = A S T , STB = BST

.

S'TL = 2LST. L*S!l'

=z

281'/;*.

Finally, let

G

X:=T

-A--R+L

-

.*I

N

and

:==

SX.

[:sing the multiplication foriiiulas stated above we obtain

Forin the countable intles set

I := ((72,

x.):

7L

L-

Then X = (&hfln.) and S we have l6mhnkl

5

0,1, .... k

= (crnlhf l k )

+ l)-'''>

= 1,

.... 3

*

P).

can he consiclered a s

(I >: I)-matrices. By 10.4.4

140

Part 2. Theory of Quasi-Normed Operator Ideals

where c is 2 positivc constant. Consequently the ( I inable operator X E B(Zl(I), 7w(I)).Since ( m k nk

=

x

I)-matrix X defines a n approxi-

+ 1' F n

it follows that

Henccl the diagonal (Ix I)-matrix8 represents a nuclear operator S E %(Zm(I), Zl(I)). Therefore N := XX E %(Zl(I), Zl(I)).Moreover, l n k nk ==

I

113 for 0 for

7% = 0 ,

>0

1%

implies trace ( N ) = A-

b,k n k f n k nk =

-

1

I

This proves the assertion, since &(I)and I, coincide.

Remark. More precisely, we have N E %fr,l,lj(Zl, ZI) for

Y

> 2/3; cf. 18.1.1.

10.4.6. Now i t follows the P r o p o s i t i o n . The operator ideal 8 i s neither injective nor surjective. P r o o f . Consider the operator N = S X constructed a t the preceding point. We have X E @(Zl, Zm) and S E %(Zw, ZI). Moreover, S has the form S ( f n )= (antn)with crn > 0. Take t h e canonical factorization

__

where Eo := Z,/N(X) and Fo := H ( X ) .By setting (A,, So):= trace (SJA,&) for

A , E B(&, F,)

we define a functional Sosuch that !<-A,, So)\5 S ( S ) $4,11;

cf. 10.3.3. Then

( X o ,So) = trace (SJXoQ)= trace (AT) = 1 .

(1)

O n the other hand, N2 L 0 implies SJXoQNe, = 0. Siiice SJ,Xo is one-to-one, we lmve QNe, = o. Therefore, if a, E;, it follows that ( a , @ XoQen,So) = trace (Q'a, 8 XJX,Qc,) = (&&'en, a,) = 0 .

10. Banach Spaces with Approximation Property

But the linear span of (Xo&el,XoQe,, for all a, E Eh and yo E F,. Hence

(Lo,8,)

=0

whenever

...}

is dense in

F,. So we get (u,, @ yo, So)

Lo t g((E’,, F,).

141 0

(2)

,\ow, from (1)a i d ( 2 ) , we see that X o cannot be approximated by finite operators. Consequently X , 6 @(,To, F,).

Since 1, has the metric approximation property, by 10.1.3 and 10.1.4, we have = R(E,, Zm). Obviously JX,Q = X E R(Zl, Zm). So the injectivity and the surjectivity of 53 imply X,Q E %(El, F,) and J X , E @(E,, ZE). If Q would be injective or surjective, then Xo E Q(E,, F,). This is a contradiction.

@(L1, Po) = R(Z1,F,) and O(E,, 1,)

10.6.7. M7e now state the main result of this section.

T he0 r em. l’here ure BnvLach spaces without approximation property. Proof. Let F, be the Banach space constructed a t the preceding point. If F, would have the approxiniation property, then X , 5 R(E’,, F,) implies X, E 6 ( E 0 ,F,) by 10.1.3. This is iiupossible.

10.5.

Notes

The (metric) approximation property was introduced in the fundamental memoir [GRO, cliap. I, pp. 164-1911. 9. GROTHEN’DIECK has HLO observed that the presence of this property yields many simplifications in the theory of tensor products. Starting with the classical basis problem [BAN, p. 1111 thcre was a long period in which most iunctional analysists hoped that all Banach spaces would have the (metric) approxiniation property. Howex-er, P. ENFLO111 found a couiiterexample in 1975. In the following time many authors tried to PELsimplify his ingenious but complicated construction; cf. A. M. DAVIE[l], T. FIGIELIB. C Z Y ~ S E I [l]. The approximation problem and related questions are extensively treated in [DIU, pp. 238-2451 and [LIT, vol. I, pp. 29-42]. Recommendations for further reading:

‘r.FIGIEL [a], T. FIGIEL/\v.B. JOHSSON [I], w.B. JOHNSOX [ 2 ] , J. LISDEXSTRSUSS P. SAPEAR[4], A.

SZAXKOWsIiI

(21.

[3],

Part III:

Theory of Sequence Ideals

11.

s-Numbers of Operators in Banach Spaces

The definition of s-numbers (singulare Zahlen) goes back to E. SCHNIDT who used t tiis concept in the theory of non-selfadjoint integral equations. The generalization to compact operators in Hilbert spaces was given in the thirties. Some years later on it becanies clear that also in the context of Banaeh spaces there are certain rules assigning to every operator a decreasing sequence of nunibera which characterize its approximation or compactness properties. In this chapter we present an axioinatic theory of s-numbers. The main examples are approximation numbers, Gelfand numbers, Kolnmgorov niinibers, and Hilbert numbers. Then we dcfine sonic special properties of s-numbers such as symmetry, injectivity, surjectivity, additivity etc. Furthermore, we determine and estimate s-numbers of diagonal operators in classical sequence spaces. Finally, the relationship between different s-numbers is investigated.

i 1.1.

s-Bumkers of Operators

11.1.1. A map s which assigns to every operator S a unique sequence ( s n ( S )is) called an s-functionif the following conditions are satisfied :

(OS,) IlSlj = s,(S) 2 s2(S)2 ... 2 0 for S E 2(E,F). (US,) s,(S 12') 5 s,(S) f jlT/i for S , T E f ? ( E F , ). (QS,) s,(RST) 5 IlRIl s,(S) llTll for I' E B(Eo,E ) , S E W E , F ) , and R E 2 ( F . Fob (063,) If S E S ( E , F ) and rank (S)< n, then s,(S) = 0. (QS,) $,(I,) = 1, where I, is the identity niap of I;. \Ve call s,(S) the n-th s-nCmber of the operator 8.

11.1.2. First we prove a result which is converse to (OS,). F r o p o s i t i o n . I t follows from

S E f?(E,F ) a d

s,(S) = 0 thnl rank (8)< n.

Proof. Suppose that rank (8)2 12. Then we can find X E f?(Z;, w t h BSX = I n . Consequently 1 = s,(In) = s,(HSX)

E ) and B E f?(F,23)

5 pq s,(S) $q= 0 ,

which is a contradiction.

11.13. Next we show that all s-numbers S,~(S) are continuous fuiictions of S. Lemma. js,(S) - sp(7')l 5 'IS- 7'

for S , T

:f ? ( B .F ) .

]'roof. Applying (OS2) we get

+

s,(S) I S , ~ ( T ) /IS - 7'1,

and

sn(5!')

5 s,(S)

+ 112'

-

811.

146

Part 3. Theory of Seauence Ideals

11.2.

Approximation Numbers

11.2.1. For every operator S E B(E, 3’)the n-th approzimation number is defined by an(S):= inf (]IS- Lll: L E g ( E , F ) and rank(L) < n).

11.2.2. First we state the Lemma. If dim ( E ) 2 n, then un(IE)= 1. Proof. Suppose that a,(IE) < 1. Then there exists L E g ( E , E ) with rank(L)
11.2.3. Theorem. The map

a:S

--f

(a,(S))

is the largest s-function. Proof. Obviously a is an s-function. In particular, (OS,) follows from 11.2.2. We now consider an arbitrary s-function s. Let S E B(E, B ) . If L E g ( E , F) and rank (L)< n, then s,(S)

Hence s,(S)

11.3.

f 8,P) 4- 11s - Ll!= I18 - 4 1 .

5 an@).

Operators in Hilbert Spaces

11.3.1. Let (9,p ) be any ineasnre space. For every p-measurable scalar funotion s and Q > 0 we put 9, := (oE fz : ]s(.r)l 2 0).The restriction of p to 9, will be denoted by pa. 8

Theorem. Let Sobe a diagonal operator acting i n &(9,p ) such that Sof = sf, where E L,(9, p ) . I’hen

$,(So)= in€ (G > 0: dim (L2(0,,p,,)) < n }

for every s-function s. Proof. We define the injection J,, i!(L,(Q,, p,,), L2(0,p ) ) and the surjection Q, E j@&Q, p), L,(Q,, p,)) in a canonical way. Then S, := Q,soJ, is invertible, and we have \/LJill] 5 u-l. If cr > slb(E,,),it €allows from (OS,) and

that diin (Lz(Qu, p,)) < n. Conversely, suppose that dim (Lz(Qa, p,,))
Theorem. Let So be a diagoml operator acting in 1, such that So(&) = (a,;,), where u1 2 0, 2 2 0. Then s,(So) = u, for ecery s-function s .

11. s-Numbers of Operators in Banach Spaces

147

11.3.3. P r o p o s i t i o n . Let S E Q(H, K ) have the Schmidt representation W

s= 2’ 0,x:: @ yn 1

with cl 2

(r2

2 . . . 2 0. Then sn(S)= c, for every s-function S.

Proof. As in D.3.3 we put M

W

X ( l n ) := 2 Enxn,

r ’ ( ~ n ) :=

,Z V n Y n ,

and So(ln) := (UnEn)J

1

1

then S = Y S o X * and So = Y*SX. Since llXll 11.3.2 that s,,(S) = s,(S,) = 0,.

5 1 and IlYlj 5 1, it follows from

R e m a r k . As a consequence of the preceding proposition we see that the Schmidt representation of every approxiinable operator is essentially unique.

11.3.4. Next vie state the basic result of this section. T h e o r e m . There i s only spaces.

one

s-function on the class of all operators between Hilbert

Proof. Using the notation of D.3.1, for every operator S E S ( H , K ) , we have an(&’) = sn(So), The assertion now iollows from 11.3.1.

11.3.6. The following leincia will be used later on. Lemma. Let S E B(H, K ) . Then, p k e n E > 0, there are operators X E 2(Z2,H ) and Y E i?(Z2, K ) such that ljXll 2 1, I/Y\l 5 1, and s,,(S)

5 (1 + E ) (SXc,. Ye,) for

n = 1, 2,

...

Proof. If S E Q(H, K ) , then we consider the Schmidt factorization described in D.3.3. Now it follows from 11.3.2 that s,(S) = s,(So) = (rn = (SXe,&,Ye,,). To treat the general casc, by D.3.1 it is enough to consider a diagonal operator So acting in L2(Q,p). Assume that Sof = sf, where s E Lw(Q, p ) is non-negative. Let E > 0 and put

i

9, := w E Q : s(w) 2 -sn(X0) I +l & }

.

Ifp, denotes the restriction of ,U to Qn, then by 11.3.1 we have dim (L2(Qn, p n ) )2 n. Hence we can find in L 2 ( 0 ,;L) an orthonorinal sequence (a,) such that u , ( w ) = 0 wherxver w cf Q,. Then

N

&,a, we define a n operator A E B(E,, L2(f2,p)) such

E”inallj7, by setting A(&$):= 1

that (1 + E ) ( S o h , , Ae,) 2 sn(So) for 1 o*

n =

1,2, ...

148

Part 3. Theory of Soquence Tdeiils

11.4.

Hilbert Numbers

11.4.1. For erwy operator S C O ( E , F ) the n-th Hilbert w,urttlier is defined by h,(S) := SUP (a,(BSX): 9 E Q ( H , h’), IIXI; 5 1, N C @ ( F ,10,IlBll 5 11. Here H and K range over all Hilbert spacees. 11. P A We now prove a counterpart of 11 2 . 3 .

Theorem. The map 18: S

+-(?L,(S))

is the smallest s-function. P r o o f . The s-number properties of ZL can be easily dcrived from those of

(1.

Let s be any s-function. If X E 6 ( H . li:) and H < Q(Y. A’)such that llXll 2 1 and IjYIl 5 1, then by 11.3.4

a,(BXX)

=

s,(nsx)5 s,(X).

Hence h,(S) 5 a,(&.

11.4.3. The following result will be used later on. Lemma. Let S E O(E, F ) . Then, givew 8

> 0, there (ire S 5 f!(& E ) and B E f!(B’, I:)

Proof. Since the case h,(S) = 0 is trivial, we i i i : i v suppose that e > 0. Choose X o f @(H, E ) and Bo E Q ( F , K ) such that l/Xoil 5 1, i,Bo/l 1, and a,(BoSLYo)2 0. Then T :-=BoSAr,,f @ ( H ,K ) . Take the Schmidt factorization T = VToU*, mhcrt. C E B(12,H ) and T7 B(Z2, K ) are partially isoinetric operators. Since To is of the form To((,)= ( ~ , ~ c ,such & ) that T~ 2 t2 2 .. 2 0, we havc = sk(To)= uk(B&3X0)2 Q for k = 1. ..., rL. Let

-

J&.

...)5,) ::

(&,

...) En, 0,o.

...)

and

Q,(&, .... 5,’) tn+l. ...)

:= ( S , ,

...

1

5,).

Then the diagram

E

xo/ H

A 9

1‘

-+F

I

+ B, ,K

L1. s-Rumbers of Operators in Ranach Spaces

coininutes. Obviously T, :-=Q,,ToJ, is invertible and jl~$!',~/1 X := XoUJ, and B :== eT,'Q,V*R,.

11 3.

149

5 1. Finally, we put

Belfand Numbers

11.5.1. For every opcrator S E g ( E , 3') the n-th Gelfund number is defined by CjL(*S)

:= U,(.r,S).

11.5.2. First we prove the P r o p o s i t i o n . Let H be

rc

IIilbert spuce and S E 2 ( H , F ) . Thew

c,(s) = n,(8) = inf (11s- Aypi]: rank(P) < ?z}, where the infimum i u tctken over all admi.ssible orthogonal projections E e ( H , I$). Proof. Given E > 0, we choose L E S ( H , F1"J) such that rank ( L )< n and lJFS - Lil 5 (1 + E ) ~ ~ ( 8Let ) . P c B ( H , H ) be the orthogonal projection with X ( P ) =- N ( L ) . Then mnk ( P )< ) I , BS well. Moreover, it follows from

/IS - SPil

= 1

iJFS(Tn-- P)ii = /'(JFS - L ) (In - P)li 5 ~IJFS - Lij

that a,(&) I 11s - S P 1 5 1IJp.Y

-

LII 5 (1 + E ) C , & ( S ) .

This completes the proof, since c,(S) 5 cr,(S) is trivial. 11.5.3. P r o p o s i t i o n . Suplmse that P has the metric extension property. Tho! c,,(S)= a,(S) for all S E B ( E , F ) .

Proof. There exists B E l ! ( P j , F ) such that BJF = I , and C,,(S')

I a,(#)

= n,(BJ,S)

$l] 2 1 , Hence

5 /p/I cc,(Jps) ic,(S).

11.5.4. Bn s-functior? s is called injective if s,,(S) = s,(JFX)for all S E D(E,p).

11.6.8. We now state the main result of this section. T h e o r e m . The map c: s -+ L;.

(Cn(S))

the Inyest injectice s-junction.

P r o o f . The properties (OS,) and (OS,) follow imaiediately from the definitionJloreover, (08,)can be checked with the same technique as used in 4.6.2. Condition (08,) is also trivial. Finally, we obtain (OS,) from ~ ~ ( 1 =,1)and 11.5.2. sin^ Pl'" has the metric extension property, by 11.5.3 we have en(& = n,(JFS) = C , ( J d s I ) Let s be any injective s-function. Then

sn(S)= s,(JpS)

a,(JpS) = c,(S) for all S E f?(E,F ) .

11.5.6. We will denote by cod ( M ) the codimension of a given subspace M of h'.

T h e o r e m . Let 8 E 2(B,3'). Then c,(S) = inf { ] ~ A Y J Lc/w~d: ( M I < n ) .

150

Part 3. Theory of Sequence Ideals

Proof. If M isany subspace of E with cod ( M ) < n, then there exists T E 2(X7F i n j ) such that T J g = J F S J z and IlTll = IlSJEII. By setting L := J,S - T we obtain an operator L E g ( E , P n j ) with rank (L) < n. Therefore cn(S) = an(JFS)5 IIJFS - LII = llVl = IlflJEII.

This proves that

.

c,(S) Iinf (iiSJ$I1:cod (a) < n)

To check the equality, given E > 0, we choose L E $(E, Finj) such that rank (L) < n and llJFS - Lll 5 (1 8) c,(X). Put M := N ( L ) .Then cod ( M ) < n and

+

IlflJgI!= IIJFXJgII = II(JFS- L) J,cll 5 IIJFS - Ljj 2 (1

+

E)

cn(S).

11.5.7. Lemma. I{ dim ( E ) 2 n, then c,(IB) = 1. ~

Proof. Let M be any subspace of E with cod ( M ) < n. Then M ~ =I IlJfJJ = ~1. This ~ proves ~ that c,(IE) = 1.

+

(0).Hence

11.5.8. Proposition. Let F be an intermediate q a c e of {Fo.F,} possessing J-type 0. I f S E S(E,P A ) then ,

C~~+,,-1 ~ (4 R F ) 5 C,~(X:E + Fo)'-ec,,(X: E -+ Proof. Given E cod ( i k f k ) < nk and

> 0, according to 11.5.6 we choose subspaccs Mk such that Fkll 5 (1 f &)

I/SJgn:.&?k

Cn,(S:

i!# -+ p k )

for

k

= 0,1

-

+

Put M := iM, n iMl. Then cod ( M ) < no n1 - 1. Moreover, it follows from //SJ$:M -+Fk//5 IISJ&: i&fk -+ $',(I! and the estimate (J) given in C.5.6 that cffo+n,-l(S: E --f F ) 5 IlSJg: M +-PI[

5 lISJZ&:Mo -+ 3',,!11-oIISJZl: dl, + Fllie 5 (1 + E ) cn,(S:E -+ E'0)l-O c,,(S: E -+ Fly. 11.6.

Holmogorov Numbers

11.6.1. For every operator S E 2(E,F ) the n-th Kolmogorov number is defined by := a,(flQL.).

11.6.2. First we prove the Proposition. Let H be a Hilbert space and X E 2(E,H ) . T h r i )

&(AS)= a,(S) = inf { IjX - PSI1 : rank ( P )< n } , where t h infimum is taken over a.11 adm.issible orthogoml p o j e c t i i m s P P r o o f . Given - L[l 5 (1

llXQB

F

> 0,

we choose L E g ( E s ~ rH, ) such tltat rank

+ E) d,(S).

B(H, H ) .

(L)< n

and

Let P E 2(B, H ) be the orthogonal projection with

11. s-Piumbers of Operators in Banach Spaces

152

M ( P ) == M ( L ) . Then rank ( P )< n, a 8 -,yc41. Jlorcover, it follows froill /IS - PSI1 = Il(1, - ) ' 1 S'QCll = Il(1,

-

P)( X Q E

- L)I, d IISQE - LII

that an(S)5 1 1 8-

5 Il8& - Ll1 5 (1

+

E)

dn(S).

This coinpletes the proof. since d,(S) 5 a,@) is trivial.

11.6.3. P r o p o s i t i o n . Suppose that E has the metric liftingp~operty.Then d,(8)=.I.,(&) S E B(E, F ) .

/or ull

Proof. Given E > 0, there exists x'E B(E, E m ) such that QES= 1, anti 2 1 E . Hence

I XI1

+

d,(S) 2 u,(S) = an(A9QEX)5 U , ( S Q E ) IlXl] 5 (1

-+

F)

d,(9).

11.8.4. An s-function s is called sarjective ii S,~(S) = s,(SQE)for a,ll S 5 C ( E , 1'). 11.6.5. We now state the main result of this section. The proof is analogous to that of 11.5.5. Theorem. l'he map

d : s4-(d,(S)) i3

the largest surjective s-function.

11.6.6. We will denote by dim (A') the diriiensioii of a given subspace A' of P. Theorem. Let S E C ( E , F ) . Then d,[S) = inf {!lQ;S11: dim (N) < n ) .

Proof. If W is any subspace of F with dim ( N ) < n and E > 0, then there exists T ~ ( E S FU )~ such , that QPT = Q$SQF and IlTll 5 (1 + E ) '[QiSll. By setting I, := SQE - T we obtain an operator L E C , ( E S U r , F ) with rank (L)< n. Thcrefo1.c d,(S)

= u,(SQE)

5 IlSQE - LI

=

:i!I!'Ii

5 (I + I ) ilQ.<Si'.

This proves that

d,(S) 5 inf {liQ.;SlI:

dim ( W ) < n ) .

To check thc equality, given E > 0, we choose L E a(Es"r, F ) such that rank (1,) llisQ3 - L' 5 ( 1 + F ) dn(S),P u t N :: X ( L ) .Them diin (AT)< n and

< 12

Ciiid

liQ-$Si= j Q$SQ,) = IIQf(SQE - L)lI

5 llSQE - L'g 5 (1

+-

F)

dn(S).

Rcnia rk. Obviously d,(S) is the 7 2 - t h Knlimogorov clianwter (width) of S(YE); d.[LOR, pp. 13%- 1491 acd [PIE, pp. 144--1401. 11.6.7. ;Ai.nin~a. If dim ( E )

n, then d,'(TE)

I.

:

P r o o f . Let N be any subspace of E with dim (A7) < n . Then N uQiIRll = IlQill = 1 by E.1.1. This prorcs that &(Ie) = 1.

+=

B. Kence

152

P a r t 3. Thcorv of Serriiciire Ideals

11.6.8. The following statement is aiialogous t o 11.5.8. P r o p o s i t i o n . Let E be an ilztermediate space of {E,, El) possessing K-type 8. If S E 2(E,, F ) , theft.

1I .7.

Symmetric s-Kumbers

11.7.1. An s-function s is called sy7nmetric if &,(AS)2 sn(S’)for all S 5 P(E, F ) . In case sn(S)= sn(S‘) the s-function is said to he completely symmetric. An s-function s is callcd regulur if sn(S)=- s,&(KFS) for all S E 2(Z,E). 11.7.2. P r o p o s i t i o n . A n s-furrctior, is complelehy symmetric if c1n11 o?&y if it is regular and symmetric. P r o o f . U w the method of 4.3.7.

11.7.3. P r o p o s i t ion. The cxpproximtrfionm~nbersare syrnirietric, but rtot regular. ProoP. Let S E B(E,F ) and E > 0. We choose L E g ( E , F ) such that rank (L)< I L and /IS- LII 5 (1 E ) o n ( S ) . Then rank (L’)< ?L and 1 18’ - L’II 5 (1 c) u,(S). Therefore cr,(9’) 5 (1 t E ) a,(&’). This pioveq that a,(&”) 5 n,(S).

+

+

Tn 11.11.9 and 11.11.10 me will show that / r n ( I : 7, + co) = 1 m d ~ , ~ (ll-+Zm)= I: 112 71 .=. 1 , where I denotes the embedding map froin 1, into co and I, respectiwly. 80 the approximation numbers cannot be regular.

for

11.7.4. We now give a n important suppleinent of the preceding result which states the coniplete spinmetry of the approximation numbers on the ideal of compact operators. Theorem. Let S

R(E, P).Then u,(S)

= un(8’).

Proof. Given E > 0, we choose L E $(E”, 5”‘) such that rank (L)< n auld jlS’ -- Llj 5 u n ( S ’ )+ F. Furthermore, there are yl, ..., ymE F with m

S( ETE) g

u {yz+

&F].

1

..

Let M be a finite dimerisioid subspace of F” containing M ( L ) and KFyl, .,Kf-q,,t. Then, by E.3.1, we can find J , E 2 ( N ,P)such that llJoll5 1 P and li,Joy” = y’’ whenever y“ A’(K17)n M . I n particular, JoKFy2= yi for i = 1, ..., 911. By setting Lox :-- J,Lk-,z for x E E we now define an operator Lo E $(E,P)with < 72, Let x E U,. Then there exists 1 ~ 6such that IISx - yili S F . Henca rank (Lo)

+

j;Sz - L0s1l

5

11S.c- ?/$

+ llyl - .&.? 5 + IIJOKFy, - JoLIiTEzil 8

-&

+ (1 +

E)

5E 5 E.

+ (I +

E ) [F

5 c:

+ ( 1 + ).

+ (1 + ).

alKpy, - ~KEXII [ilK*Wz - KFSXl1

la,(,”)

+ IlKfiSX

+ l/ij””KEZ - LK a ~ l i ]

+ 2&1.

- /.li‘Ezlll

11. s-h'nmbers of Operators in Rannch Spaces

We get

a,(~5 )

(1s- ~ ~ 5l (1 i + E ) (a,(S")

+ 28) +

153

E.

CI,~(S") 5 u,(S') 5 n,(S). This proves that u,(S') = a,@) for

Consequently a,(&) all s E R(E, P).

11.7.5. P r o p o s i t i o n . 1'Ae Gelfund numbere arc regular, but the Kolmogorov numbers are not. Proof. The ro@arity of the Gelfand numbers follows froin their injectivity; cf. &o 11.5.6. 0 1 1 the oth(.r hand, the embedding map from 1, into co shows that the Kolmogorov numb-rs are not. regular; cf. 11.11.9 and 11.11.10. 11.7.6. P r o p o s i t i o n . Let S E O(E, 3').Then c,(S) = d,(S')

a,(&) 2 C,(S').

ft?d

Proof. 8inct. Ji, is a metric surjection, the surjectivitv of the Kolniogorov numbers iiiiplies d,(S') = d,(S'J;.) 5 U,(S'J&) 5 U,(JFS) = C,(S). (1) Since Q i is a itirtric injection, the injectivity of the Gelfand nurrihcrs yields

C,(S')= C * ( Q ; S )

5 fl,'(Qki?) 2 U,(S&e) = d,(S).

(2)

Gsing 11.7.5 and (2) we obtain c,(S) = C,(sr,S)

I L.7.7. Wo

I ~ O Wgive

= C,(S"K,)

5 cn(s") 5 d,(LY).

(3)

a n iliiportmt supplenient of the preceding result

Theorem. Let S E &(E, F). Then c,(S) = dE,(S') nnd

(En(#) = ~~(8').

Proof. Observe that (Ear)' has the metric extension property. Hence, by 11.5.3 and 11.7.4, d,(Sf = a,(S&)

= a,(Q;S)

= cn(Qg7) 5 c,(S').

(4)

11.7.8. T h e o r e m . The Hilbert numbers w e completely symmetric.

Proof. Let S E S(E, F ) . If X E @(El, E ) and B E @(F,K ) such that llXll 2nd jlBll 1, t-hen h,(S') 2 u,(X'S'B')

51

= CL,(BSX).

This proves that hn(S') 2 hn(S). On the other hand, given

E

> 0, according t o 11.4.3 we choose Y E g(Z;. F') and

Obviously there is B E e ( F , 2 ; ) such that Y = B'. Moreover, by E.3.2 we can find S f S(lg, E ) with X'S'Y = AS'Y and llXll 5 1 + E . Hence PI, = X ' S ' B . We now

154

Part 3. Theorv of Seauence Ideals

obtain PI, = BSX, as well. Consequently

1

h n ( 8 ' ) = Q = hn(e&J 14-8

I I141 h,(S) IIXII 5 (1 4- &) U S )

so h,(S') 5 h,(S). 11.7.9. Proposition. There exists an s-function whic7~i s injective, .surjectizw, and completely symmetric. Proof. Put s,(S) := n,(JFSQE)for S E B(E, F ) . Since the other s-number properties are evident, we only check (OS,). For this purpose let E be a Banach space with dim (It)2 n. Supposs that L E ~ ( E S W Ein , j ) and rank (I;) < n. Tlieii, given E > 0. 1 by E.1.2 we can find .co E E s ~ rsuch that //zoI/5 1 -& E and IIJEQEzO - yII 2 I + & 1 for all y f N ( L ) .Hence IlJEQE - LI[2 -This proves that s,(IE) 2 1. (1 + e),'

The injectivity and the surjectivity can be checked as in 11.8.5 m d 11.6.5: respectively.

Let K denote the canonical injection from (3")"' Then QF.= JhK. Therefore a,(#')

= c,(h"Qp)

z=

= Zl(

tTF,) into (PW)'>= 11(UFr)''.

c,(S'J;h') 5 C n ( 8 ' J i , ) 5 dn(Jps)= S , ( 8 ) .

This shows the symmetry.

11.8.

Additive s-Numbers

11.8.1. An s-function s is called additive if

+

s?&*+"z-l(sl 8,)5 s,l(S,)

+ snz(S,)

for all

s,,s, E O W , F ) ,

Remark. This condition contains (OS,) as a special case. 11.8.2. Theorem. The approximation numbers. Gelfand numbers, Kolmog~rovnzmbers, a d Hilbert numbers are additive.

Proof. Given 118, - Lkli 5 (1

E

+

anl+n,-l(sl

> 0, we E)

a,,(&)

+

82)

choose L,, L, E g ( E , F ) such that rank (Lk) < n, and for k = 1, 2. Then rank (L, L2)< n, n2 - 1 and

+

+ - (L1 + 4 ) I I 5 llsi - LlI! + 5 (1 + [a,l(Sl)4-%l(&)I.

2 Il(S1

82)

+

1182

- &ll

E)

The additivity of the remaining s-numbers can be derived from that of the approximation numbers.

11.9.

Multiplirative s-kumbers

11.9.1. A n s-function s is called mnultiplicative if .sm+,,(ST) 5 s,(S) s,(T) for all T E O(E, F ) and S E 2 ( F , G ) .

11. $-Numbers of Operators in Banach Spaces

155

R e m a r k . This condition contains (08,) as a special caw.

i 1.9.2. T heo rein. The approximation numbers, Gelfand numbers, and Kolnqoroc numbers are multiplicative. Proof. Given E > 0, we choose L E 5(E,17) and K E S ( F , G ) such that rank(L)
+

+

+

+

The multiplicativity of the remaining s-numbers can be proved by the help of 11.5.6 and 11.6.6, respectively.

11.9.3. P r o p o s i t i o n . T'he Hilbert numbers are nol nmlliplicntive. Proof. The assertion follows from n-llZ 5

h,(I: co -+ co) 5 cn-112,

where c is a positive constant. This estimate can be checked by using results from the theory of absol~t~ely h u m m i n g operators; cf. 22.4.2 (remark).

11.10. Maximal s-Numbers 11.10.1. An s-function s is called maximal if

sn(S)= SUP {s,(BSX):x' c Q(E0, E ) , IIXIl 2 1, B i B(F, Fa), ilBll d 1) for all S E 2 ( E , P ) . Here Eo and Fa range over all Banach spaces.

11.10.2. Using the same technique as in 8.7.4 and 8.7.5 wv obtain the P r o p o s i t i o n . An s-function s is rnnxinzal i f and only i f

s,(S) for all

= sup

(s,,(&f;SJ&): M E Dim ( E ) ,N

c Cod ( F ) ]

X E B(E,F ) .

11.10.3. For later use we state the L e m ~ i i a .Let R be n maximal s-function. Suppose that 8 c 2(E,F ) am? (zn)E co with 8 2 z, 5 1. Then, qiven E > 0, thew are X E Q(Eo,E), IjXIl 5 1, and B E @ ( F , F 0 ) , IlBIl 5 1, such thnt zn s,(BSX) 1sn(S) for

rt =

1 + E

Here Eo and Fa arc szcitable Banach spaces.

1, 2,

...

156

Part 3. Theory of Sequence Idea,ls

P u t E0 :== ll(E,) and F, := l,(F,). Then the desired operators can be defined by thc equations

Here J: and Q: arc the canonical limps defined in C.4.1 and equipped with sonie obvious superscripts. U4ng t,B,,SX, = QffBSXJf we have 7 , s,(BSX) 2 s , ( & ~ B S X J2 ~ )z,s,(B,SX,) 2 s,(4

I+&

*

11.10.4. The next result is analogous to 8.7.11.

I'roposit ion. Euery mnxinurl s-function i8 regular.

11.10.6. An s-function s is called ultrastable if s,2((Si)li) 5

liln sn(Si) 11

for every bounded family (Si)of operators SiE Z(Ei. P,) and every ultrafilter

U.

11.1Q.G. The following result is analogous to 8.8.7.

P s o p o s i t io n. Every ztltrastLrble regular s-f uiact ion i s masirrinl. R e m a r k . Let us mention that, conversely, every iriasiiiial s-function is ultrastable and regular; K. D. KURSTENEl].

3 1.10.7. ,4n s-function s is called weakly lower semi-continuous if

sn(S) lim inf s,(S,) v

for every direct,ed family of operators X, 5 f?(E.3')converging to S weak operator topology.

E B(E,F ) in the

11.10.8. The next statement is similar to 8.7.17. P r o p o s i t i o n . Every maximal s-function is weakly lower senti-continuous.

11.10.9. Leinma. Let ( L a )be a bounded famihj of operutors Li E S(Ei,F E ) .Then mnl; (hi) 71. inplies rank ((Li),)2 n. Proof. By B.4.8 the operators Lican be represented in the forin I1

Li

=

(lik

'3v*

k=l

such that $zikll 5 1 and llyikll 5 IILill. Let a, be the functional on (Ei)ndefined b? (E, a k ) := Iiin (xi,u i k ) U

for x = (si)u

11. s-Numbers of Operators in Banach Spaces

157

and write yk := ( Y ~ ~ It ) ~follows . from

that

Therefore

So we have shown that I, :== (L,)uadmits t h e representation n

L=

a&‘1;y k . 1

Hence rank ( A ) 2

71.

Wc :$re now able t o check the Theorem. The approximntion i~uinbersare ultrastuble. Proof. Let (8,) be a bounded family of opcrators 8, t 2(E,,P,). (hven E > 0 choose Li E g ( E , , F , ) such that rank ( L , ) < TZ and llLT2 - L,ll 2 ( 1 E ) a,(S,). I t follows from

+

rie

llLtll 5 IF, -

+ IIJ.~’A 5 (1 f &) 4 8 , )+ lifl,Il

i(2

+

2)

lls81:

that the family (hi)is also bounded. Hence rank ((LJU)< n. Moreover, we have

This completes the proof.

ll.LO.10. As a n immetli~ieconseqwnce of 11.7.3 and 11.10.4 we get the P r o p o s i t i o n . The appro~
fll.10.11. Using the method of 8.8.8 we can check the Theorem. The Gelfund numbers and Kolmogorov ?mrnbers are ultrastnblc.

1 1.10.12. As a supplement of the preceding result we obtain the T h e o r e m . The Gelfand numbers are maximal, but the Kolmogorov 7Lurnbw.s uie not. P r o o f . The maxiinality of Gelfand numbers follows froin their regularity and ultrastability; cf. 11.10.6. The Kolniogorov nunibers are not ni~ximalby 11.7.5 and ll.JO.4.

11.10.13. The last statement of this section is a n iiuinediate consequence of 11.3.4 and the definition 11.4.1. Theorem. The Nilbeit numbers ore rnu.t.irna1.

168

Part 3. Theory of Sequence Ideals

11.11. s-Numbers of Diagonal Operators 11.11.1. I n this section we determine or estimate s-numbers of diagonal operators where 1 (= u, v 5 co. For this purpose it ii-ill be supposed that S E %(la, S(E,) = ((r,En) for (6,) E I, and cAL: 0: = 10.

z),

11.11.2. We define

and

JR(E1,

..*,En)

&%(El,

--

Pn(E1,

->

9 m . J

...1,

:= ( E l , * * - , E n , 0,

tn, Enll, En, E n + l ,

--a)

:= (tly

..-) := (51,

-

* .7

En) >

-.*,E n , o ,

***).

The choice of the underlying Banach spaces will depend on the specid situation. 11.11.3. Theorem. Let 1 5 u 5

00.

Theit

a,(S I , + I,) = c,(S: I,, -+ 1,) = &(S:1, + I,,) = 0s.

Obviously rank (L) < n. Hence

P r o o f . P u t L := SP,,.

a,(S: I , +)2,

5 /(AS - L: 1, + lull = 0,.

Suppose that a, > 0. Then 8, :=QnSJn is invertible and IIS;': 1: -+Zz11 = a . ' , Therefore, using 11.5.7 we obtain 1 = cn(In:1; -+ 1;) (= c,(S,: 1; -+ 1:) llS;1: 1;

1:z

3

5 cn(S:1,

-->p

0 , ' .

This proves that o,, 5 cn(S:1, -+ 1,)

S an(&1,

+ 1,)

5 an.

The Kolmogorov numbers can be treated by the sane nlethod; cf. 11.6.7.

11.11.4. To prove the next theorem we need some leinmas. with cod ( A ) < n. Then there em& Lemma 1. Let N be a subspace of = 1) 2 112 - n 1. e r= ( E ~ ..., , cn,) E hrsuclt that llelioo = 1 und card ( k :

+

Proof. We fix some extreme point e of U,. Put

11 := (2:Ek = 0 for k E K ] . K := ( k : /&kl = i] and 2 Clearly card ( K ) f dim ( N )= m. Suppose that card (I<) 2 m - n. Then dim ( M ) 2 n and therefore M n N (0).Hence we can find 2 c M n N with 11x11, = 1. Since

+

6 := 1 - sup ( j & k l : k

6 K ) > 0,

it follows that e f 6x c UAh-. So e cannot be an extreme point, which is a contra1. diction. Consequently we have card ( K ) 2 m - 12

+

Lemma 2. Let 1 d then,

z1


a. I f , u l ,..., ,u,+~

> 0 and IEntll I nlin (IflI, ...,

11. a-Numbers of Operators in Banach Spaces

159

Proof. Put

We are now prepared to check the Theorem. Let 1 5 27 < u

a,(&: I,

--f

I,)

5 00

= cn(S:I ,

-+ lu) = d,(S: I ,

Proof. First we treat the case 1 Hence

We now suppose that

D(&, ...)&n)

G,

and 1/r = l / v

< u < 00.

- 1/u. Then

41,)

=

2'al;

(nm

ll".

P u t L := SP,,. Obviously dini(L) < %.

> 0 for some rn 2 n. Then by setting

...)o,""Em)

:= ( a p E l Y

we dcfine aonc-to-oneoperatorD from 1; onto lz. By Lemma 1 for every subspzeeH of 1: with cod ( N )< n there is e = (cl, ...,8,) E D ( M ) such that llelim = 1 and I, where K := ( k : lckl = 11. P u t S, := Q,SJm anddeiiote the card ( K ) 2 n~ - N injection from M into 1; by J. If x := D-le, then Lemma 2 iinplies

+

160

Part, 3. Theorg of 8eyuence Ideals

Consequently from 11.5.6 we obtain

c,(S: I,,

--f

Z") 2 c,(S,,,:1:;

--f

1

);2

This proves that,

To treat the case u = co w e only need Lemmti 1. The Kolmogorov numbers can be deterniined by dixality; cf. 11.7.6.

11.11.5. L e m m a . Let 1 u n ( I ::1

-+

4v I :u 5 co and

Zy) ==-

(in - n

+

IIr

/or

1)1/7

= 1/r 7~

= 1.

- l/u. Then

.,., /ti .

Proof. By 11.11.3 and 11.1 1.4 the assertion follows from an(I:1: ->

);z

= n,,(P,,,: I,

3

Z,).

11.21.6. In the following we deal with diagonal operators froiii I , into lp. For this purpose some informations ahout orthogonal projections in 1, are required. Lemma. Let (z,)be a scular sequence. Then there exists an n-dime~isionnlo r t h q m l projection P E 2(12, 2,) urifh IIPeilJ,= 3;for i = 1 , 2. ... i f and only if co \7

U

7c?

I

= 1L

0

rtnn

s z, 5 1.

1

Proof. Since every n-dimensional orthogonal projection P can be written as I&

P

=

2-z5

..., z,) is an orthonormal family, it follows that

xk. where (z,.

1 N

'u

1

i-1

2 1.

Clc*arly0 5

n

I1

5 l/Pei/l:= 2' k_1: ifer, =l

xk)j3

/irk!/; = n.

= 1

f

Conversely, let ( x i ) be a scalar sequence satisfying the above conditions. Then the existence of P can be checked by induction. If ?L = 1, then P := x* x := (z,, z., , ..) is the desired projection. We now suppose that the assertion l i a h been proved for s m i e n. Let W

2 ni =:

11

+1

and

I 2 z1 2 n2 2

1

Then there exists a natural number k such t.hat h

h-,l

')

2 x9 < 1 -=, ; n;. 1

I -

1

Pnt

Ixi

otherwise.

-

0.

161

11. 8-Numbers of Operators in Bmach Spaces

0bviously k

00

znii=n,

En;;=1, 1

and O ~ n o i ~ l .

k f l

Therefore we can find a n orthogonal projection P with

llPeillz = 0 for i = 1, ...,k and llPeillz == zOi for i = k

...,

+ 1, ...

Let zo:= (nol, n o k , 0, ...). Since 1 1 ~ ~ 1 1 , = 1 and Pxo = o tc.e see that the operator Po := : Z @ xo P is a n orthogonal projection with llPOeajjp = noi for 1. By setting i = 1,2, Let 0 5 (Y

+

...

u: :=

1

( 1 - a2)1/$ek -&ek

+ cxek+l

for i = k ,

f (1 - a2)l1'ek+1 for i = k f 1 ,

otherwisci

ez

we define a n orthonormal sequence (ui).Since the operator 00

U,:=E

ef @ u:

1

is unitary, it follows that P, := U ~ P o Uniust , be an orthogonal projection. Observe that Pa is the above Po if LY = 0. Clearly we have IIPaeil12 = zoi= ni for i k and i k 1. Moreover, l/Poekll2 = nOkand iiPlekll2 = Zok+l. Since I,P,eklIpdepends continuously on the parameter a and 7dok 2 n k 2 zbl 2 z o k + l , there exists a. with /IPu,ekllz= nk.Then it follows from

+

=+ +

C llPaee~ll~ =n + 1 W

1

that ll~abek+l~12 = 7dk+l. s o Pao is the desired projection.

11.11.7. T h e o r e m . If (q&) E co, theit n,(S: I ,

--f

7,) = sup

)2: &

h = ?L, n

Uk

Proof. By 11.6.2 we have

i

+ 1, ... .

an(S:I , + I,) = inf { 1 1 8 - PSI/:dim (P)< n ), where the infimum is taken over all aciinissible orthogonal projections. I n view of : S - PSll = sup Il(8 - PS) e&., the preceding lemiria iniplies

11 Pictsch. Operator

162

Part 3. Theory of Sequcnce Ideals

Otherwise let

oh,,

:= 0. It follows from

sup (1 - ,f)l’2 u p i 1 2 (1 - ?

C p

i

that h

h

i

(1 - 3 : ) 2 h -n

1

sup (1 - z;)o ; 2 ’ o i 2 1

-+ 1.

1

Consequently

sup ( I - n 5 ) 1 / 2 oi 2 ohn for 72, = n, n.

+ I , ...

i

This proves that an(&El -> Z2) 2 sup (ahn: h = n, n

+ 1, ...).

Obviously

Hence 0

5 lini sup ohn

oi

ahon= SUP (ohn:h = n, n

ho-n ha-1

2 oi2

and therefore lim ohn = 0. so we can find ho 2 n with h

h

+ 1, ...]. By an easy coinputation, it follows from

s h,-n+l

2

oi2

1

1

and

h,-n+

1

h.

5-

Y

oi2

1

thk

1

(1 - ,;)1’2

Cri

=

ohen

oi

for i = 1, ...,h,, for i = 71, 1,

+ ...,

2 -

h,+ 1 - n + 1 9

he+l

z‘ 1

Oi2

163

11. a-Numbers of Operutors in Banach Spaces

11.11.8. Lemma.

Proof. By 11.11.7 the assertion follows from

un(I:I;" -+ Z?) = a,(P,: I ,

ZZ).

--f

11.11.9. Proposition. U , ~ ( I,I :--f co) = 1 for n = 1, 2, ... Proof. Obviously

an(I:I, -+ co) 5 111:I ,

--f

coil = 1 .

I n order to check a n ( l :I , -+ cg) 2 1 we consider L E g(Zl,co). Then, given there exist y, .,ym E co such that

..

rn

L(u1) S U k/i 1

E

> 0,

+

Here we denote by U , and U , the closed unit ball of I, and,,Z respectively. Write yi = (rib)and choose n with lqinl 2 E for i = 1, ...,m. For some & we have ((Len- yi,llco 5 E. Hence

IiI - L:4

+-

coll L IlIes - Lenll, L Illen - ~ 2 11 - ??i,J - & 2 1 - 2 E .

This proves that 111 - L : I ,

--f

cojj 2 1 and therefore

11.11.10. Proposition. an(l:I , -+2,)

lbi0- Lenlloo

i o ll ~

as(n(lT:I , -+co) 2 1.

= 2,3,

= l / Z for

...

Proof. Put

1

Lo := y x o @ x,, where xo := (1, 1, ...). z

Then rank (Lo)= 1 implies an(I:ZI 3) ,2

5 111 - Lo:I,

->

1,Z

= 1/2.

Suppose that there exists L E @(I1, I,) such that 111 - L: I , + Z,11 denote the E-th coordinate of Lei. We now obtain

- likl5 ] ] l e i- Leill, 5 111 - L: I ,

3

< 112. Let ti,

Z,11.

If i =+ k, then IlLei

- hek]], 2 I&

- tkkl= 11 - iljk

- (1 - &)I 2 1 - 2 [I1- L: I ,

+ Z,1

3 0.

This proves that L( U,) cannot be preconipact, which is a contradiction. So we have 1 1 1- L:I , -+ E,IJ 2 lj2. Hence crn(I:I , -+ E,) 2 1/2.

11.11.11. In the following we use logarithms of base 2. Sublemma. There are xl,...,x, t 2; such that

11*

112i1i2=

1 and

164

Part 3. Theory of Sequence Ideals

.

Proof. If m 5 12, then we can take a n ortlionorinal family (xl,..,xm).Let us now suppose that for soinc m 2 n there are xl, .. ., xmE 1; which have the above property. Put p := log m. If cp denot,es Khintchine's constant, then

+

cp 5 ( [ p / 2 ] 1)112

5 (log m)1/2.

It follows from m

that, for at least one e E P, we have nr

2' \(Xi7 ellp 5 $m1 Put xm+,:= d 2 e . Then

.., rn. So the assertion is proved by induction.

for i = 1, .

We are now able to check the important

L e m m a.

un(l:1';

ZZ)5 3

4

for n

=

1, ..., m.

.., xm E I; as constructed in the suhlemma. Write x i = (Eil, ..., tin)

Proof. Take xl,. n

P u t L := 2 z ! @ x!, where x! := (tlr, ..., f m f ) . Clearly rank ( L )5 n. Moreover, 1

the operator L is represented by the matrix ((xi,xk)). We now see from

C L ~ + Zy ~ (+ I : Zg) 5 )!I- L : Z';" -+ZZli 5 2

log m

This completes the proof. R e m a r k . For nz 2 2 we have

ul(I:1';" --f l g ) == 1, a2(I:1': + 1:) = 112, and u m ( I :1';

--f

Zg)= l / m .

However, it follows immediately froni 11.11.8 that a&:

1';"

4

I",") 2 cntllz.

Using number-theoretical methods R. S. ISMAGILOV [ 2 ] proved that

11.11.12. It seeins very likely that the preceding lemnia can be extended in the following way. For a somewhat weaker estimate we refer to B. CARL/A.PIETSCH [2].

11. s-"nmbers of Operators in Banach Spaces

C o n j e c t u r e . Let 1 5 u 5 2. Then there exists some constant e

165

> 0 such that

11.11.13. Finally, we mention a very striking result which has been recently proved by B. S. KASHIN[3] and B. S. MITJAGIK[a]. Lemma. There emysts 0 such that

ZF) 5 p

c,,(I:2;" -+

11.12.

[log (7n

+ 1)]3'2

nI/2

for n = 1,

..., m .

Relationships between s-Plumbers

11.12.1. As a consequence of 11.2.3 and 11.4.2 we have the Theorem. Let S E O(E,3').Then US)

s c,(&

d4s)

and

h,(4

5 4 2 ( W 5 a,,(S).

11.12.2. In the followiiig some converse estimates are checked. Theorem. Let S E O(E, F ) . Then a,(S) 5 Zn1/2c,,(S) and a,(&) 5 2dkI,,(S).

Proof. Given F > 0, we chooseanoperator L E g(E*u, F ) withrank ( L )< n and - I,]/ 5 (1 + E ) d,(S). By B.4.9 (remark) there exists a projection P E B(P,P) such that M(P)= M ( L ) and llPil 5 (n - 1)1/2. Obviously rank (P)< n. Moreover, it follows from

IP - PSI1 = iiVF - P ) SQEII= IIUF - P)(SQE- OII 5 (1 + ( n - 1)1/2) llSQE - L!l that a,(#)

p /IS- 15s;;5 (1

+ (n - 1 ) q

IISQE

- LI/ 5 (1

+

E)

2n1'2dn(s).

The remaining estimate can be proved by the same technique. R e m a r k . If 0

< a < 112 and p > 0, then estimates of the form

a,(S) 2 gnv,l(s) or a,(S) 5 ,oned,(X) cannot hold for all S E B(E.F ) ; cf. 11.11.8 and 11.11.13.

11.12.3. Theorem. Let S E ,f?(E,3').Then n

n ck(S)5 1

> 0,

fl hk(8)

n dk(S)5 enn!fl h k ( 8 ) . n

11

18

e%!

and

1

1

1

-&(A').

Then there exists z1E E with I+& ilzll/ 5 1 and IISxlll 2 al. By induction we can find zl,.., x,,E E such that llzkll 5 I and Il&;,Sxkll 2 6, for k = 1,. .., n. Here Nkdenotes the linear span of (19.z~: i < k]. P r o o f . Given

E

we put

BK :=

.

166

Part 3. Theory of Sequence Ideals

Next we choose b: E (PINk)‘with 1lb:ll 5 1and I(&;,Sxk, b;)l 2 dk. If bk := (Q”,)’bE, then 1(Szk,bk)l 2 dk and (Sxj, bk) = 0 for i < k. Hence the matrix ((Sxi,a,) has superdiagonal form, and it follows that n

[det ((8%h))l==

fl I(fl%

n

6k)l

n8k 1

1

By setting

2

n

X(Ei):= 2’ tizi and By := ((y,

a,)

1

we now define operators X E f?(g,E) and B E 2(F,Zi). Clearly llXll 5 d 2and l[Bll 5 n1I2.Put So := BSX and consider a Schmidt representat,ion n

so= 2’ a&

@

Vj

1

such that al 2 ... 2 equality (27.2.3) froin

CT,~

2 0. Then ai = aj(So)5 nhj(S). Using Hadamard’s inn

(Szi, h) = (Soei,4 = 2 (ei, uj) aj(vj, ek) j=1

we get

n 1

r n” n hi(&). n

A

5

of

1

Clearly 12’ 5 enn!.This proves the second estimate. Finally, the first estimate follows by 11.7.6 and 11.7.8.

11.12.4. Problem. Does there exist some constant e

> 0 such that

c,(S) 2 enh,,(S) and dn(S)2 e?zh,(B) for all S E f!(E,P). Remark. Weaker estimates have been proved by W. BAUHARDT [l].

11.13. Notes I n the context of integral operators the concept of s-numbers first appeared in a paper of E. SCHMIDT[I]. The generalization to approximable operators in Hilbert spaces probably snd R. SCHATTEN [l]. An excellent treatise of this theory is goes back to J. VON NEUMANN the monograph [GOH, pp. 24-64]. In Banach spaces there are many different possibilities [l] has introduced the to define some equivalents of 8-numbers. At first A. N. KOLMOGOROV so-called n-th diameters (widths) of bounded subsets, and then I. M. GELFANDsuggested that there is also a dual concept. The approximation numbers have been studied by A. PIETSCII [2]. The reader is warned that the axiomatic theory of s-numbers presented in this chapter differs from that developed by the authors in [18]. There is an extensive literature devoted to computations and estimations of 8-numbers of diagonal operators in sequence spaces and of embedding maps in function spaces.

11. s-Numbers of Operators in Bmacli Spaces

167

flccommendrttions for further reading:

[DUN, [LOR], [PIE], [SAT,], [SENl, exp. 18, 191, [SE-,

exp. 171.

W.BAUHARDT [l],N.S. BIRMAB/N. Z. SOLOMJAK [l],B. CARL/A.PIETSCE (21, M. FIEDLER/ V. P T a K [I], E. D. GLUSKIN[l], I. C. GOHBERG/M. G. KREJN[l], R. A. GOLDSTINE/R. SEAKS 111, C. W. HA [l], C. V. Humox [l], C. V. HUTTON/J. R. MORBELL/J. R. RETHERBORD [I], R. J. ISMAGILOV [l], [2], P. D. JOIINSOB [l], B. S. KASHIN[l],[ 2 ] , [3], H. KONIG[a], K. D. [CURSTEN [I], G. G. LORENTZ [l], V. E. MAJOROV [l], A. S. NARKCS [l], B. S. MITJAGIN [l], i4], B. s. MITJAQIN/G.31. CHENKIN [I], B.s. &fITJAGIN/A. PEECZYP~SKI [I], B. 8. MITJACIIN/ V. $1. TICHOMIROV [l], A. PIETSCH[20], 8. J. ROTFELD[i], M. Z. SOLONJ.AK/V. M.TICHO.~IIP.OV[l],V. M. TICHOJI~KOV [l], [2], H. TRIEBEL[l], [2], [3], [4].

12.

Entropy Numbers of Operators in Banach Spaces

I n the folloaing we introduce the so-called entropy numbers of operators in Banach spaces. This concept is more suitable for generating operat,or ideals than the c-entropy defined by L. S. PONTRJAGIN and L. G. SCHNIRELMANN in 1932. First we state the basic properties of entropy numbers. Then, as an elementary example, diagonal operators in classical sequence spaces are investigated. Finally, we deal with the relationships bet,weenentropy numbers and some special 8-numbers. In what follows all logarithms are to the base 2. For simplicity we consider real Banach spaces only.

12.1.

Outer and Inner Entropy Numbers

12.1.1. A map s which assigns to every operator S a unique sequence (sn(S)) is called a p8eudo-s-function if the following conditions are satisfied: (OS,) liSll = sl(S)2 sz(S)2 ... 2 0 for S E B(E,P). (08,) S n ( S T)5 s,(s) IlTll for 8, T E B(E,P). (OS,) s,(RST) 5 IlRll s,(S) IlTII for T E B(Eo, E ) , S E B(E,F),R E B(F, Po)Concepts such as injectivity, surjectivity, additivity, multiplicativity, and maximality can be carried over from s-functions to this more general situation.

+

+

12.1.2. For every operator S E B(E,F ) the n-th outer entropy number en(S)is defined to be the infimum of all o 2 0 such that there are yl, ..., yq E F with q 5 2*-l and 4

fwd s u

+

{?A

1

O77FI.

12.1.3. Theorem. The map e: S -+ (en(#))

i s a pseudo-s-function. Proof. First we check IlSll = e,(S). It follows from

lSl1 := inf {o 2 0: S(UE)G C u p ] that el(#) 5 $Sil. We now assume that S(U,) C= yo + aU, for some yo E F . If x E UE, then there are y+, y- E UF with +Sx = yo + ay, and -Sx = yo + q-. Hence 2 IlSxll = c j/y+- y-11 5 20. So we have l/SIl2 o and therefore IISll 5 el@). This proves property (OS,). The remaining conditions (OS,) and (08,) are special cases of 12.1.4 and 12.1.6.

12.1.4. Theorem. The outer entropy numbers are additive. Proof. Let that

s,,8,E g ( E ,F ) . If

sk(UE)

‘t

5

ak

> e,,(sk), then there are yik)’,.. ., yb: E F such

4k

u {y:k’+ O k u F ) i=l

and qk

for k = I, 2.

12. Entropy Numbers of Operators in Banach Spaces

169

u,, we can find i k with s k x E ?,Jif’ f (TkuJ?for k = 1, 2 .

Hence, given x E

This implies

(81f

82)

x t ?At’ f y:f) f (el

+

62)

K’F

and therefore v,

+ S,) ( U c ) s u

Pa

u iL=l

(81

{ykf’

i,-1

+ Ylf’ + + c?.)U p ) . (61

Xoreover, qlqe 5 2(nl+n*-1)-1. Hence en,+n,--l(S1+ 8.J 5 el the proof.

+ cz. This completes

12.1.6. Analogously we obtain the T h e o r e m . The outer entropy numbers are multiplieutive. 12.1.6. For every operator S E P(E, F ) the n-th inner entropy number fn(S)is defined to be the supremum of all e 2 0 such that there are xl, ..., xp E with p > 20f-’and

rJE

k. ] / S x i - Sxkll > 2e for i =i= 12.1.7. The next result can be established easily. Theorem. The map

f: 8

+

( f n m )

i s a p8eudo-s-fumtion.

12.1.8. The following statement is also trivial. P r o p o s i t i o n . The inner entropy numbers are injective and surjective. 12.1.9. P r o p o s i t i o n . The inner entropy numbers are maximal. Proof. Let e < fn(S).Then we can find xl. ...,x,.,E U, such that p > 2R-1and / I ~ z ,- Sx,ll > 2e for i + k. Choose b1k E UF,with l(b’xz - s x k , bzk)l = \lsxA- SX,]~. Let M be the linear span of xl, ..., xp and put

N := {2/ E F : (y, bL,:)= 0 for i, E

= 1,

...:p } .

Then

liQ$SJ$x, - Q$XJE,x,ll 2 I(Sx, - Sxk. a,,)]

> 29

for i -j= k.

Hence fi,(S) = sup {fn(Qf8J,”,):ill t Dim ( E ) ,N E Cod ( F ) } .

12.1.10. We now compare outer and inner entropy numbers. Theorem. Let S t B(E,3’).Then fn(S)5 en(@ 5 2fn(S). P r o o f . Suppose that :md yl,..., y4 E F with

6

IjSx, - Sx71j> 20

> e,<(S)and

p

< f n ( S ) .Then we 9

for i $; j

znd

S ( U B )G

can find xl, ..., rP

u {yL+ C T U ~ ~ ) ,

&-I

Yp

170

Part 3. Theory of Sequence Ideals

whcre p > Zn-l 2 q. So there must exist different elements Sx,and Ssi which belong oU,. Consequently 2~ < j\Szi- SzJ 5 20. This proves that to the same set 7jk f n ( S ) 2 e,(Q Given e > f,&(S), we choose a maximal family of elements rl, ..., xp E UE such that l\Ssi - Sxk\\> 2e for i f E. Clearly p 5 2-1. Moreover, for L U Ewe can find borne i with jlSx - Sxi/l 5 2p. This means that

+

P X(uE)

!G U {Ssj 1

+ ~QU,).

So %(AS’) 5 2~ and therefore e,(S) 5 2fn(S). I2.1.11. P r o p o s i t i o n . Let E be an intermediate space of {Eo,E l f possessing K-type 0 I f S € 2( EZ,F ) , then G.+~,-~(S: E -+ F ) 5 2enO(S: Eo + F)l-e e,,(S: El

3

F)O

p r o o f . Suppose that crk > e n r ( 8 : Ek .+F ) . Then we can find pk

S(uE,) 5 u (!/‘,“’ h=l

+

+

+ (1 + e ) := (1 +

s u u kJoyIp.’ + ely;:’

+ 2(1 +

ho=l

E F with

be given. By C.5.5 there are 1-8 R c1 I/zljll5 (1 E ) c0 cl.

+

+

0:-’0TuF

Hence

Qa

S(U,)

>0

oo llzollo

u;~cTA-‘IY~.

for some hk, where @k

....’:y

GkuF)

m d qh 5 28e-1 for k = 0, 1. Let 2: E UE and E zo E Eo and x1 E E, with ro x1 = z and Obviously we have

Sxk 6 @kyLt’

ZJ;’),

E)

91

hl=l

S o w it follows from qoql 2

E)

o;-60!UF1.

that

ea,+nl-l(S:E -+ P ) I 20;-eu!. This completes the proof.

12.1.12. P r o p o s i t i o n . Let F be an intermedinte space of {F0,PI}possessing J-t?jpe 6 . I f S E B(E, F A ) ,then en,+nl-l(S:E

--f

Proof. Suppose that with

F ) 5 2ens(S:E -+ Po)‘-’ enl(S: E + F,)’. crk

c= u {Yp’ +

> e,,(S: E .+Fk). Then

we can find y:”,

..., ?/it’E Fn.

4k

S(U,)

0kUF,I

h=l

and Put

k

t&

5 2mk-1 for k == 0,1, Let xI,..., zpE I , := (i:8x6 E yp’

+ a0Upo}.

uEbe

given, where p

> 2(”0in1-1’-1.

12. Entropy Kumbers of Operators in Banach Spaces Ps

card (I,)2 p

Since

171

> yoql, we have card (I,,) > q1 for some ho. Hence there are

&= 1

i, j E Ih, such that Sx,and Sxi belong to the mme set yit' -+ glUF,.This mcans that ' / S X~ Sxl//k5 2ck for k = 0, 1. Now C.5.7 yields jlSx, - SXJi 5 2o;-'ei. Hence fnoL,z,-l(S E: -> F ) 5 IT;-'CT?.

So, by 12.1.10, we obtain the dvsired estimate. 12.1.13. Finally, it is shown that the outer and inner entropy nnnlbers do not have all s-number properties. P r o p o s i t i o n . If dim ( E ) = m ,then

Proof. Observe that there is a translation-invariant Bore1 measure p on E . Obviously we may assume that p ( U E )= 1. Then p(OUE)= em for all c 2 0. Let P

UE

U [xi

+ oUE)

and q

5 P-1.

1

it follows from Q

1 =p(uE) 5

/.&(xi

0 u B ) = qcm

1

that u 2 q - 1 ' ~ 2 2-(n-l)/m. So the first estimate is proved.

+

e)/e = 2(*l)lVn.Since 77, is compact, we can find a Let e be determined by (1 maximal family of elements zl, .... xp g UE such that IlSzi - SZ~II> 2e for i k. Then the sets x i eUE are disjoint and we have xi & (1 f @)uE. Hence

+

13.2.

+ ,puE

Entropy Numbers of Diagonal Operators

12.2.1. First we deal with identity maps between finite dimensional spaces. Lemma. Let 1 5 u, v 5 co. Then

+-

172

Part 3. Theory of Sequence Ideals

P r o o f . Let us denote by Suppose that 9

L7:

arid U y the closed unit ball of 1; and l?, respectively.

+ (iUy}

t7: 2 (J (yL

and q

2-l.

1

Then 9

i(UZ)5

Z(yt I

+ UUY) = qo"i.(U;"),

where h is the Lebesgue measure on W71'. Now A(Ug) = 2" and /?(UT)= P / n z ! imply U* 2 m!/2m-1. Using e:%! > m* we get (i > m / 2 e . Therefore

_- Zy) 2 m / 2 e .

e,JI:: 2

Finally, it follows from

Zz

em(I:

--f

Zy) 5 111:1; -+ZTii e n a ( IE':," - m 1 L l ' U - l ' V e m ( I : I? <

--f

Zc) 111:ZF -+ZTll

-+ ZF)

that

,

e r L ( I1;: +Zr)

[

5 4 8 log (v;+

1)1°C'" for n = 1, ..., m.

Proof. First we suppose that u = 1 and If vz 1 4 ,then

71

= 00. Let U r and U z be as before.

Put

K ( x ) := { k : !$,(i > (i} for x = (tk) E

777.

We have Itk 1 5 110 < rn.

card ( K ( z ) )<

Wz)

Let 9 denote the collection of all sets K O K : = { x 6 t;g:&.=O

Then 2

Hence

{ 1,

.. ., wz} with

card ( K ) < l/o and put

if k G K ) .

+

U H ( 2 ) UUZ for all x E U?.

u;l s u {r;, +- 0u;i. R

Clearly, we can find y:K) E ZE such that 9h

c

U,

+ oUZJ

U (yjK) i-1

and qK 5 (I/o -!- l)ra1d(2i).

12. Entropy Numbers of Operators in Banach Spaces

Consequently there are yi E 9

u;nz u

{?Ji

173

Zz with

+2 d Z )

1

and

So we get e#:

I;"

-+

Zz)5 20 5 8 log (rnn+ 1)

Obviously this estimate is also true for 72 = 1, 2 , 3. The general case 1 2 ZL 5 v 5 cu can be treated analogously. However, we use interpolation methods. It follows from 12.1.11 and 12.1.12 that

2e,(I: Zy --f ZY)1'9 e,(I: E;" + ZE)l-lis

e,(I: I;"

3

e,(I : 1:

i

and where l / u = ( I - 0 ) j l

e,(I:

Ze,(I : Z?

):Z

+ O/v.

3

ZF)1- el(T ::2 + IF)",

Since (1 - 0) (1

ZE + Zr) 5 4e,(I: Z;"

--t

-

l/v)

= l/u -

l/w, we have

Zz)li*-l'u.

This completes the proof.

12.2.3. L e m m a . Let 1 5 v 5 u 5

e,(I:

00.

Then

+ Zr) 2 m l / a - l i u ,

Proof. The assertion follows froni

e,(I : IF

111 : 1:

-+) :2

-+

ZF,/.

12.2.4. We now estimate the entropy numbers of a diagonal operator 6 E 2(l,, Zv). AS in 11.11.1 it is supposed that ol 2 u2 2 2 0. P r o p o s i t i o n . Let 1

5 u.Y 5 w. Then

1 en(#: Z, + I,) 2 - n1/0-1'*0~~. 2e

Proof. P u t S n ( f l ,.. ., E n ) := ( o l f l ,..., 0 ~ 6 , ) Then . i t follows from 12.2.1 that 1 2e

- n l / ~ - l< i ~c,(Z: ~

-+ Z;)

5 118;:'

5 o;'e,(S: 1, -+ l o ) . This completes the proof.

+ Zzll en(#,:

I:

3

I74

Part 3. Theorv of Seauence Ideals

12.2.5. We now state the main result of this section. T h e o r e m . Let S E

B(&, I,)

such that A S(:,)

= (en&,) and (en)E

co. Then

where p p is a positive constant. Proof. Without loss of generality we may suppose that el 2

2 . - -2 0. Let

6,

E ( E ):= max (k:ek(S) > e ) for 0 < e 5 al. We now show that

Put nz := max { k :c k 2 E ) and 8, := &,SJ,. We denote by U r and U g the closed unit ball of 1r and EZ, respectively. If y t Sm(Up), then there exists a n integer valued vector g = ( y l , ..., y m ) such that y

Sinco el 2

E Ern-llZ(2g

+Uz]

+ EUF

2~rn-1129

... 2 a,,, 2 E , we have

+

~m-92g Uz]

S m ( U r )+ 2~m-112UZ

3Sn1(U.3.

Let (gl, ..., go] be the collection of all gr = ( y i l , ..., y i m ) with

&m-~~Z(2gi -t U E } G 3Sn1(UF). Clearly 9

( , ?!A

ur)& u

(2ET?2-112 gi

1

+ EUrl

and therefore q

+

S ( U 2 )& JmSm(UF) em+lU,

u (2emn-llzJmgi+ ~ E U , ) , 1

where U , denotes the closed unit ball of I,. On the other hand,

+ uz}]5 zrn1b(

q

p[&m-'"lrnA( uz) = 2 I[&m-112[2gi 1

m

U r ) flak

9

1

where I is the Lebesgue measure on z". Using Stirling's forinula we get elI'(t

Hence

+ 1) 2 (2n)1/2tttll2

for O

< t < 00.

12. Entropy Numbers of Operators in Banach Spaces

-.1+q -

P

log ( 1 j t ) d t

P

175

[: ] . o;

0

This completes the proof.

12.3.

Relationships between Entropy Numbers aiid s-Numbers

12.3.1. We begin this section with an easy result. Theorem. Let S g B(E, F ) . Then h,(S) 5 2e,(S). Proof. By 11.4.3, given z.

lIXl1 5 1, IIRll 5 1, and BSX that cn(In)2 1/2. Hence

> 0, there exist X = PI,, where

B(Z!, E ) and B E B(P,1:) such that

1 e =h,(X).

I+&

1 ha(&)= p 5 2e,,(PI,) = 2e,(BSX) 5 2 e , ( ~ ) . I+& This completes the proof.

M7eknow from 12.1.13

176

Part 3. Theory of Sequence Ideals

12.3.2. Theorem. Let S E B(E,F ) . Thew c,(S)

5 ne,(S)

ccnd

d,(S)

5 me,(&).

Proof. If 0 < g < c,(S), then we can inductively find xl,..., x, E U E and ..., b, E up such that l(s.rk, bk)l == IlsZkll > @ and 2, &!k. Here the subspacelVk with cod ( f x k ) < k is defined b y Mk :=(x E E : (sx, b,) = 0 for h < kt. 61,

denote the set of all vectors e = ( E , ,

Let

Obviously card (8,)= 3. If e'

. .., E,) with

q = f l . Then

=+e r r , then we put h := niin ( k : + E;

E;).

Since

it follows that I/S.z,. - Sze.ii > 2e/n. Therefore e,(S) 2 f n ( S )2 p/n. This proves ne,(S). that c,(S) The second estimate can be checked analogously.

12.3.3. T h e o r e m . Let S E S ( E , B). Then

Proof. Given (= (1

//S -- L II

L

X

t

L

+

E

> 0,

E)

= JLoQ be

we choose a n operator L E g ( E . F ) with rank ( L )< m arid

cc,(S). Then

the factorization described in A.3.4. It follows froin 12.1.13 that

This completes the proof. R e m a r k . The case m = 1 is trivial, since e,(S)

5 IISlJ.

12. Entropy Numlicw of Opcrntors in R m a c h Spaces

12.4.

177

Notes

The notJionof &-entropyfirst appeared iri t,he classical paper of L. S. PO'ONTRJACIN~L. G. SCHXII:E:LNIAN [i]. Many results concerning the &-entropyof special subsets in function spaces can be found in A. N. KOLMOGOROV/V. 31.TICHONIIROV [l] and G. G. LORENTZ ti]. We also refer to the monograph [LOR, pp. i5O-i67]. As B. S. MITJAGIN[t] observed in 1961 the &-entropy is n powerful tool to dinracterize nuclear locally convex spaces. I n the same paper he studied the behaviour of this function for ellipsoids in Hilbert spaces. The theory of the so-called entropy numbers has been developed by the aut,hor and is here presented for the first time. However, certain functions inverse to the &-entropy already tbppear in B. S. MITJAOIS/ A.PEEEZY&SEI 111 and H. TRIEBEL[3]. tkcommendations for further rexlitig :

[SEM,, exp. 18-201, [TRI].

R. lr. DUDLEY[I], 31. B. R. OLOBE L3], H. TRIEREL [1].

K.CARL/A.PIETSCH [I], (I'NYIOXIROVEl],

J[tRCUS

[I], B. S.NIT.JIGIN/S'. M.

13.

Sequence Ideals on the Scalar Field

I n this chapter the concept of a sequence ideal on the scalar field is introduced. The most important examples are the classical spaces 1, with 0 < p < co. Roughly speaking, we can say that the theory of sequence ideals is a simple counterpart of the theory of operator ideals. I n the following we only give some basic definitions and formulate many results without proofs. This is motivated by the fact that the analogous statements have been checked in the context of operator ideals. Furthermore, we mainly treat quasinornied sequence ideals. The modifications for the general case are left to the reader. Finally, we describe further examples due to G. G. LORENTZ, W. L. G. SUGEKT and W. ORLICZ.

13.1.

Sequence Ideals

13.1.1. Let us recall that 1denotes the set of all bounded scalar sequences. Moreover, := { 1, 2, ...). we put A sequence ideal a on the scalar field is a subset of 1satisfying t,he following conditions :

dv

(NI,) el E a, where el is the 1-st unit sequence. (NI,) It follows from xl,x2 E a that x, x2 E a. (NI,) If s E 1 and x a, then sz t a. (NI,) Let z be any one-to-one map from .N into itself. Then

+

J,x E a a.nd Q x x E a for all x E a. R e m a r k . Scalar sequence ideals will be denoted by small Gothic letters. R e m a r k . I n place of (XI,) we may also assume that a is permutation-invariant. However, to get analogy with (QNI,) we prefer the above version.

13.1.2. We mention the trivial P r o p o s i t i o n . Every sequence deal i s a linear space.

13.1.3. For later use we define the map D which assigns to every sequence z = ([I, 62, ...) the doubled sequence Dx := (&, &, E,, [,, ...). L e m m a . Let a be a sequence deal. Then x E a implies Dx E a.

D

P r o o f . P u t .;(n):= 2n - 1 and B(n):= 2n. Then the assertion follows froin = J, JB.

+-

13. Sequence Ideals on the Scalar Field

179

13.1.4. A sequence s = (5,) is called finite if it has only a finite number of coordinates tn 0. The set of all finite sequences is denoted by f.

+

13.1.5. T h e o r e m . f i s the smallest sequence ideal. 13.1.6. The set of all zero sequences is denoted by c,. 13.1.7. A sequence ideal a is said to be proper if a

+ 1.

13.1.8. T h e o r e m . c, i s the largest proper sequence ideal. P r o o f . The ideal properties of c, are evident. We now consider a proper sequence ideal a. Let us suppose that a $ c,. Then there exists x c a \ c,. Consequently we can find E > 0 such that i l l := ( n :\tn12 E ) is infinite. Obviously the sequence s = (c,) with cn := 5;' if n c $1 and a, := 0 if n 6 ikt is bounded. Hence efif = sx E a. P u t e = (1, 1, 1, ..-). If n is any one-to-one map from N into itself such that n ( N )= M , then it follows froin (XI,) that e Q,eM E a. So Q = I, which is a contradiction. ---J

13.2.

Qua,si-Kormed Sequence Ideals

13.2.1. Let a be a sequence ideal on the scalar field. A map a from a into 9?+ is called a quasi-norm if the following conditions are satisfied. (QNI,) a(el) = 1, where e, is the 1-st unit sequence. (QNI,) There exists a constant tt 2 1 such that

+

+

a(xl z2) 5 x[a(xl) a(xz)] for xl,x2 E a. (QNI,) If s E 1 and x E a, then a(sz) 5 ( ( ~ ( 1a(%). (QN13) Let z be any one-to-one map from JV into itself. Then a(Jzz) (= a(x) and a(Q,x) 5 a(x) for all

5

E a.

R e m a r k . Quasi-norins on scalar sequence ideals will be denoted by sinall Roman let,ters.

13.2.2. P r o p o s i t i o n . Let a be a sequence ideal with a quasi-norm a. Then a is a linear topological Hausdorff space. 13.2.3. A quasi-normed sequence ideal [a, a] is a sequence ideal a with a quasi-norm a such that a is complete. 13.2.4. P r o p o s i t i o n . Let [a, a] be a quasi-normed sequence ideal. Then 1]x1]5 a(x) E a.

for all x

Proof. It follows from

that 11x11 = sup n

13*

5 afs).

180

Part 3. Theory of Sequenrr Ideals

13.2.5. L e m m a . Let [a, a1 be a quasi-noinzed sequence idctrl. Then a(Dx) 5 2a(x:) jor all .c f a. Proof. Using the notation of 13.1.3 we get a(&)

=

a(J,c + J p ) 5 ?a(%).

13.2.6. A quasi-norm a on the sequence ideal a is said to lw a $1-norm (0 < p if the p-triangle inequality holds: a(zl

If p

+

.zr)p

= 1, then a is

5 1)

5 a(xl)p + a(.x,)p for xl,xz u .

simply called a norm.

13.2.7. A p-wormed sequence ideal [a, a] is a sequence ideal a with a p-norm a such that a is coniplete. If p = 1, then [a, a] is said to be a n o r m d sequence ideal. 13.2.8. The following statement is analogous to 6.2.5. T h e o r e m . Let [a, a] be n quasi-normed sequence idecil. T ~ Pthere ~ Lexists an equivalent p-norm ap.

13.2.9. Theorem. [c,, 13.2.10. Let 0 < p is denoted by I,. we put

\].I\] i s a noriried sequence ideal.

< co. The

set of all absolutely p-summable sequences z = ( f n )

R e m a r k . It is convenient to define [Iw, l,] := [l. li.\i].

13.2.11. Theorem. El,, lP] i s a nonned sequence ideal / o r p 2 1 and a pa-normed sequence ideal for 0 < p < 1. 13.2.12. Let [a, a] and [b, b] he quasi-normed sequence ideals. Then [a, a] means that a b and a(z) 2 b(s) for all z E a. 13.2.13. P r o p o s i t i o n . Let 0 13.2.14. We put I, :=

< p , 5 p , 5 ce. T h e n

ip,]

r-

[h, h]

[fpz, IPJ.

n I,. P>O

13.2.15. P r o p o s i t i o n . 1, is a seqzieilre ideal.

13.3.

Procedures

13.3.1. A rule

new: [a, a] -., [anew, anew] which defines a iiew quasi-normed sequence ideal [anew, anew] for every quasi-normed sequence ideal [a, a] is called a procedure.

13. Sequence Ideals on the Scalar Field

R e m a r k . The quasi-nortned sequcncc ideal as [a, a]new.

13.3.2. We now list

seine

[anew.

181

anew] will frequently be writtcii

special properties :

(31) If [a, a] & [6,b], then [a, a]new C, [b, b]new (monotony). (I) ([a, a]new)new = [a, a]new for all [a, a ] (idempotence).

A monotone and idempotent procedure is a hull procedure if [a, a] & [a, aInew and a kernel procedure if [a, a ] 2 [a, a]new for all quasi-nornied sequence ideals. respectively.

13.4.

Ninimal Quasi-Xormed Sequence Ideals

13.4.1- Let [a, a] bo a quasi-nornied sequence ideal. A sequence x belongs to the ?iLi?zimd kernel a m if x = tx, with t E co and x, E a. We put amin(x)

..-

inf llltll a(J-ll)I,

where the infinium is taken over all possible factorizations.

13.4.2. The following statement is similar to that of 8.6.2. T h e o r e in. [amin, amin] .is a. quasi-normed sequence ideal.

13.4.3. Moreover, we have the Theorem. The rule

ntin: [a, a] + [amin, amin] .is a h n e l procedure.

13.4.4. For every sequence x = (En)we set

Analogous to 10.3.3 we now prove the Theorem. Let [a, a ] be a quasi-normed sequmce ideal. Then amin consists of those sequences x E a such that

x = a-lim Pnx. n

Proof. First observe that a m h ( z ) = a(x) for all x E f. If x E a m h , we can find a factorization x = txo, where f E co and zo E a. Then a(x - P,lx)d !It - P,tll a(ro). Consequently x = a-lim Pnx. n

182

Part 3. Theory of Sequence ldeals

We now consider a sequence x E a with x = a-limP,x. Then (P,x) is a n n

a-Cauchy sequence. It follows from amin(P,x - P,x) = a(P,z - P,x) and 13.4.2 that (Pnx)converges in a m h . Since x is the only possible limit, we obtain x E ami". R e m a r k . The above statement means that amin is the a-closure of f in a.

13.4.5. A quasi-nornied sequence ideal [a, a] is called minimal if [a, a] = [a,

&Imin.

13.4.6. We now give the main examples. P r o p o s i t i o n . T h e quasi-normed sequence ideals [lp,lp] with 0

[c,, 11.111 are minimal. 13.6.

< p < co and

Maximal Quasi-Normed Sequence Ideals

13.5.1. Let [a, a] be a quasi-normed sequence ideal. A sequence z belongs to the maxim1 hull amax if tx E a for all t E c.,

We put amax(x) := sup (a(tx):t E c, and lltil 5 1 ) .

13.5.2. The proof of the following statement is similar to that of 8.7.2. T h e o r e m . [amax, a""]

i s a quasi-normed sequence ideal.

13.5.3. Moreover, we have the T h e o r e m . T h e rule

max : [a, a] --f [amax, amax] i s a hull procedure.

13.5.4. Analogous to 8.7.5 we now prove the T h e o r e m . Let [a,a] be n p-normed sequence ideal. T h e n sequemes x E I such that (P,x) is a-bounded. I n this m e ,

amax

co?zsist8 of those

amax(x) = sup a(P,x). n

P r o o f . If z E

amax,

then we have a(P,x) 5 amax(.).

Conversely, suppose that a(P,x) 2 a(Pn(tz)- P,(tx)) 2 /jP,t

-

e for n = 1, 2, ... Let t E c,.Then P,tll a(P,x) 5 I'P,t - P,,,til Q for n > m.

Hence (P,(tx)j is an a-Cauchy sequence endtherefore tz ;a. This proves that x E amax. Moreover,

-

a ( t z ) = lim a(P,(tz-)) = sup a(P,(tx)) 5 IItII sup .(Pnz) n

This proves that

amax(z) =

n

sup a(P,.r). n

n

13. Sequence Ideals on the Scalar Field

183

13.5.5. A quasi-nornied sequence ideal [a, a] is called maximal if [a, a] = [a, a1-X.

13.5.6. We now give the main examples. P r o p o s i t i o n . The quasi-normed sequence ideals [I,, l,] with 0 niaximal.

< p 5 co w e

13.6.7. Let a be a quasi-norm on the sequencc ideal a. Then a is said to bc weakly lower semi-continuous if a(.-)

2 lim inf a(x,)

for every directed family of sequences x, E a coordinatewise converging to x E a.

13.5.8. As a counterpart of 8.7.17 we have the P r o p o s i t i o n . Let [a, a] be a maximal p-normed sequence ideul. Then a i s weakly louer semi-continuous.

13.6.

Adjoint Kormed Sequence Ideals

13.6.1. Let [a, a] be a normed sequence ideal. A scquence .- = (6,) belongs to the ndjoint sequence ideal a* if there exists a constant o 2 0 such that

1:2’ I

lnvn 5 as(y) for all 1~ = (vn)E a.

We put

a*(x) := inf o . R e m a r k . The adjoint normed sequence ideal [a*, a*] will frequently be written as [a, a]*.

13.6.2. The following statement is analogous to 9.1.3. Theorem. [a*, a*] i s a normed sequence ideal.

13.6.3. The next result is evident. P r o p o s i t i o n . Let [a, a] and [b, b] be normed sequence ideals. Then [a, a] & [6, h] implies [a, a]* 2 [b, b]*.

13.6.4. Moreover, we have the P r o p o s i t i o n . Let [a, a] be a normed sequence ideal. Then [a, a] E [a, a]**.

13.6.5. The main exaniples are given in the following P r o p o s i t i o n . Let 1 5 p 5 co. Then [l;, If] = [Ipt, I,*].

13.6.6. A nornied sequence ideal [a, a] is called perfect if [a, a]

=

[a, a]**.

13.6.7. P r o p o s i t i o n . The normed sequence idenl [lp, I,] i~ perfect for 1 5 p 4 ob. 13.6.8. A normed sequence ideal [a, a] is called self-adjoint if [a, a] = [a, a]*.

184

Part 3. Theory of S r q i i c n w Ideals

13.6.9. Theorein. [I,, l,] is the mcly self-trdjoint normed sepirnrt’ ideal. Proof. By 1 3 . 6 3 we have [I,, 1,]*

[I,, 12].

: :

Let us now suppose that the norrnod sequence ideal [a, a] is self-ndjoint. Then it follows that

>;

u2

W

IEnl~=

En& F a(%) a(.?-)== n(.c)2

1

Hence x

for

5

a.

1

1, and 12(z)5 a(%).This proves that [a. a ] E [I,, I,]. By 13.6.3 we obtain

[a, a] = [(I, n]* 2 [I,, l,]* Consequently [a, a] = [I,,

-=

[12, 1J.

41.

13.6.10. FinalIy, we formulate

B

result whicli is analogoils to 10.3.8.

Theorem. Let [(I, a] be a minilnnl normed sequence ideal. I f y

a*, the72

00

(x,

Y> :=

s

SnTn

1

defines a functional y on a. The correspondence obtained in t h i y w a y i~a metric Gomorphimu between [a, a]* arLd

ra, 4‘.

13.7.

.+Numbers of Sequences

13.7.1. For every finite sequence x by card (2).

=

(6,) the cardinality of (n:l,, =+= 01 is denoted

13.7.2. A map s which assigns to every sequence x I a unique sequence (s,(z)) is called an s-function if the following conditions are satisfied:

--

(NS,) llxll = sl(x) 2 s2(x)2 * 2 0. (NS,) s,(x y) 5 s , ( T ) $- j/y//for x. y E I. (NSd s , ( ~ )5 sn(xlIIyIi for 5, Y E I. (NS,) If x E f and card (5) < n, then s,(x) = 0. (NS,) Let M be any set of natural numbers. Then card (31)2 n implies s,(eM)= 1.

+

We call sn(x) the n-th s-number of the sequence x. 13.7.3. For every bounded sequence x the n-th approaimcltion tiurtder is defined by a,(%):= inf (/!z- I / / : I E

13.7.4. Obviously we havc the ‘I’ht.oreni. The mop

a:z -+ (a&)) is an s-function.

i

and card ( I )


185

13. Seauence Ideals on t!hc Scalar Field

13.7.5. The next result is analogous to 11.3.4.

T h e o r e m . There i s only one s-function on the set of bounded sequewcs. Proof. Let s bc any s-function and let x E I. l f E E f and cnrd (2) < n, then sil(x)5 sn(Z) j/x - 111 = ijx - 111. Hencc s,~(z) 5 nn(x). Write x == ( t i ) and put -12 := (i:sn(x) E 5 Iti]),where E > 0. Let y := ( 7 1 ~ )with tii := if i M and q e := 0 if i M . Then eLll= q.It follows from

+ +

(;’

that card ( M ) < n. If I := zeM,we obtain card ( I )

< n and

+

Ib - 1115 sn(r) E Consequently o n ( ~5) sn(x).This proves that sn(z)= a&). an(x) 5

13.7.6. We now describe an easy method how to compute the s-numbers of a given sequence. Theorem. Let x = (fn) E co. Then there are a one-to-one map n from .Ninto itself as well ns sequences s = (a,) and t = (z,) such that (rl 2 crg 2 2 0, i ~=~jtZll = ... = 1, and z = J,(st). MoTeover, ~ ~ (=2 on ) for 72 = 1,2,...

---

R e m a r k . According to the preceding characterization the sequence (s,(x)) is called the non-ixcreasing rearrangement of x.

13.7.7. Finally, we mention without proof the following P r o p o s i t i o n . Let z = (En)E I. Then n

2’ g ( x ) = sup 13.8.

Iti/:card ( M )

1.

n

.4f

1

Domination Property

13.8.1. A norined sequence ideal [a, a] has the domiitation property if, given and y E a , then n

B

1

1

1s&) 5 z . s k ( y )

for

?L =

2:

E1

1, 2, ...

implies z E Q and a(s) 2 a(y).

13.8.2. First we prove that a, weaker. property holds for every normed sequence ideal. Lemma. Let x , y E f and n

11

sk(x) 5 2’ sk(y) for 1

Then a(z) 5 a(y).

1

7~

= 1 , 2,

...

186

Part 3. Theory of Seyuenre Ideals

Proof. Write x = (&,) arid y = (17%). Without loss of gmerality wc may suppose that >, t22 .-.2 0 a n d q1 2 qz 2 ... 2 0.

c1

We now choose no such that t, = 0 and ?in = 0 for n > no. Let n range over all permutations with n(n) = n for n > no. Furthermore, let t = ( t t L be) a n arbitrnry sequence such that 5, = i.1 for n, 5 no and tn= 0 for n > no. Then the convex hull C of all sequences tQ,y is compact. Suppose that x 6 C. Using a separation theorem we can find a finite sequence a = (a,) such that m

m

2 antn> 1

and

1

2 anyla5 1

for all (7,)6 C .

1

Choose ~t and t such that s,(n) = z,a,(,). Then m

n

m

s (tn-

1 < 2 a,,tn = 1

tn+d

2 mi

n=l

i=l

m

la

f 2 ( t n - En+,) 2 si(a) n=l

2=1

=

tnsn(a) ==

Z (sn(a) - s n + l ( a ) ) 2'

n=l

1

(%(a) - s,+&))

n=l

-

t i

2=1

n

cu

2 2' -

n

m

m

&YV8

i=l

m

m

X sn(a) T n == 2

7naz(n)p17,

1

1

which is a contradiction. Hence x

5 1,

E C . So there exists a convex combination

m

x ==

&tkQxJ1

Since Al,

..., I.,

m

2 0 and

?Lk

=

1, it follows that

1

m

a(%)5

&a(&,$)

= a(Y).

1

113.8.3. P r o p o s i t i o n . Every minimal normed sequence ideal hns the donLinntion l?roperty. Proof. Let [u. a] be minimal. Suppose that n

2 s&) 1

2

E I, y E a, and

n

5 2 sk(y) for n = 1,2, ... 1

Siiice a is ininimal, we have y E co. Moreover,

implies x E c0. Hence, without loss of generality, we can assume that

El 2 & 2 ... 2 0 and vl 2 vz 2 --.2 0.

13. Sequence Ideals on the Scalar Field

Then, by 13.8.2, we get a(Pmx) a(Pmy)for m = 1, 2, ... Given with a(y - Pm0g)5 E and m m.+l

E

187

> 0, there is ?no

m

tk5 2' qk + E

for m

> mo.

mo+l

Ot.herwise, we could find m,

< ?nl <

such that

This would imply

which becomes false if p tends to infinity. -4pplying (*) and 13.8.2 we obtain

+

5 2s for

a(P,nx- Pm,x)5 a(Pmy- PnL,y

m

2 mo.

Consequently (Fmx)is zlii a-Cauchy sequence. Since x is the only possible limit, it follows that x E a. Moreover, a(%)= lim a(P,,x) 5 lini a(Pmy)= a(y). m

m

13.8.4. P r o p o s i t i o n . Every muximal nornzed sequence ideal htrs the domination PVPWtY. Proof. Let [a, a] be maximal. Suppose that x E 1, y E a, and n

n

1

1

2 sk(s)5

sk(y) for n = 1, 2,

...

By 13.7.7, given no and E > 0, there is m, such that n

?t

2 sk(y) 5 (1 + E ) C sk(Pm,y) for n = 1, ...,no. 1

1

Using n

n

n

2 sk(Pn,X) 4 1

5 2'

1

gk(~)

1

we obtain n

sk(Pn*z) 2 (1 1

+

11

E)

sk(Pm,y) for n = 1, ..., no.

1

n

2 sIl(Pn,x)is constant whenever n > no, the above 1 = 1, 2, ... Consequcntly, it follows from 13.8.2 that.

Since

Finally, by 13.5.4, we get z

inequality holds for all

a and a(%)= sup a(Pnz)5 a(y). n

13.8.5. A normed sequence ideal without domination property has been constructed by G. I. Russu [l], [2].

188

Part 3. Theory of Sequence ldeals

13.9.

Lorentz Sequence Ideals

13.9.1. Let 0 < p < M and 0 < q < 00. Then the Lorentz sequence ideal I(,,, consists of all sequences L $ I such t hat

:{

a3

f ( , , q , ( := z)

Cn the case where 0 < p l(p,&)

<

[ n l / ? ~ - l / ~ ~ ~ ~ ( 2co. ')1q~'*

< 00

and q = c13 the above condition is replaced by

:= sup (nl'Ps,(x):n

=-

1, 2 , ...) < a .

Remark. Obviously [I(,,,,, l(,,,,] = [1,, I,].

13.9.2. Theoreni. Let 0 < p < 00 a ~ 0d < q and nuzximal quasi-rtornied sequence ideal.

< 03.

Then [I(,,,,, I(,,,,]

13.9.3. Theorern. Let 0 < p < M. I'hen [I(,,,, l,p,,,] sequence ideal which fails to be minitml. Remark. A sequcnce z E I belongs to ):;;1

is a minimnl

is a nzaxintal qwzsi-nor?iied

if and only if liin n%,(z)

= 0.

n

13.9.4. The collection of all Lorr.ntz sequence ideals is lexicographically ordered. P r o p o s i t io 11.

= [4P,&,), l ~ P s . P d l = [l(p,qo)' l(p.qp)l

[ 4 p z . q 1 b '(Pz,4,)1

for 0

[l(p,q,,,1(,,,)1

for 0

p1

< Q1

< $12 5 00, < ( I 2 5 00.

13.9.5. Under special assumptions the Lorentz sequence ideals can be normed. Proposition. Let 1 < p

< 00 and 1 2 q < m. Then

In the cme where 1

&$nes an equivalent norm on e q u i m norm .is qiven by

< p < 00

13.10. Sargent Sequence Ideals 13.10.1. Let @

=I

(p,) be a sequence such that

>= p2 2 -.-2 0, C pn = 0 0 , OQ

1=

1

and lim yta= 0. R

and q = 09 an

13. Sequence Ideals on t,he Scalar Field

189

Then the Surgent sequence ideul 3, consists of all sequences x E 1 with

R e m a r k . If @ = ( n - l ' p ) with 1 < p

13.P0.2. T h e o r e m . [&,,s,]

< 00,

then 5, and f, coincide.

i s a trtaximul normed seque?~ceideal which fails to be

,,li,tirnZ.

n

R e m a r k . A sequence r E 1 belongs to

I:

?$" if and only if lini 2' sk(x) 71

I

rpk = 0.

13.10.3. P r o p o s i t i o n . T h e (idjoint normed spquence iden1 [G:, s:] em-ists of all sequences x 5 I such thut m

&5)

y&c)

:=

< 00.

1

13.11. Orlicz Sequence Ideals 13.11.1. Let @ be aiiincreasirigconvex continuous function on 9+ with @(O) = 0 and @(1) = 1. Then the Orlicz sequenee idenl 1, consists of all sequences x = (En) such that m

2' @(ltn//~) 5 1 for some G > 0 . 1

we put

Remark. If @ ( f ) := < p and 1 2 p

13.11.2. Theorem. [I,, l,] 55

i.7

< cu,then 1,

and I, coincide.

a maximal normed sequence idenl.

R e m a r k . If @ satisfics the so-called (&)-condition, then [l,, lo] is minimal, as ell.

13.11.3. Finally, we nieiition that 1; is also an Orlicz sequence ideal.

13.12.

Notes

scguence ideals, that means normal permutation-invariant linear spaces of bounded sequences, \ \ w e investigated by many authors. The most results of this chapter belong t o the folklore of functional analysis, and we are riot able t,o give exact references. The int,erest,ed reader should consult D. J. H. GARLING[l:].The concept of a minimal and maximal normed sequence i!lcitl can be found in B. S. XITJAWX [2]. The domination property --as introduced by P. A. V-ILDERON [l]; see also B. S. MITJAGIN [3]. Further information about Lorentz sequence i(1eals are given in [BUT, pp. 181-1871 and [TR.I, pp. 131-1351. For the theory of Sargent sequence ideals we refer t o [GOH, pp. 139-1501. and Orlicz sequence ideals are treated i n LLIN, p. 561. The standard reference for Orlicz function spaces is the monograph [KRA]. ~~ecominendations for further reading-:

J. I. GRIBANOVI l l , G. KOTHE/O.TOEPLITZ 111, G . I. Itc-ssu [ll, 121, W. L. C. SARGENT [l].

14.

Operator Ideals and Sequence Ideals

I n this chapter we deal with operator ideals generatt-d by a n additive s-function and a scalar sequence ideal. The class Gf) consists of all operators S such that (Sn(S))E a. For special s-functions the above definition goes back to I. A. NOVOSELSILIJ and the author (1963). The main interest is devoted t o those operator ideals (5); which correspond to the classical ideals I,. We state their basic properties and investigate the relationships between operator ideals GF) defined by approximation numbers, Gelfand numbers etc. Moreover, the same method is used for generating operator ideals by the entropy numbers. Conversely, for every operator ideal 8 and fixed exponents u and v we may consider the ideal of all scalar sequences such that the corresponding diagonal operator 5). This leads to the interesting concept of a limit order. belongs to We also define so-called small operator ideals and give the most important examples.

14.1.

GP’-Operators

14.1.1. Let s be an additive s-function and let Q be a sequence ideal on the scalar field. An operator S E B(E,F ) is called an G,bs)-operatorif (an(&)) E a. The class of these operators is denoted by St’.

14.1.2. Theorem. Gf) i s an operator ideal.

+

Proof. Let S,, S, E G f ) ( E ,F ) and put S := S , S,. Then it follows from 13.1.3 that, (s,(SL), s,(Si),s,(S,),s2(Si),...)E a for i = 1,2. Cy S , ~ - ~ (5S sn(SI) ) sn(S,) we hare (‘I(’)?

sl(s)> ‘3(’),

s3(S)7

’**)

+

E a.

Hence (sdf% s,(S),s3(S),%(S), * * .)

This proves that XI evident.

E a*

+ S, E GF). So (01,)is satisfied. The remaining properties are

14.1.3. P r o p o s i t i o n . For every (compbtely) symmetiic additice s-function €he operator ideal Gf) i s (completely) symmetric, as well. 12.1.4. P r o p o s i t i o n . Por every regular additive s-ftinction the operator ideal G!’ regular, as well.

14.1.5. P r o p o s i t i o n I . For every injective additive s-fwnction the operator ideal 6;’ i s injective, as well.

14. Operator Ideals and Sequence Ideals

191

P r o p o s i t i o n 2. For every surjective additive s-function the operator ideal G t ) is surjective, as well.

14.1.6. Let [a, a ] be a quasi-nornied sequence ideal on thc scalar field. Then we put

Sf’(S) := a(s,(S)) for S 6 Gf). 14.1.7. Using the method of 14.1.2 it can easily be seen that S t ) is a quasi-norm. However, we do not know whether G:’ is also complete. P r o b l e m . Is [G:), SF’] a quasi-normed operator ideal?

14.1.8. We now give an affirmative answer to the preceding question in a rather general case. T h e o r e m . Let [a, a] be a maximal sequence ideal on the scalar field. Then [Gr’, SF)] i s a qwi-normed operaior ideal. Proof. By 13.2.8 we may suppose that a is a p-norm. Therefore the criterion 13.5.4 can be used. Let (8,)be a n SF’-Cauchy sequence of operators 8, E G:’(E, F). Then there exists 8 E e ( E ,B’) with 8 = ~ ~ . ~ ~8,. - l iIt m follows from k

\$‘e now obtain ( ~ ~ -( Sk)) 8 E n and a(s,(S - 8,)) 5 & for k 2 ka. Therefore S 6 Gf’(E,F ) and X = S&s)-li~:i S,. So the completeness of [Gf’, St’] is proved. k

14.1.9. The next result is an improvement of the preceding theorem. P r o p o s i t i o n . Let s be a maximal additive s-function and let [a, a] be a maximal quasi-normed sequence deal on the scalar field. Then the quasi-normed operator ideal [Gf), St’] i s maximal, as well. Proof. Suppose that S E 2(E,P)belongs to the maximal hull of Gf).Let (tn)E c0 with Itn/5 1. Theii, given E > 0, by 11.10.3 we can find S E 6(Eo,E ) and B E % ( F ,Fa) such that ( ( X /5 ( 1, ((B(( 2 1, and

192

Part 3. Theory of Sequence Ideals

Now it follows from (s,(RXX)) < a that (rnsn(S))E a even for all ( ~ ~ (E8a )and ) thcrefore S’ E G f ) ( E ,F).Moreover, we have

(T,)

E co. Iiencr

S 3 S ) = a(sn(S))= sup { a( ( w, ( S) ) ) :(z,) E co, I~,I 5 11

5 (I 5 (I where

+ E) sup {a(s,,(BSX)): //XI15 I, /IS// 2 1) $- s) sup

{S&S’(BSX): IIXIl I 1, 1 1 ~ 1 15 11,

X @(€to, E ) and R E @(F,Fo).This completes the proof.

14.1.10. P r o p o s i t i o n . Let s be a n ultrastable ndditive s-function and let [a, a] be n ,tiaximal quasi-normed sequence ideal on the scalar field. Furthermore suppose that a i s weakly lower semi-continuous. Then the quasi-normed operator ideal [Gt’, Sf’] i s ultrastable, as well. Proof. We consider a n SF)-bounded family of operators 8, E GF’(Ej)F b ) . Put where & := lini sk(S,).Then

.c = (&))

U

Using 13.2.8 and 13.6.4 we obtain z (8,)n E G ~ ) ( ( E ~(F,)tl). ) ~ , Moreover,

St)((S,),)

=I

c a. Now

it follows froin S ~ ( ( S ,5 ) ~&) that

a(SL((S,)U)) 2 liin a(sk(S,))= lim S!$(S,).

u

11

14.1.11. Finally, we deal with thc operator ideal @(’) generated by ail additive s-function s and the scalar sequence ideal c0.

14.1.12. Theorem. TIP qerator zdeul

is closed.

Proof. Suppose that 9 t Z(E,3’)belongs to the closure of W). Then there exists E0 f @(*)(E, F ) with I/S - Sol/5 E . We now choose a natural number no such that s,(S,) E for n 2 no. Clearly, s,(S) 5 sn(So) l/S - Sr,Il 5 2s for n 2 no. This proves that S E @(”(E,F ) .

+

14.1.13. P r o p o s i t i o n .

==

6. WC) =R, and W d )= R.

Proof. The first equation is evident. The remaining ones follow froin @ma= R and @sup = 8 as well as from the fact that @ ( c ) and @(d) is the injective hull and the surjective hull of @fa), respectively. R e m a r k . A direct proof of the last equation can be found in [PIE, p. 1461.

14.2.

%,-Operators and @,-Operators

14.2.1. Let s be a n addit,ive s-function and 0 < p is called a n Gt)-operator if (s,(S))E I,. We put

The class of these operators is denoted by G!).

< m.

An operator S < Z ( E , PI’)

14. Operator Ideals and Sequence Ideals

14.2.2. Theorem. [GF), SF)] i s

LC

193

quasi-normed operutor idenl.

Proof. The assertion follows immediately from 14.1.8. However, we are interested in estimating the constant in the quasi-triangle inequality. For this purpose let S,, S2 E G:){E, F ) . Put x := Zl’P niax (S’F1,1). Then

s X[St’(S,) + s;’(s*)]. Remark. We see from 6.2.5 and the preceding estimate that [GF), St’] is q-normed for l / q := 111, + 1 and 1 5 p < 00. This result, cannot be improved; cf. 18.5.1 and 18.6.1 (remark).

14.2.3. Theorem. Suppose that s is n multiplicative and additive s-function. If < p , q < 00 and 11‘ = l / p l / q , then GE’ o Gf’ E GP).

+

0

Proof. For S

c GF)(F,G ) and T E G:’(E, F ) we have

This proves that ST E GF’(E,G). 12.2.4. Let us agree, for simplicity, that [Up, A,] denotes the quasi-normed ideal of GF’-operators generated by the approximation numbers. Operators belonging to this ideal are frequently called Up-operators.

14.2.5. Theorem. The quasi-normed operator ideal [Up,A p ] is completely symmetric.

Proof. The syinrnetry follows from a,(S‘) 5 u,(S); cf. 11.7.3. We now suppose that S E aF“‘(E,F).Then 8‘ E B(F’,E’) and therefore S E a(E,P).So: by 11.7.4, we get a,(&’) = w,(S). This proves that By]= [Up,Ap1. 14.2.6. As a consequence of 11.10.9 and 14.1.10 we have the Proposition. The quasi-nornted operator ideal [a,, Ap] i~ ultrcistable.

12.2.7. Theorem. The quasi-normed operator ideul

[Up. A,] is muxirnul.

Proof. We know from 14.2.5 that [a,, Ap] is regular. So, the maxiniality is implied by 8.8.7.

14.2.8. Lemnia 1. For every S E Up@, E’) and that Ap(S - L ) r F . 1s Yietsch, Operalor

E

>0

there existx L E

a@,F ) such

Part 3. Theorv of Seguelice Ideals

194

Proof. Choose m with

Then 2m

mazm(S)P 5 2’ a k [ S ’ ) p

<~p/4.

m+l

Consequently, we can find an operator L E g ( E , F ) such that rank ( L )< 2m and m jjS - L / / p < @/4. Then ak+2m-l(#

- L ) 5 ak(S)

Hence

+

3n

A,(B - L)P =

ak(S - L)P

k= 1

= ak(S)* 03

+

k=rn+2

u~+,,(S - L)’

5 3m 1;s -Lp +z ’U k ( S ) f ’ 5 .&, CQ

m+2

Lemnia 2. If S E g ( E , F ) , theiz _4?(5) = A p ( S ) . P r o o f . Givcii E > 0: we choose an operator L, g ( E , F ) with rank (L,) < n and jlX - L,J 5 (1 + E ) n,,(iS)for n = 1, ..., r , where r := rank (8). Put

N :=: N ( 5 ) n S ( L , ) n and

+-

M := 111(5) N ( L , )

.- n W(L,)

+ - - - + M(L,).

Then N E Cod ( E ) and iM E Dim ( B ) . Obviously there exist unique operators Xo, Llo, .,L,, E g ( E / N ,M ) such that the following diagram commutes:

..

Since the operator norin is injective and surjective, we have

C o n j e c t u r e . The quasi-nonned operat,or ideal

[up,A,]

is minimal.

1%

14. Operator Ideals and Sequence Ideals

R e m a r k . Since S E U,(E, P) can be represented in the form S = AJim 8, n

with S, E g ( E , P), it follows from A?'"(S, - 8,) = A,@, - S,) and the com= SC,. But we do pleteness of a,"i"(E,H) that S E n,"l"(E,P). Consequently not know whether the quasi-norms and A, coincide on the whole ideal. This lack is %I consequence of the quasi-norm catastrophe; cf. 6.1.9.

%rin

14.2.9. The next result is implied by the definition of Gelfand numbers and Kolmogorov numbers, respectively. Theorem.

[a?, Ainj] P = [Gt), SF)]

and

[Uy,A y ] = [Gkd),Sr)].

14.2.10. By 11.7.7 we have the Theorem. The quasi-norm& operatw ideals [@$, AFj] and to each other.

[WF, A r ] are dual

14.2.11. P r o p o s i t i o n . Suppose that P is a n internwdiaie space of {Po,F,] posse-ssing J-type 8 . Let 0 < po,p , < CQ and l i p := (1 - 8)/po+ e / p l . I f S E E(E, HA),then S 6 %Ej(E, Po) and S E UkJ(E,F,) imply S E U:j(E, 8). Moreover,

AFj(S: E + F ) 5 2llpA;j(S: E -+Fo)l-o A z j ( 8 : E +-F,)*. Proof. By 11.5.S we have

14.2.12. Using tho same method from 11.6.8 we get the P r o p o s i t i o n . Suppose tlzat E i s an intermediate space of {Eo,El} possesSing K-type 0. Let 0 < po,p 1 < cy) and l / p := (1 - 8)/po 8/pl. I f S E B(E,, F),then s' E %Er(Eo, F ) and S E %:,'r(El, F ) imnply S E U F ( E ,F ) . Aloreover,

+

AF(S:E

3

F) 5 2llpA3S: E ,

3 F)l-O

Ar(S:El+ F ) e .

14.2.13. TOprove the following theorem we need an eleiiieritary 01

Leiiinia. Let 0 < p < p o < w and l / q = I l p - lipo. If 2 0, then (nl/poon)E 1., 2 o2 2 Proof. Using

13*

(ol$)E

1, such that

196

Part 3. Theory of Sequence Ideals

we get

As a consequence of 11.12.2we now have the T h e o r e m . Let 0 < p

< 2 and

l / q = 1/p - 1/2. !"en

5 a,, and

S C P 5 2Iq.

R e m a r k . It will follow from 14.4.9 and 14.4.10 that, given p , the exponent q cannot be improved.

14.2.14. Lemma. Let 0 < po < w. If (en)E I,, ( n l k o ,E) G.

and el 1 c2 2

--.2 0,

then

Proof. Choose m such that

Then, for n 2 2m, we get n

nc?

5 2(n - m ) crp 2 2 z afG"5 B e p ~ . m+l

This proves that lim n%sn = 0. I

As a supplement of the preceding result we now have the Theorem.

ulp G 0 and U y 5 0.

R e m a r k . Enflo's counterexample yields OinJ$ 8 and GSur$ 0 . Using the same construction it is possible to show that even $ (5 and $ 0 for 2 < p < m; cf. K. D. E ~ S T E [l]. N

up

14.2.15. Let

11s

agree, for simplicity, that

[Bp,Hp] denotes the quasi-normed

ideal

of Gr)-operators generated by the Hilbert numbers. Operators belonging to this

ideal are frequently called &wperatms.

14.2.16. As a consequence of 11.7.8and 14.1.3 we have the T h e o r e m . The quasi-normed operator ideul

[ap,€Ip] is completely symmetric.

14.2.17. The next result follows from 11.10.13and 14.1.9. Theorem. The quasi-ncurmed operator ideal

[a,, €IP] is maximd.

14.2.18. A proof of the following result is given in [GOH, p. 371 and [HAR, p. 89.1 Lemma. Let & 2 E2 2 ?a

n

1

1

--.and q1 2 yl2 >= -.such that

x l k s z ? ?f ko r n = l . 2 , ... If @ is a conwx function on the real line with lim @({) = 0, then .Lt-

A

n

X@&) 5 .Z @h) for 1 1

n = I,%

...

m

11. Operator Ideals and Sequence Ideals

197

We are now prepared t,o check the Theorem. Let 0 < p

1

Lot

@(E)

l / q = l/p - 1. Then

&:, E U$ and Sj, 5 UF.

&,(It, P ) . We know from 11.12.3 that

Proof. Let S n

< 1 and 98

log [k-lck(S)] 5 2 log [ e 7 ~ ~ ( 8 ) ]for n = 1, 2, ... 1

:= exp

( p t ) .Then it follows from the preceding lemma that

ra

n

1

1

3 [ k l c k ( S ) ] p5 e p 2 hk(S)p

for n

= 1,

2,

...

Using 14.2.13 (lemma) we now get (ck(S)) I,. By 14.2.9this means that S E U t j ( E ,3‘). The second inclusion can be checked in the same way. Hence Sj,

%rj.

14.3.19. C o n j e c t u r e . Let, 0 < p

< 1 and l/q = l/p - 1. Then Sj, & U,.

14.2.20. Finally, we mention that the more general quasi-norined operator ideals generated by the quasi-nornied Lorentz sequence ideals [l(p,9)7 l,,,,] play an important role in the theory of Sobolev’s embedding maps. We mention in particular the case where q = 0 0 ; cf. [TRI].

[6&,, S&]

14.3.

(E,-Operators

14.3.1. An operator S

B(E,F ) is called an @:,-operatorif

(en(&))E 1 , with 0 < p

< m.

We put

Here the norming constant

E,

is choosen such t,hat Ep(Zcf) = 1.

The class of these operators is denoted by @ ., 14.3.2. Analogously to 14.2.2 we have the Theorem. @ [,

E,] is a quasi-nmed operator ideal.

R e m a r k . In the above definition we can also use the inner entropy numbers. Then we get the same operator ideal, and

yields a n equivalent quasi-norm which is sometimes more convenient to work with ; cf. 14.3.4 and 14.3.5. Here the norming constant rp:, is choosen such that Fp(Zx) = 1. 14.3.3. The next result follows froiu 12.1.6. T h e o r e m . If 0

< p , q < cx) and l J r = l/p + l/q,

then (E, o (E,

14.3.4. We now state an immediate consequence of 12.1.9. Theorem. The operator ideal @,

is

wmximd.

(Er.

198

Part 3. Theory of Sequence Ideals

14.3.6. Applying 12.1.8 we get the Theorem. The operator ideal Cr?, is injective and surjectiw.

14.3.6. Until now no relationship between e,(S) and e,(S) semis to be known. P r o b l e m . Is the operator ideal Gp completely symmetric?

14.3.7. For completeness we mention the following interpolation property which follows from 12.1.11. P r o p o s i t i o n . Suppose that E is an. intermediate space of {Eo,E l ] possessing K-type 8. Let 0 < p Q , p l< 00 and l/p := (1 - 8)/pQ O/pl. If S E E(E,, P ) , then S E (.fpo(Eo, P)and S E @p,(E1,P)imply S E GP(E,F ) .

+

12.3.8. Using 12.1.12 we analogously get P r o p o s i t i o n . Suppose that F i s an intermedinte space of {PQ,PI) pamessing J-type 8. Let 0 .< po, p1 < 00 and l/p := (1 - 8)/po 8/pl. If S E E(E,F A ) , then S E (.fP,(E, Fo)and S E (.fp,(E,F1)imply S E e P ( E ,F ) .

+

14.3.9. Obviously 12.3.1 yields the T h e o r e m . Cip 5 $jpfor 0

< p < 00.

14.3.10. The following inclusions can be derived from 14.8.15 and 14.3.9. See also 12.3.2. T h e o r e m . Let 0 < p

< 1 and

l/q = l/p - 1. Then

14.3.11. Finally, we state a converse inclusion. Theorem. Let 0

< p < p < m. Then aP& eq.

Proof. Suppose that S E Mp(E,F). Then there exists a positive constant c with

arn(S)d For every n

C

mllP

for

ni. =

1, 2, ...

> 1 we determine m,by

n-1

plog?l.

<msp-

n-1

log n

+ 1.

It now follows from 12.3.3that

Hnece (.tg(S))E Iq and therefore S E G,,(E, P).

14. Operator Ideals and Sequence Ideals

Remark. Since Gqis injective and surjectivc, we also have UF

14.3.12. Conjecture. Up 5 Gpfor 0 < p 14.4.

199

B, and U r 5 Bq.

< 00.

Limit Order

14.4.1. For every operator ideal 9l the limit order A(%, u, v) is defined to be the and 1 2 u,v jbo. infimum of all ii2 0 such that Dl E U(Z,, l"), where O&) :=(r2tn) 14.4.2. The importance of the above concept comes from the following property. Proposition. Let % be an operator ideal. Then A(%, u,v) is the infimum of all l / r 2 0 such that every diagonal operator of the form AS&) = (on&,)with (on)E I, helongs to Q). Proof. Suppose that r has the above property. If 3. therefore D1 E %(la, 5). Hence l/r 2 A(%, u, a). G~

Conversely, let (on) I, with l/r > 2 o, 2 2 0. Then it follows from

---

A(a,u, v).

> llr,

then ( r 2 E 1,) and

Clearly we may assume that

that on 5 an-llr. So D,,, E ?ll(l,,1,) implies S E N(lu, l,,).

14.4.3. The next characterization is very useful in computing special limit orders. Theorem. Let [a,A] be a qwcsi-normed operator ideal. Then infimum of all ii 1 0 such that

A ( I : 1: --f 1:) en1 for n = 1,d, where e 2 0 is some constant.

?,(a, u, v) is

...,

the (* 1

Proof. According to 6.2.5 it is enough to check tho assertion for every p-normed operztor ideal. First let us suppose that (*) holds. Put

Nk:= ( n : 2k 2 n c 2k+l}, and let Qk be the diagonal operator generated by the characteristic sequence of X k . Then A ( Q k : 1, -+ 1,) 5 e [card (Nk)ll= ~ 2 ~ ~ . For E

> O we get,

2' A(2-(14-&)kQk)p 5 Q P x 2-8pk < m

03

0

0

00.

Consequently 03

:=

2-(i+dkQk 0

Telongs to %(la, 5). Since n-(l+&) 5 W J + E for ) k n E N,, we have DJ+* E 1) This means that i. E 2 I.(%, u , v).

+

5).

200

Part 3 . Theory of Sequence Ideals

Conversely, let A. > I.(%, u, v) and put r := l/l.. Then the inap 13 assigning to every sequence (an)E I, the diagonal operator S E 91(Z8, 1,) defined by S(&J := (a&) is closed. So there exists a constant e 2 0 such that for all (a,,) E 1,.

A(&: I , -> 5) 5 &(a,) In particular, if

...,En, t&, ...)

P&,

:= (51,

... l a , 0, ...) )

>

then we have

A(]: 1; -+ Z;!)

=

d(P,:I,

--f

1,)

5 en1'' for

n = 1 , 2. . ..

This proves (*).

14.4.4. We now consider a first example. Lein ma.

Hence we have the Proposition.

14.4.5. In what follows the behaviour of A(%, u, v) is graphically represented by inearis of diagrams in the unit square. We shall use the coordinates l/u (horizontal) and l / v (vertical). I n the left-hand diagram we plot the level curves, while the algebraic expressions of I(%, u;v) are indicated on the right.

For the ideal of all operators we have the diagram:

14.4.6. The next result is trivial. Proposition. Let

A(%

0

and ?8be operator ideals. Then

d,u , 10) 5 I,(%,

21:

v)

+ I.(%,

c, 2 4 .

Remark. In particular, we have

q2€,21, v ) I q2,u , uo) + I.(%,

uo, %lo)

+ 42,

2'0, 2')

This estimate is the basic tool for all computations which will follow later on.

14. Operator Ideals and Sequence Ideals

201

14.4.7. Proposition. Let 2I be an operator ideal. Then I.(%mi", u, v) = I.(%-,

u, v) = ).(a,u, v).

Moreover,

A(%-',

u,w) = A(%, w', u').

14.4.8. Let us mention that the limit order is a continuous function of 1/u and 1j.l.. More precisely, we have the

Lemma. [A(%, u, v) - A(%, uo, vo)l S jl/u - l/uol

+ Il/v - 1/d.

Proof. The inequality follows immediately from 14.4.4 and 14.4.6 (remark). 14.4.9. Proposition.

for l d u 1 2 ~ v I c o .

5 IL 5 cc,then by 11.11.5 we have

Proof. If 1 5 ak(I:

):Z

3

= (12

-- k -+ 1)l'r for k

= 1,

...,

72:

where l/r := l/v - l/u. Consequently

Here x means that both sidea are of the same order; cf. index of symbols. Therefor

A(%,,, u, v) = 111, - liu We know from 11.11.8 that

+ l/v

for 1 5 v

u S

00.

202

apm

Part 3. Theory of Sequence Ideals

The above result is represented in the following diagram :

1 -

2

P

1

Remark. It has been recently proved by B. CmL/A. PIETSCH [2] that in t,he case where 0 < p < 1 and 1 5 u 2 2 2 v 5 00 we have

A(%p, u, v) = max (1/p - l/u

+ 1/2, 1lp - 1/2 + l/v) .

The same result should be true even for 0 < p

A(%p, u, v) = max (l/p - 1/u whenever 0 < p

<

< 2.

+ 1/2, l/p

--

Moreover, we conjecture that

1/2

+ l/w, 0 )

and 1 2 u

5 2 5 w 5 00. 14.4.10. Proposition. Let 0 < p < 2. Then 00

l/p - 1/24

+ l/v

for

1 2 v 5 u 200,

forz~Uu.vv0O,

A(*$,

u,e) =

+ 1/71 for 1 5 u 5 v 5 2 , 1 l / p - l/u + 1/2 for 1 2 u 5 2 Iv 5 l/p - flu

Proof. The formulas l.(%rj, u,w) = l / p - 1/u

+ 1/v

for 1 2 v 2 u 5 00

and

A(*$,

u, w) = l/p

can be checked as in the preceding proof.

By 11.11.13 we have

for25uSv$oo

0O.

14. Operator Ideals and Sequence Ideals

203

Therefore

Finally,

A(W$, u. v) 5 A(W2, u,3) and

I.(W$,

2,.)

+ I@,

2, v)

2 i(2, 2, u)+ Qly,u,v)

imply

A(W2, 24, v) = 1/p - l i z c

+ 112

for 1 5 u 5 2 5 21 5

This completes the proof.

SOwe have the diagrani :

14.4.11. Obviously 142.10 and 14.4.7 yield the Proposition.

A(%F,u, v) = I(U'"j,d,u ' ) . 14.4.12. P r o p o s i t i o n . Let 0

A((&,

< p < i. Then

u,v) = l / p - 1/24

+ l/v.

Proof. By 12.2.1 we have I 2e

ek(I:lk+ly) 2 - n

for 1;-= 1, ..., n.

Therefore

EJI : Z k -+ ):Z

1 2eep

2 -nl'P-1.

This means that A(@p, 00, 1) 2 l/p A ( @ p , O0,

l ) 5 A(f?, co,u , -k A(ep>v,

itnplies l/p

-

+ 1. Now

l/u

+ '(g,

>'

+ l / v 5 Acep,u, v).

On the other hand, we have seen in the proof of 12.2.2 that e k ( l : +-Zk) 5 8

log (n

k

-+ 1)

for k = 1,

...,n .

=.

(4)

u)4

Part 3. Theory of Sequence Ideals

Xoreover, en+*(I:1% +-1:) = 112. Using 12.1.5 we get

s cnl’P-1 log (n + 1) , where c is some positive constant. This yields A(@,, 1, 00) 2 1/p - 1. from 1, =) - A(2,m, v) A(@,, u, v) 5 A(2,u,1) that A&, u, v) 2 1/p - 1/u l/w.

+ +

it f ~ h w s

(2)

The above result is represented in the following diagram :

14.4.13. Analogously we get the

Proposition. Let 1 5 p

< m. Then

u,v) = max (1/p - l/u

+ l/v, 0 ) .

This yields the diagram :

14.4.14. The limit order of the operator ideal 4, with 0 < p < 1 has been computed by B. CARL/Q.BETSCH [2] recently.

14.5.

Small Operator Ideals

14.5.1. An operator ideal is finite dimensional.

is called mudl if %(E,F ) = B(E, F ) implies that 1or F

14.5.2. Proposition. Let s be an injective,surjective, and additive s-fuwcthz. T?wk Gg) .is ,mall.

14. Operator Ideals and Sequence Ideals

205

Proof. If G,a")(E, B') = B(E, F) then there exists a constant e 2 0 such that S:)(S) 5 e llSll for all S E O(E, F).Assume that E and P are infinite dimensional. By E.2.2, for every natural number n, we can find A, E O(E,lg) and Y , E O(%, F) with 1 5 q(A,) 5 /[An[[ 5 2 and 1 5 j ( Y , ) 5 l[Ynll 5 2. Applying B.3.3 and B.3.7 we have the obvious diagram

where 1IA;Jl

5 1 and 1lY;~Il (= 1. Put AS, := Y,I,A,. Then

sk("9R)

= S d J n YnolnAnoQn) = 8 d Y n o l n A o )

I IIYTJI 8dYnJnAnd IIAGVI I din) == 1 for k = 1,

...,n. This yields nl'v 5 S!)(&)

2 e [IS,,ll2 4p , which is a contradiction.

14.5.3. The following main result is an immediate consequence of 11.2.3, 11.5.5. and 11.6.5 as well as of 11.7.9. Theorem. The operator ideals Up,4[2, and

%r are small.

14.5.4. Finally, we state a problem which goes back to A. GROTHENDIECK.

Conjecture. The operator ideal !It is small. Remark. Some partial results are due to W. J. DAVIS/W. 3. JOHNSON [I] and I. I. TSEITLIN El].

14.6.

Notes

J. W. CALKM observed that operators on the separable Hilbert space belonging to a given ideal can be characterized by the asymptotic behaviour of their a-numbers. Since the concept of 8-numbers has been extended t o operators between Btlnach spaces, the same construction yields operator ideals on the class of all Banach spaces. Starting from the Kolmogorov numbers and the approximation numbers I. A. NOVOSELSEIJ [I] and A. PIETSCH [2] have defined special kinds of Gf)-operators. See also W. B s n H s s D T [l]and B. CARL/A.PIETSCH [I]. Interpolation properties of Bas)-operators are extensively treated in [TRI, pp, 107-1171. The [14]. Furthermore, A. PIITSC'H important notion of a limit order first appeared in A. PIETSCH [19] proved that all operator ideals Up with 0 < p < 00 are small. Recommendations for further reading:

CH. CONSTANTIN [ 2 ] , W. J. DAVIY/W. B. JOHNSON [l], R. A. GOLDSTEIN/R. SAEKS[1], C. V. HUTTON [Z],H. KOXIQ[l], A. PIETSCH [18], [20], B. ROSENBERGER [3], I. I. TSEITLIN [I].

15.

Operator Ideals on Hilbert Spaces

The theory of ideals in t-he operator algebra of the separable infinite dimensional Hilbert space was created by J. W. CALEIN in 1941. Excellent preseatations are G. KREJNand R. SCHATTEN. This given in the monographs of I. C. GOHBERG~M. chapter deals with operator ideals on the class of all Hilhert spaces. First we show that each proper operdtor ideal is uniquely determined by its component on 1,. Then a one-to-one correspondence between scalar sequence ideals and proper operator ideals is established. Unfortunately, we do not know whether this relationship also liolds in the quasi-nornied case. So we must restrict ourselvcs to nornied ideals haviiig the domination property. The main part of this chapter is devoted to so-called Qp-operatorsfirst defined and R. SCHATTEN about 1946. In the case where p = 1 these by J. VON NEUMANN are the operators of tracc class (nuclear operators), and p = 2 yields the classical ideal of Hilbert-Schmidt operators. Finally, we extend operator ideals from the class of Hilbert spaces to tho class of all Banach spaces. Xost of them have an infinity of different extensions. On the other hand, we will see later on that the ideal of strictly nuclear operators admits a unique extension. All concepts in the theory of operator ideals on Hilbert spaces are analogously defined as in the case of Banach spaces.

15.1.

Operator Ideals

15.1.1. Proposition. Every (quasi-normed) operator ideal on Hilbert spaces 6 conzpletely symmetric. Proof. The asscrtion follows from D.3.1 and the fact that So = 8; for every diagonal operator Soacting in L2(f2,p).

15.1.2. The next result states that every (quasi-nornied) operator ideal on Hilbert spaces is injective; cf. 4.6.9. Proposition. Let U be an operutor ideal. For every injection J E 2(Ko,K ) and every operator So E 2(B,K O )it follows from JSo E U(H, K ) that Xo E % ( H ,K O ) . Proof. Put Qy:= J-lPy, where P is the orthogonal projection from K onto H ( J ) .Then QJ = I,*. Xence So = Q(JSo)E % ( H , K O ) .

15.1.3. The surjectivity can be checked in the same way. Proposition. Let 3 be an operator ideal. For every surjection Q E i?(II,H o ) and every operator 8, E 2(Ho,K ) from SoQE %(H, R)it follows that So E %(I& K ) .

15.1.4. In the theory of operator ideals on Hilbert spaces there are further simplifications due to the fact that the metric approximation property holds; cf. 10.3.

15. Oyeutor ldeals 011 Hilbcrt Spaces

15.2.

205

Proper Operator Ideals

13.2.1. We start this section with a useful characterization. Theorem. An operator S E B(H, K) isapproximableijalzd onlyif ( ( S X e , , Ye,,))c co for all operators X E B(12, H ) and Y E B(l2,K ) . Proof. Suppose that the above condition is satisfied. Given E > 0, we choose maximal orthonormal families ( x i ) and ( y i )with l(Szi,yi)l 2 E for all i E 7. Observe that I is finite. Otherwise we take il,i2,.. . I . Let m

I ( &:= ) 2 taxi,, and

m

Y(7,‘):= 2 vnyi,.

1

1

Then it follows that lim (Sxi,, yi,) = lim (SXe,, Ye,) = 0 , n

n

which is false. Put

A := 2 xt @ xi and B :=2’ y: @ yi, I

I

l i ( I K - B ) S ( I H- A ) ] ]> E . Then thereare x E H and y f K with [((IK- E ) S(IH- A ) x , y)I > E 11x11 11y11. Put xo := x - A z and yo := y - By. Obviously, we may suppose that ll.rOl]= 1 and llyoll = 1. It follows from 111, - All = 1 and llIK - Bl/ = 1 that (/zjj2 1 and Jjy/l2 1. Hence /(Sxo,yo)) > E. So we e m

Let us assume that

enlarge ( x i ) and (yi)b y adding zo and yo, respectively. This contradiction implies

11s - S A - BS + BSAll = /1(19- B ) S(IH - A)ll 5 E . Hence S E @ ( H ,K ) . Conversely, let S E 6 ( H , K ) , X E 2(Z2,H ) , and Y f f?(Z2, K ) .Then Y*SX E a(&, &) yields Y*SX E B(Z2, Z2). Since (en)is a weak zero sequence, we get lim jl Y*SXe,,jl = 0. n

Using I(SXe,, Ye,)l 5 IIY*SXe,,II it followsthat lim (SXe,, Y e , ) = 0. n

15.2.2. The next r e s d t is of great importance. Theorem. Every proper operator ideal on Hilbert spnce.s colzsists of approxirmble operaiors only. Proof 1. Let 2l be a proper operator ideal. Then it follows from %(Z2, 7,) =/= f?(Z,, Z2) and 5.2.2 that 9€(Z2, 1,) G 6(12, Z2). Consequently, for S E %(H, K ) , X E e(4,H ) , and y E B(12, K ) we have Y*SX @(&, Z2). Finally, 15.2.1 implies liin (Y*SXc,, en) = 0 and therefore S E @(H,K ) . So % ($5. n P r o o f 3. Let U be a proper operator ideal. Then SOW 5.4.6 implies 8c - ~ U C l O S= $p= gc10s = @ *

2lclm

is also proper by 4.2.8.

15.2.3. P r o p o s i t i o n . Let U be cc. proper operator ideal on Hilbert spaces. Then S E 2 ( H , K ) belongs to 2l if and o?tl!j if Y*SX E a(&,I,) for nll operators X 2(1,,H) ancl Y E f?(12, K ) .

208

Part 4. Theory of Sequence Ideals

Moreover, for every quasi-norrn A defined on 91 we have A(8) = sup (b(Y*SX):!/XI/5 1 and liYI] 5 1).

(*I

Proof. Suppose that S E E(H, K ) satisfies the above condition. Then S is approximable by 15.2.1 and 15.2.2. Consider the Schmidt factorization of S described in D.3.3. Then So = Y*SX E 'u(Z2,I,) implies S = YSoX* E n ( H , K ) . This shows the sufficiency of the given property. Its necessity is trivial. The forniula (*) is an inimediate consequence of D.3.3 and (QOI,).

16.2.4. By the same considerations as in the preceding proof we are also led tQthe Proposition. Let% be aproper operator ideal on Hilbert 8paLe8. Then 8 EE(H, K ) belongs to U if and only if 8 = YS,X* with So E a(&, Z2), X E f!(Z, H ) , and Y E B(h, K). illoreover, for every quasi-norm A we have

A(S) = inf IlYll AWO) IlXllY where the infimum is taken mer all possible fnctorizatims.

16.2.5. As a consequence of 15.2.3 or 15.2.4 we state the fundamental Theorem. Every proper (qua-G-nornzed)operator ideal 012 Hilbert spaces i s uniquely determined by its component on 1,.

16.2.6. Using the characterizations given in 15.2.3 or 15.2.4 as a definition we can extend every proper operator ideal a(12, Z2) to the class of all Hilbert spaces. SO we obtain the following improvement of the preceding result. Theorem. There i s a one-to-one correspondence between the proper (quasi-izornied) operator ideals an the class of all Hilbert spaces and the proper (qua8i-normd) operator ideals on the separable infinite dimensional Hilbert space.

15.3.

Operator Ideals and Sequence Ideals

15.3.1. Let U be an operator ideal on the class of Hilbert spaces. Then 1% denotes the set of all sequences (u,,) E 1 such that the operat,or S E B(Z2, I,) defined by S(t,) := (u&J belongs to 'u. 15.3.2. Obviously we have the Theoreni. 1%i s a sequence ideal.

15.3.3. Let a be a sequence ideal on t,he scalar field. Then e, denotes the class of all operators S E E such that (s,,(S)) E a. Remark. Let us recall that, by 11.3.4, there exists only one s-function on the class of Hilbert spaces which is of course completely symmetric, maximal etc.

15.3.4. The next statement is a special case of 14.1.2. Theorem. B, is an operator ideccl.

15. Operator Ideals on Hilbert Spaces

209

L5.3.5. Summarizing 15.3.2 and 15.3.4 we obtain the main result of this section. Theorem. The rules

?.+la

and

a+f,

define a one-to-one correspondence between all proper operator ideals on the d m s of €filbert spaces and all proper sequence ideals on the scalar field. Proof. The assertion will follow from the fact that S E U(R,K ) if and only if

(4s))E b. To check this conclusion we consider the Schiiiidt factorizations S = YSoX* and So= Y*SX described in D.3.3. Then, by on = s,,(S0)= sn(S), the following statements are equivalent: s E ?.(IT, 10, soE ?.(12, 12), (a,) E IS, (s,(s))E 1%. R e m a r k . Clearly there is a one-to-one correspondence between all operator idcals in the separable infinite dimensional Hilbert spacc and all sequence ideals on the scalar field; cf. 15.2.6.

15.3.6. Let [a,A] be a quasi-normed operator ideal on the class of Hilbert spaces. Then we put la(s) := A(S) for s = (a,) E 1%, where the operator S E 2(Z2,1,) is defiiied by the equation S(:,,) := (antn). 15.3.7. Obviously we have the

Theorem.

[Is, lA]is a quusi-normedsequence ideal.

15.3.8. Let [a, a] be a quasi-nornied sequence ideal on the scalar field. Then we put := a((s,(S))) for S t Ga.

Sd@)

15.3.9. As a special case of 14.1.7 we now forinulate the open P r o b l e m . Is [Go,S,] a quasi-normed operator ideal?

15.4.

Domination Property

15.4.1. A normed operator ideal [a, A] has the domination. p r q e r t y if, given S E O(H, K ) and T E U(H, K ) , then n 1

iiiiplies S E

n

sk(S)5 2 sk(T) for n = 1,2, ... 1

a(H,X )and A(S) 5 A( T) .

15.4.2. Obviously we have the Theorem. Let [a,A] be a normed operator ideal with domination propert!!. Then [la- 1 ~ i ]s a normed sequence ideal possessing the dominution property, a.s well. 00

15.4.3. Lemma. I f S E G1&, Z2), then 1

I(Se,, en)i 5 S,(S).

Proof. Consider the Schmidt representation

1d

Yietecli. Ooerator

210

Part 3. Theory of Sequence Ideals

with c1 1 0, 2 -. 2 0. By 11.3.3 and Bessel’s inequality it follows from 00

(Sek, 9)= 2‘ n=l

a,(ex-, 5,) ( Y n , el,)

that m

W

5 2’ un

I(Sek, k= 1

n=l

w

2 I(%, k=l

5,)

ek)i

m

s z: 0, = S,(S). n=l

We are now able to check the fundamental Theorem. Let [a, a] be a w m d sequence ideal with domnimtbn property. Then an qerator S E B(H, K ) belongs to 6,i f and only i f ((SXe,, Ye,)) E a for all operators x E 2(Z2, H ) and Y E f i ( Z 2 , K ) . I n this m e ,

SE(X) = SUP (a((SXe,, Ye,)): IIXII 5 1, IIYII 5 I}. P r o o f . Suppose that S E G J H , K ) . Moreover, let X E B(Z2, H ) and Y E S(l,, K ) with llXl[ 5 1 and [[Yl[5 1. Form the sequence t := ((SXe,, Ye,)). Put PMx:= eMz for x € 4, where M is any finite set of natural numbers with card ( M ) 5 n. Since Y*SXPM E Gl(12,12), by the preceding lemma we have n

n

Therefore it follows from 13.7.7 that n

n

21 gk(t)2 2’ sk(S) 1

for n = 1, 2,

...

So we get ((SXen,Ye,)) E a and a((SXe,, Ye,)) 5 S,(S). Conversely, let S B(H, K ) such that the above condition is satisfied. Given E > 0, by 11.3.5 there are operators X E B(12,H ) and Y E B(&, K ) with l[Xi 5 1, IlYll 5 1, and s,(S)

2 (1

+

E)

(SXe,, Ye,)

€or n = 1,2, ...

Hence (s,(S))E a and ASE G,(H, K ) . Moreover, we have

S,(& = a(s,(S)) 5 (1

+

6 ) a((JXe,,

Ye,)).

15.4.4. As a consequence of 15.4.3 we obtain the Theorem. Let [a, a] be a mrmed sequence ideal with domination property. Then

[Go, S,] is a norrned operator ideal having the dominatim property, a8 well. Proof. Using

Sa(S) = SUP {a((#xe,, Yen)):IlXll I1, IIYII 5 I} we see that f& is a norm. To check the completenesswe consider an S,-Cauchy sequence of operators 8, E B,(H, K ) . Then there exists 8 E B(H, K ) such that S = [[.ii-lim8,. k

15. Operator Ideals on Hilbert Spaces

21 I

Let X E B(Z2, H ) and Y E 2(Z2,K ) with llxll 5 1 and llyll 2 1. Put tk := ((SkXe,, Y e , ) ) and t := ((SXe,, Ye,)). Given E > 0, we choose ko such that S,(sh - 8,) 5 E for h > k 2 ko.This means that a(tn - tk) 5 E for h > k 2 k,,. Hence (tk)is an a-&uchy sequence which obviousIy tends to the limit t. So we have t E a. Therefore S E Ba(H,K ) . Moreover, it follows from a(t - t k ) 5 E that Sa(S- 8,) 2 E for k 2 ko. Consequently S = Sa-limSk. Using D.3.3 it can easily be checked that [Go, S,] has the doniik

nation property.

15.4.5. Summarizing 15.4.2 and 15.4.4 we obtain the main result of this section. Theorem. The rules define a one-to-one correqondeme between all normed operator idmls with dorniwh property on the class of Hilbert spaces and all normed sequence ideals with domination property on the scalar field. Remark. Let us mention that minimality and maximality are carried over in both directions.

15.4.6. We now check the compatibility of the *-procedures for normed operator ideals and sequence ideals. Theorem 1. Let [a, a] be a nomzed sequertce ideal having the dOmination property. Then lea*,%*I = [e:, 8 3 Proof. I n the following we use criterion 10.3.6. Take the Schmidt factorizations L = X L o Y * Let S Go*@, K ) and L E S ( K , a). and Lo = X*LY, where Lo E 8(12, 4) is of the form Lo(&)= (&En) with (A,) E f. Since the maximal normed sequence ideal [a*, a*] has the domination property, we have ltrace (SL)I = Itrace (rSXL,Y*)I =

1:

(SXe,, Y e , ) 2,

- a*((SXe,, Ye,))a(&) 5 S,.(B) <

Sa(L).

Consequently S E G*,(H,K ) and Sz(S) 5 S,,(S). This shows that [G,., S,]

& [G:, SX].

To prove the converse inclusion we first consider an approximable operator S E GX(H, K ) and its Schmidt factorizations 8 = YSoX* and So= P*SX. For (An) E f we put Lo(En) := (3’,tn). Then

1;

I

u,,~,, =

/trace (S~L,,)I = ltrace (Y*SXI=,)I

5 Sit(&) Sa(L0) = S,*(S)a(&)and a*(o,) 5 Sx(S). It follows from o, = s,(S) that S € G,*(H, 9)

Hence (0,) E a* and S,,(S) 2 St(S). In particular, we have S,*(S) = S:(S) for all elementaryoperators. Consequently 8.7.5 (remark) implies [G,., S,*] = [G:, S 3 .

By 15.4.5 and Theorem 1 we immediately obtain Theorem 2. Let [%, A] be a n o r 4 operator ideal with domination property. Then

[I%*, I**]

.

,

= [lg, 121*

212

Part 3. Theory of Sequence Ideals

15.5.

G,-Operators

15.6.1. An operator AS’ E i!(H, I<) is called an Bp-operatorif (ss(S))E I, with 0 < p We put

;(

S,(S) :=

< w.

8.(SyJ)llp.

The class of all B,-operators is denoted by B,. Remark. It is convenient to define [G,, S,] := [i?, [;.I!]. 15.5.2. The next result follows immediately from 11.3.3. Theorem. Let 0 < p < w. I’hen S E O ( H , K ) belongs to B, if and o d y if there

e.mkts a Schmidt representatioa

sqzh that

( ~ i )E

Ip(I).

In this w e , S,(S) =

k

3- l a a l p 1’p.

)

15.6.3. Theorem. [Bl, S,] is a normed operator ideal. Moreover, on the closs of Hilbert v, have [Gl, Sll = [%, El. Proof. Clearly 15.5.2 implies [a,, S,] [%, N]. To prove the converse inclusion we consider S E %(H, K ) . Since X E @ ( H ,K ) , t*hereexists a Schmidt representation

s = 2 uix; @ yi. I

Moreovcr, given 6 > 0, we can find a nuclear representation W

AS’ = x u : 0v, 1

with

2 ll%!l IPflll s (1 + E ) W ) . rn

1

It follows from (AS’q, yi) = ui that 00

luil I .Z ((zitun)

Ydl -

9 . v (

fl=1

Using Bessel’s inequality we obtain W

x bil r z’2 I(%’

ufl)

I

(vfl,Yi)l

Ifl=1

W

s z Il%ll Ilvnll 5 (1 4-

E)

W).

fl=l

Consequently S E G,(H, K ) and S,(S) 2 N(S). This proves that [GI, S,] 2 [%, N].

16. Operator Ideals on Hilbcrt Spaces

213

15.5.4. L e m m a 1. Let 0 < p 5 2 a d As‘ E B(H, K ) . If (Ilflqll) E &(I) for Some w h o iwrzal basis (xi)of H , theji. S € G J H , K ) and

(2; ]]s.ztp)

SJS) 2

I’P.

> 0, wc choose i E 3 ( l )such t,hat

Proof. Given E

P u t P := 2; xr @ x i . Then i

Hence ]IS- SPll 5 E . So we l a v e shown that S is approximable. Consider a Schmidt. representation

s=

u p ; @ 2:j J

0. Then

such that ai

liSxi12 =

0 ;

l(x2.uj)j2 for all

EH.

5

J

Using Holder’s inequa1it.y with the exponents 2 / p and 2/(2 - p ) we get uiy =

uip

J

=

r~

5

(F /(xi,

Ui)/2)

2’ ap I ( X i , U j ) l P 1(5i,uj)l2-p

c (E u; I

!(Xi,u#)PA

J

( X 1(5i, U q o P ) ’ 2 J

I 2 r 2 ! ( P i ,.Ui)12 - I ( ?

j

)

‘

=

2 II8PillP. I

This proves that S E Gjp(H,K ) and

S,(S) =

(2upjl‘p 5 (2j!SZillPjl/P. I

J

< cy, md S E G J H , K ) . Then

L e m m a 2. Let 2 5 p

(T IIS~illP)1’P d

SpW)

f o r every orthonormal basis ( x i )of H . Proof. Since fl E Q(H, K ) , we can find a Schmidt representation

s = 2 up; J

vj

214

Part 3. Theory of Sequence Ideals

Hence

15.5.5. If p = 2, then the preceding lemmas yield a n important T h e o r e m . An operator S E 2 ( H , K ) belongs to G, i f and only i f (jlSxi/l)E I,(I) fw every (some) orthonormal basis (xi) of H .

Ir, this case, S,(S)

=

z

lSSi112 1'2.

( I

)

R e m a r k . Usually G2-operators are called Hilbert-Schmidt operators. The basic examples are operators S in L,(Q, p) possessing the form Sf = g with dL")=

J sb, B ) f ( B ) WB) >

s)

where the kernel s belongs t o L2(Qx Q, p

x p). Then

.

For further informations we refer t o [DUN], [GOH], [RIAU,], and several other Hilbert space textbooks.

15.6.6. Theorem. [G,, S,] i s a normed operator ideal.

Moreover, the norm S2 can be generated by the scalar product

(8,T):= X (&xi,T x j ) for S , T E G,(H, K ) , I

where ( x i )i s any orthonormal basis of H . Proof. The existence of (8,T) follows from

2 I ( S X i , TZi)j 5 (? I!SXijlZ)1'2 I

(T

I l W 2 ) 1 / ~= wf9

SAT).

Obviously the above equation defines a scalar product such that S,(S) = (S.S)ll'. R e m a r k . Since all components G , ( H , K ) are Hilbert spaces, we see that the S,] is self-adjoint. normed operator ideal [G2,

15. Operator Ideals on Hilbert Spaces

215

16.5.7. Using anot,lier method we prove a special case of 15.4.3. Theorem. Let 1 5 p < 00. An operator S E B(H, K ) belongs to 6, if a d o d y if ((XX%,Ye,)) E 1, for all 0pera.tor.sX E B&, H ) and Y E E(Zz,K). In th& case Sp(8) = SUP {lp((SXen, Yen)): IlXll 5 1, IlYIl 5 1). Proof. Suppose that the above condition is satisfied. Then by 15.2.1 we have S E @(E,9).Consequently there exists a Schmidt representation

s=

yi.

GjXf I

Without loss of generality we may assume that I = { 1,2,

...).Put

X(Ei):=rl: Eixi and Y(qi) := rl: qiyi. I

Then

~i

I

= (SXei, Yei). Hence (ai) E I,(I). This proves that 8 E G J H , K) and (SXei, Yei)).

S p ( 4 5 I,(

Conversely, let S E G,(H, K ) , X E B(&, H),and Y E So := Y*8X E Gp(Z2,Zp). Take a Schmidt representation

8,= 2 .ja$

B(&, K).

@ yi

I

such that ui 2 0. It follows from

2 ail(% en)12= rl: Gi I(% I I

I(%

e,)P

e,)lZ/P’

5 {$’ 0’ ~(zi, e n ) I z p {.$‘ I(zi, en)12)”*’

that

{F

~i/(xi,e n ) 1 2 r

en)12p.

5 {+‘ 0: ih,

Analogously we obtain

{+‘ ui I(Y~,

en)Iz)plz

{+’

0; I(yi, en)12ya

Noreover, t(&en, en)i 2

S Finally, we get

LI ‘4”I(si, e,)I ~i”’ Kyi, e,)I

k

{F

bi I(zi, en)12)llz

Ibi,

en)12y*

Then we have

216

Part 3. Theory of Sequence Ideals

15.5.8. We now formulate the main result of this section.

T h e o r e m . [GP, S,] is n n,ornied operator S e a l for p ;2 1 n?zd ideal for 0 < p -==c1.

CI

p-nornted operator

Proof. We know from 14.2.2 that [Gp, S,] is a quasi-nonned operator ideal. If p 2 1, then the triangle inequality follows from

S,(@ = slip (l,((SXe,, Ye,)):IIXII I 1, IIYII I I}. On the other hand, 18.5.1 and 15.5.2 iinply the assertion for 0

< p < 1.

15.5.9. The next. statement is also of great importance; cf. 14.2.3.

< p , q < CQ and l/r = 1/p + l/q.

Theorem. Lrt 0 [Gpj Spl 0

[Gq,

Sql

r z

Then

[Gr. S r l *

Proof. If S E G,(H, K ) , thcm wc consider a Schmidt representation

s = 2' a& 0yi I

such that (ai) E 1,(1)and ui 2 0. Put

S(&) :=

and

of:u&q

Y(qi):==

@'iliyi.

I

I

Then S E Gq(Z2(1), H ) and it follows froin

S q ( X )= that Hence

5.'

(7

4

d

llg

Y E Gp(Z2(I), K ) . Obviously we have 6 = Y X * . Moreover, and Sp( Y ) =

s, 0 S,(S) 5 ( c u; I 1 [Gr,

2 aT ( I

I)

= S,(S).

Srl E [Gpt S p l * [Gq, 891.

We now check the converse inclusion in four steps. By 15.2.5 i t is enough to consider operators acting in s fixed Hilbert space H . (1) Let S E G2(1$,13):\rid T f G,(H, H ) . Take a Schmidt representation

T =-:

2' T~X: '9y i Ja

such that t i 2 0. Choclse an orthonormal basis (xi)with i E I which is an extension of the orthonormal family (xi) with i E Io. Then Tzi = tiyi for i C I . and T x , : o for i @ I,. Consequently

+

where 1Jr = 1/52 l / q . Sinoc 0 < r 5 2, by 15.5.4 (Lemma l ) ,we have ST E G,(H, 11) S2(S)SJT). This proves that and S,(sT)

rB,, S,1

o

En,S,1 E 1 6 , S,]

with

llr = 1/2

+ l/q-

15. Operator Ideals on Hilbert Spaces

21 7

(2) According to 15.1.1, by (l),we obtain

[G,, S,] o [G2, S,] 2 [G?,S,]

l/r = l/p

with

+ 112.

(3) Suppose that S E G,(H, H ) and T E G g ( B ,If), uliere 2 5 p , q < my and 1 ‘r = lip f l/q. Then 1 5 1’ < co. Let X E 2(Z2,If) and Y E 2(ZZ,H ) such t h t X!1 5 1 and JJYI!5 1. By 15.5.4 (Lemma 2) we have

5 S,(S*Y) S,(TS) 5 SJS) S,(T). ”her? 15.5.7 implies SI’

E G,(H, H ) and S , ( S T ) 5 S,(S) S,(T). Hence

[Gp,S,] o [G,, S,] C= [G,, S,l with l / r == lip

-t-

l/q.2

5 p, g < w.

(4)To treat the general case we choose integers rn a i d n snch that l / p - 11.2 2 mi2 < lip and

l/q

-

112 5 n / 2 < l;q.

If 3/po:= lip - m/2 and l/qo := l/q - n/2, then 2 5 po, qo < S :G,(H, H ) arid 1’ E G,(H, H ) , there are factorizations

S 15

ASra.. . Slk90 a d

T

00.

Given

..

= llolll. T,,

ith So t G J H , H ) and

S,,(So) = S p ( S ) P ’ p a ,

To f G,,(H, H ) and

Sq6(Y0)= Sq(T)4’90,

Sh E G,(H, U ) arid

S,(S,)

= S,(S)p’*,

’r, E G,(H. E I )

S,(T,)

= Sq(T)Q‘z,

and

where h = 1, ..., m and k = 1,

and

ST

= (S,,,

- a *

...,n. Now

S,(SOl’o) T1

S,(ST) 5 S,(S,)

*. *

T,$) f G,(H. H )

S,(S,) S,.(SO)

- Sp(S)P/2+-’PP

it follows from ( I ) , (2), and (3) that

PlPO

S,@(TO) S,(T,) .’. S,(Y1n)

84 ( q q ” - “ .

q/Z+qlq,

= S,(S) Sq(2’).

This completes the proof.

15.5.10. Finally, we mention an iiiiiiiediate coiisequcnce of 12.2.5 and 12.3.1. T h e o r e m . Let 0

< p < m. Then G,

= @, on

the class of Hilbeit ~prrces.

Remark. Obviously the quasi-noriiis S, and E, do not coincide, however, they are eauivalent..

218

Part 3. Tlieorv of Seauence Ideals

15.6.

Extensions of Operator Ideals

15.6.1. In this section U, denotes some operator ideal on the class of Hilbert spaces. An operator ideal U defined on the class of all Banach spaces is called an extension of f& if U(H,9) = U,(H, K ) whenever H and K are Hilbert spaces. 15.6.2. An operator S E B(E,F ) belongs to the superior extension Urp if it holds BSX E 910(H,K ) for all X E B(H, E ) and B E O(P,K ) . 15.6.3. We have the simple

a,. Proof. Obviously UEupis an extension of a,. Now let Theorem. Urp is the largest extension of

bc an arbitrary extension. Suppose that S E %(E,F). If X E B(H, E ) and B E B(F, K ) , then it follows that BSX E U(H, K ) = %,(H, K ) . Hence S E U:up(E, F).This proves that E UFp.

15.6.4. An operator S c C ( E , F ) belongs to the inferior extension Up‘ if there exists a factorization S = YS,A, where A E B(E,H ) , S o €U,(H, K ) , and Y E B(K, 2;”). 15.6.5. Analogous to 15.6.3 we obtain the Theorem. i s the smallest extension of 8,. 16.6.6. An operator S E C ( E , F ) belongs to the right-superior extension sZgp if, given B E B(P, K ) , there exists a factorization RS = SoA, where A E O(E,H ) and So E Uo(H, K ) . 15.6.7. Theorem. U r p 6s an extension of Uo. Proof. Let S,, S, E Urp(E,F ) . Given B E O(F, K ) , we can find factorizations BSi = So+4i,where A , E B(E, H , ) and So, E %,(Hi, K ) . Let H := H , x H2 be the Cartesian product wit.h the Euclidian norm. Then

+

+

+

B(fJ1 S2) = (SOlQ1 So,&,) (JlAl J2A2). Now it follows from SolQ1 + S,Q, E U,(H, K ) that S, -+S, f Urp(E,F). This proves (OIl). Since (01,) is evident, must be an operator ideal, Obviously U%P(rr,K ) = U,(H, K). 15.6.8. An operator S E E(E, F) belongs to the right-inferior edension 9lF if there exists a factorization S = Y A , where A E B(E, H ) and Y E B(H, F ) such that BY E a@, R)for all B E B(F, K ) . 15.6.9. Theorem. i s an extension of a,. Proof. Let S,, S, E Up(E, F ) . There are factorizations Si = YiAi, where A( E O(E, Hi) and Yi E B(Hi, F ) such that BYi E %,(Hi, K ) for all B E B(P,K ) . Let €1 := H , X H z be the Cartesian product with the Euclidian norin. Then 8,

+ 8, +

=

(Y1Ql

+ Y?Q,)(JiA, +

J2-4)

+

and B(Y,Q, Y2Q2) E 910(H,K ) for all B E S ( F , K ) . Hence S, 8,E Up(E, F ) . This proves (Orl). Since (01,)is evident, U r n must be an operator ideal. Obviously Ut;’”(H,K ) = A o ( H ,K ) .

15.6.10. An operator S E f?(E,F ) belongs to the left-superior extension UzlUp if, given X E B(f.1, E ) , there cxists a factorization SX = YS,. where So E a0(H, K ) and Y E O(K,F).

15. Operator Ideals on Hilbert Spaces

219

15.6.11. Analogously to 15.6.7 we have the

Theorem.

atpis an extemhn of a0.

15.6.12. An operator i3 6 O(E,P) belongs to the left-inferior extension 2Ip if there exists a factorization S = Y A , where A E B(E,9)and Y E B(K,F ) such that A X E a0(H,R ) for all X E B(H, E). 15.6.13. The next result is analogous to 16.6.9.

Theorem.

is an eztension of 910.

15.6.14. Clearly the class of all extensions becomes something like a complete lattice with respect to the natural ordering. In the following diagram the arrows point from the smaller operator ideals a t the larger ones.

T Remark. The extensions conctructed above do not coincide in general. However, there are operator ideals on the class of Hilbert spaces admitting a unique extension. A trivial example is the ideal of finite operators. See also 18.7.9. 15.6.15. Using the technique of 8.4.9 we can prove the

Proposition. The operator ideals a?, Up, and

@f:

are injective.

15.6.16. Analogously we get the

Proposition. The operator ideals

arp,Cap,and @cfare mrjedive.

15.6.17. We mention without proof the following result.

Proposition. The operator ideal

2 completely symmetric.

15.6.18. We now formulate the open

Problem. Is the operator ideal

Urpcompletely symmetric?

Remark. If all extensions are restricted to the class of reflexive Banach spaces, then we have the following relations: (%rp)du.%l = a s u p 0

(Ul‘’P dual

T=

(%p)dusl

=

0

)

UruP 0

0

win

dual

)

(%?)dual

0

(gy)aual = ginf 0

’

= ulUP 0

=

alin

220

Part 3. Theory of Sequence Ideals

It can be shown that, the right-hand forinulns hold in general. The last equation is equivalent t,o the assertion of 15.6.17. 16.6.19. If I is any extension of

a,,

t.hen

ugl' = 8-10 a 0 9-1 at;'"= (80 U) 0 8-1 a""= sj 0 (80 9-1) !@f= 9 0

%;;UP = 8-1 0

(U 0 8)

%.,i= ll (8-1 0 U) 0 sj

u 8, 0

where 8 denotes the ideal of all Hilbert operators; cf. 6.6.1. If [a,, A,] is 8 quasinormed operator ideal on t,he class of Hilbert spaces, fhen the shove extensions become quasi-normecl operator ideals in a canonical way; cf. 7.1.2 and 7.2.2.

15.7.

Notes

J. W. CALKIN observed in 1941 that theic is

it one-to-one correspondence between operator ideals on the separable Hi1bci.t space and certain sets of non-increasing sequences of nonnegative numbers. See also [SAT2, p. 261. According to D. J. H. G-ARLIXG[2] it is possible to replace this so-called characteristic sets by sequence ideals. The operator ideals G p correspond[I]. ing to the classical sequenee ideals I, have been studied by J. VON x E U M A N N / R . SCHATTEN The case p = 1 was already treated in a paper of F. J. MURRAY/J. vos NEUMANR [l], and [l]. Full treatements for p = 2 one gets the ideal of Hilbert-Schmidt operators; cf. E. SCHXIDT of this theory may be found in the monographs [GOH, pp. 91-95], [RIN, pp. 75-1073 and [SAT,, pp. 29-43] as well as [DUN, pp. 10%- 11001. The problem of extending operator ideals from the class of Hilbert spaces to the class of Banach spaces has been extensively studied by [7], [lo], [21]. the author; cf. A. PIETSCE

Recommendations for further reading:

A. BROWN/C. PEAP.CY/N. SALIEAS [l], CII. C>ONSTAXTIN [l], I. C. GOHBERG/M.G. BEJN (11, J. R. HOLUB[4], [S]. H. J. JUKEK[l]. A. S. XARKUS [l], K. MAURIX[l], C. A. NCCABTHY[l], C. MERRECI/PHAXTEE LAI[l],D. MORRIS/N. SAUNAS [l],F. J. MWRBAT/J. voa NEUMANN [l], w.OOBTESBRIh-K [l], -4.PEECZP~~SKI [4], -4.PIETSCH [121, D. J. VOX RIEMSDIJK [I], s. SALINAS[l], [2], [3], 13. SPRATTEE [l], K. TSUJI[i]. See also the bibliographical notes of Chapter 28.

Part IV:

Basic Examples of Operator Ideals

16.

Family Ideals on Banach Spaces

In the following we introduce the concept of a family ideal on the class of arbitrary Banach spaces. Most results obtained for scalar sequence ideals could be carried over to this general situation. However, we only f0rmulat.ethe definitions and a few elementary statements.

The main purpose of this chapter is to treat some basic examples which we need later on. The ideals of absolutely and weakly p-sumniable families are of special importance. Furthermore, we consider ideals of so-called mixed (s,p)-summable families.

16.1.

Family Ideals

16.1.1. Let us recall that I denotes the class of all bounded families in arbitrary Banach spaces. A family deal

is a subclass of I such that the components

a(E, I) := a n I(E, I) satisfy the following conditions:

(Fb) e, E a, where eo = (1)is the single unit family. (FI,) It follows from "E, z,E a(E,I)that x1+ x2 E a(E,I). (FI,) If s E l ( I )and z E a(E, I),then m E a(E, I). (F13)Let z be any one-to-one map from I, into I. Then J,x E a(E, I) for all 3~ E a(E, I,) and Q,x E a(E,I,) for all z E a(E, I). (FI,) If S E 2(E7F ) and x E a(E, I),then Sx E o(F, I). Remark. Family ideals will be denoted by bold small Gothic letters. Remark. For the components on the index set M = (1,2,. ..] we shall simply write a(E).

16.1.2. Proposition. Let a be a family ideal. Then all components a(E, I)are linear qme-s.

16.1.3. Let be a family ideal. A map a from a into W+is called a quasi-normif the following conditions are satisfied :

(QFI,)a(eo)= 1, where e, = (1) is the single unit family. (&PIl) There exists a constant x 2 1 such that

+

a(xl

8,)

+ a(xz)l for

5 x[a(z,)

a,,xzE a@, I).

(QFI,) If s E l(I)and a E a(E, I ) , then a(sx) I ((slla@). (QFI,)Let n be any one-to-one map from I, into I. Then a(J,z) a(z) for all z E u(E,I ) . x E a(E, I,) and a(&#) (QFI,) If S E 8 ( E ,3') and z E a(E, I),then a(&) 5 llSll a@).

a ( a ) for all

324

Part 4. Basic Examples of Operator Ideals

Remark. Quasi-norms on family ideals will be denoted by bold sinall Roniaii letters.

16.1.4. Proposition. Let a be a family ideal zuithaquasi-norm a. 1’henallcon~ponent.s a(E,I ) are linear topological Hausdorff spaces. 16.1.5. A quasi-nornied family ideal [a, a] is a family ideal a with a quasi-norni a such that all linear topological Hausdorff spaces a(E, I)are complete. 16.1.6. Proposition. Let [a, a] be a qunsi-normed family ideal. Then 1 1 ~ 1 15 a(x) E a.

for all x

16.1.7. A quasi-norm a on the family ideal a is said to be a p-norm (0 < p 5 1) if the p-triangle inequality holds:

a(xl

If p

==

+ a,), 2 a(al)p + a(xz)p for 3cl,x2 E a(&‘,I).

1, then a is simply called a n m w .

16.1.8. A p-normed family ideal [a, a] is a family ideal a with a p-noriii a such that all linear topological Hausdorff spaces a(E, I) are complete. If p = 1, then [a, a] is said to be a norm& family ideal. 16.1.9. Let [a, a] and [b, b] be quasi-normed family ideals. Then [a, a] E lb, B] means that a & b and a(a) 2 b(x)for all x E a.

16.2.

Absolutely p-Summable Families

16.2.1. Let 0 < p 5 00. A family x = (xi), where xi lutely p-wnamable if (ilxi[l)E & ( I ) .We put

E for i E I, is called abso-

l,(xi) := lp(ilxill). The class of all absolutely p-summable families is denoted by I,. Remark. Obviously [Irn,1-1 = [I,

11.11].

16.2.2. Theorem. [I,, l,] 12 a normed family ideal for p 2 1 nnd a p-normed family idealfor 0 < p < 1. 16.2.3. Proposit,ion. Let 0 < p1 5 p, 5 00. Then

u,, lP,l G [I,,, 1PJ 16.3.

Weakly p-Summable Families

16.3.1. Let 0 < p 5 00. A family z = (xi), where xi E E for i E I, is called w ~ U Y p-summable if ((ai, a)) E I,(l) whenever a E E’. We put

w,(q)

:= sup {ip((xi,a ) ) :Ilall 5 11.

The class of all weakly p-suminable sequences is denoted by t~,.

16. Family Ideals on Banach Spaces

228

Remark. The existence of wp(zi)follows from the closed graph theorem or the principle of uniform boundedness. Remark. Obviously [w,,

w,l

:= [I,

11.111.

16.3.2. Theorem. [mP,wp] is a normed famity ideal for p < p < 1.

>= 1

and u p-normed

f.miZy deal for 0

16.3.3. Proposition. Lei 0 [WP,, WP,]

16.4.

< p , 5 p2 =( 00. Then

s b p , , WP.1.

Mixed (s,p)-Snmmable Families

+

16.4.1. Let 0 < p 2 s CQ and determine r by 1/r 11s = lip. A family x = (xi), where xi E E for i E I , is called mixed (s, p ) - s u m d l e if it can be written in the form z,= zjxtwith (ti) E l,(I)and (z!) E m,(E, I ) .

Put m(s,p)(xi) := inf & ( t i ) w&%)I,

where the infimum is taken over all possible factorizations.

The class of all mixed (8, p)-summable families is denoted by m(s,p). Remark. I n the boundary cases we obviously have [ r n ( p , p , , In(p,p)l

= [Wp, Wpl

and

[ t t f m , p , , m,m,p,I

= [ 4 7 > 1Pl.

16.4.2. Theorem. [tq,,,, m(s,p)] , is a normed family ideal for p 2 I and a p-normed family ideal for 0 < p < 1. Proof. We only mention that the p-triangle inequality follows from the next criterion.

m(s,p)]can be considered in an obvious way, see 7 . 1 2 ,

Remark. Clearly as the product of [lr, l,] and

[a,, w,].

16.4.3. We now give an important characterization. (8,

Theorem. Let 0 < p < s < 00. A family (xi), where xi E E for i E I , ia mixed p)-summable if cind d y if

((

\(xi,u)ls dp(a))1'8)E $,(I) whenever p E JC'(UEp).

UE'

Here we suppose that UEt is equipped with the weak E-tOpdO[y?J.

1%this case, m(s,p)(zi) = SUP

{[+(rjLJ .>I") I I(xi,

dp(a)

:P E

1-

JV-~UE,)

Proof. Let (x,)satisfy the above condition. Suppose that there are pn E W ( U E 3 ) with

15 Pietsch, Overator

226

Part 4. Basic Examples of Operator Ideals m

By setting p := 2 2-n,u,,we obtain ,u E W (UEs)such that 1

This contradiction shows the existence of

Put u := r / p a n d v := slp. Then l / u compact convex subset

K:={

+ l l v = 1. We now

consider the weakly

(ti):z E r 5 op and t i 2 0

1

I

of &(I). Observe that the equation

@(ti) :=

i

+

(ti

E)-O

J [(xi, UP;*

.>I" d ~ a, )

where ,u E W(U,,), E > 0, and i E 3(I),defines EL continuous convex function @ on K . Take the special family (ti) with

Then (ti)E K and @(ti)5 o p . Since the collection 3 of all functions @ obtained in this way is concave, by E.4.2 we can find (6;) E K such that @(&;) 5 o p for all @ € ,F simultaneously. In particular, let S(a)be the Dirac measure at the point a. Then we obtain 2 ($ .)-" ](xi,a y (1op

+-

t

for E > 0, jjalj 5 1, and i E 3(I). If xi 0, it follows that 5: 0. Hence we can put zi := ]Epll/Pand x: := t;'xi. In case xi = o let. ti := 0 and x! := 0. Then xi = zixf.Moreover, we have

+

+

(q

lril')l/r

d (2Igp)l.

5 f+

I

and

for i E % ( I ) and l]u/l 5 1. Hence lr(ri)w,(xp) S cr. This proves the necessity of the above condition.

Conversely, suppose that (xi)is niixed (s,p)-summable. Take any factorization

xi = TjZp with (zi) E &(I)and (xf)E m,(E, I ) . Applying Holder's inequality we obtain

16. Family Ideals on Banach Spaces

227

I l,(Zi) w,(z;) whenever p E W(U,.). This prores the sufficiency of the above condition.

16.4.4. The next inclusion follows immediately from 16.4.3. However, there is also a n eleiiientary proof; cf. 17.1.4.

16.5.

Notes

The definition of 8 family ideal seems to bc given in this monograph for the first time. The special ideals of weakly and absolutely p-summable families have been investigated by several [5]- The only non-straightauthors. I n particular, we refer the reader to A. GROTHENDIECK forward result of this chapter is a powerful characterization of mixed (s,p)-summable families the proof of which is mainly taken from B. XAUREY [2]. See also [SEN,,exp. 151.

Recommendations for further reading:

H. APIOLA[l], M.E. MUNROE[l], 51. S. ~ K L C P R[~ l],I LD. RUTOVITZ [l].

17.

Absolutely Summing Operators

This chapter is devoted to the theory of absolutely summing operators first defined by A. GROTHENDIECK in 1955.

.'r '1r

We deal with the following operator ideals :

%.p)

Pr

absolutely (r, p , q)-summing operators,

absolutely (7, p)-summing operators, (p, q)-dominated operators,

absolutely r-summing operators,

%

p-dominated operators.

The basic concept is that of an absolutely r-summing operator due to the author (1966). As the main tool we use a characterization of those operators by means of some fundamental inequalities. The definition of (11, q)-dominated operators goes back to S. KWAPIE& who also proved the formula B(p,qb = @:"&' o !&,.

17.1.

Absolutely (r,p , q)-Summing Operators

17.1.1. Let 0 < r , p , q co and 1/r 2 l/p + l/q. An operator S E E(E, F ) is called absolzltely (7, p , q)-summingif there is a constant u 2 0 such that lr((flsi9

bi)) 5 Q wp(si)wq(bi)

for all finite families of elements xl, We put

P(r,p,ql(~) := inf

...,x,

E E and functionals b,,

...,b,

€ F'.

(T

The class of all absolutely (7, p , @-summingoperators is denot,edby ?&r,p,qb.

17.1.2. The proof of the following result is straightforward; cf. 6.5.2. Theorem. [!$(rmp,q), P(r,p,qJis a nornaed operator ideal for i d f dfW 0 < 7 < 1.

7

2 1 and an 7-nom7d

opWd07

17.1.3. We now state a generalization of 8.7.8. Theorem. T?Mquasi-wmtd operator ideal [?&r,p,qb, P(r,,,,qJ-is mccxim~l.

229

17. Absolutely Summing Operators

Conjecture.

@(r,p.q)

is proper if and only if 1 / r > l / p

+- l / q - 112.

Remark. I n 17.2.7 we will show that the answer is affirmative for p (1 =

00.

= 00

or

230

Part 4. Basic Examples of Operator Tdcnls

17.2.

Absolutely (r,p )-Summing Operators

17.2.1. Let 0 < p 5 r 5 00. An operator S E O(E, F ) is called absolutely ( r , p ) summi?q if it belongs to the quasi-normed ideal [(P(r,p)>P(r,p)l:= [ ( P v , p , o o ) * P(r,p,oo)l-

17.2.2. We now give a more direct characterization. P r o p o s i t i o n . An operator S E 2 ( E , F ) is absolutely (r.p)-summing if and only if there .is a constant cr 2 0 such that

l,(SXi) 5 CT WJXj) for all finite families of elements xl,...,x, f E.

In th& case, P(r,p)(S) = inf u. Proof. The sufficiency of the above condition follows iniinediately from

lA(Sxi9 b i ) ) 5 MllB4l IIbill) 2 1ABXi) ww(bi) 5 wp(xi) wea(bi)*

To prove the necessity, given xl, ...,x,,, f E,we choose b,, @xi, bi) = IISxill.Then lr(SZi)

= lr({S~i> 6,))

...,b,,, E U,. such that

5 P(r.p)(S)Wp(zi)wm(bi) 5 P(r,p)(S)mrp(ai)*

R e m a r k . The preceding criterion shows that the absolutely (1,l)-summing operators coincide with the absolutely summing operators introduced in 6.5.1.

17.2.3. The phrase “absolutely (r,p)-summing” is derived from the following property. P r o p o s i t i o n . An operator X E B(E, F ) aS absolutely ( r , p)-summing i f m d only i f every weakly y-summable sequence i s mapped into a n absolutely r-mmmable sequence. Proof. The necessity of the above condition follows by applying 17.2.2 t o the finite subfamilies of (xi)E w p ( E ) . Conversely, if every weaklyp-summable sequence (xi)is transformed into an absolutely r-summable sequence (&‘xi), then we get a closed linear map from mP(E)into I@). Hence, there exists a constant u 2 0 such that lr(Sxi)I;c r ~ ~ ( x ifor ) all (xi) E wp(E).I n particular, this estimate holds for finite families.

17.2.4. The next statement is a consequence of 17.2.2. T h e o r em. The quasi-normed operator ideal

[?&r,p,,

Y,,,,,] is injective.

17.2.6. It follows from 17.1.3 that the quasi-normed ideal of absolutely (r,p)-sunlwing operators is maximal and therefore ultrastcable;cf. 8.8.7 (remark). For a direct proof we refer to 8.8.11. Proposition. The quasi-normed operator ideal

[Ti(,,,), P,,,,] is ultrastable.

231

17. Absolutely Summing Opcrators

17.2.6. As a special case of 17.1.4 we have the Froposition. Let rl 5 r2, p1 5 p 2 , and lipl - l/rl [ P ( r 1 , p 1 ) 9Y(r,,p,)l

1/p2 - l/r2. Then

-

[ P ( r 2 , p , ) j P(r2,ps)l

17.2.7. Theorem. The operator ideal (P(r,p) is proper if and only if l/p - I / r < p 2 r < 00.

< 112

and 0

Proof. It will be shown in 22.1.12 that the identity map of Z2 belongs to for 1/p - l/r 2 1/2. Moreover, we have I p ( m , p ) = 2 for all p

@(r,p)

> 0.

Suppose that E is an infinite dimensional Banach space. Given a natural number 12, hy 33.2.1 we can find x,, . ., 2, E such that IIzl/j = -.. = llz,Jl = 1 and w2(zi) 5 2. So ?@(r,2) is proper for 2 5 r < 00. Clearly !&r,p) & ?#(7n2) for 2 5 p 5 r < 00. for 0 < I, 5 2 and I/p - 1/r = 1/2 - l/ro. Hence the Noreover, ?@(r,p) E operator ideal ?@(T,P, with 0 < p 2 r < 00 and l/p - 1/r < 1/2 is proper.

.

L7.2.8. Let us define the space ideal P(7,P)

:= Space (%*p))*

In particular, a Banach space is said to have the Orlicz property if it belongs to

Po,,).

17.2.9. We now give an interesting example which has bcen investipted by

s. KwAxmlil/A.PEwzP6sKI [11.

Proposition. The operator S E B(E,, co) defined by

(T t i ) a3

S(&) :=

is absolutely (r, l)-sumniiny for 1 < r 5 00. Remark. It can be shown that the operator S defined above is absolutely (r,p)suinming even for 0 < p < r 5 bo. On the other hand, we have S 6 ?D(Z,, co). Hence ?&r,p) $ ?D. However, it will be proved in 17.3.12 that !@(r,r) c ?D for 0 < r < 00. 17.2.10. Pinally, we mention a trivia.1,but useful Proposition. Suppose that F 13 a n intermediate space of (F,,, F,) possesSing J-type 6. Let 0 < p 5 ro, rl 5 00 an.d l/r := ( I - 6)/ro i3/rl. If S E 2(E,FA), then S E ?&,o,p)(E, F,) and 8 E P(rl,p)(E, F,) imply S E ?@(r,p)(E, F ) . Moreover,

+

17.3.

Absolutely r-Summing Operators

17.3.1. Let 0 < r 5 00. An operator S E 2(E, F ) is called absolutely r-summing if it beloiigs to the quasi-normed ideal

Pr1 := [V(7,r,m), P(r.r.oo)I* I l e u ~ a r k .Obviously [?&,P,] = [!&r.r), P(r.r)land [Pm. P,] = [e, 11.111I3377

17.3.2. We now prove the fundamental characterization of absolutely r-summing opcrators.

232

Part 4. Basic Examples of Operator Ideals

Theorem. Let 0

< r < 00.

An operator S E a ( E , F ) is absolutely r-summing if a conatant cr >= 0 and a probability p c W(Ur) such that

and ody if there ezc&

llfwl 5

fJ

{ J I(%, .)I'

1

4-44

for xE8'. Here we suppose that UEcis equipped with the weak 3 - t q o l o g y . Uh'?

I n tlink m e

P,(S) = inf u. Proof. The sufficiency of the above condition is evident, since for xl, ..., x, E E we have

ju

{ J z'I+.;, UE

a)l7

a&)}

1lr

I B Wr(2i).

1

Conversely, let S E @,(El3') and put c := P,(S). Take C(UEp)'equipped with the weak C(ZT,,)-topology.Then W(U,,) is a compact convex subset. For any finite family of elements xl, ..,x, E E the equation

.

defines a continuous convex function @ on W(UEp). Choose a, E U p such that

If S(a,) denotes the Dirac measure at the point a,, then we have m

@(&%))

=

z [11~41-7

fJr

l(zi, ao)lr]= Ir(8xi)r-

5 0.

1

Since the collection 9 of all functions @ obtained in this way is concave, by E.4.2 there is po E W (U E j such ) that @(po)5 0 for all @ E .Fsimultaneously. In particular, if 0 is generated by the single family (x),it follows that ll8Zll' - fJr

j1(~,41r &,(a)

I0.

UEI

This completes the proof. Remark. Observe that there exists po E W(U,.) for which the best possible value of c, namely P,(S), is attained.

17.3.3. For operators actringon a Banach space C ( K )the preceding crit,erioncan be stated in a simpler form. The proof is completely analogous. Theorem. Let 0 < r < 00 and let K be a compact Hausdorff space. A n operator 9 E Q(C(K),F ) is absolutely r-summing if an.d only if there exist a constant c 2 0 and

17. Absolutelv Summing Operators 18

233

probability p E W ( K )suc?~that

ilsfll5 0 { J If(oJ)lrdp(a)}l’r

for ~ Z fZ E c ( K ) .

K

I n this cme,

P,(Is) = inf

0.

17.3.4. Let p be a probability on the compact Hausdorff space K. If 1 2 r then the canonical map from C ( K )into L,(K, p ) is denoted by J,.

< 00,

17.3.5. Proposition. Let 1 5 r < 00 and let K be a compact Hnusdorff space. An operator S E B(C(K),F ) .iS abdutely r-summing i f and d y i f there ex138 a commufative diaqrnm

cuith s m e probability p 5 W ( K )and Y E B(L,(K,p), F ) .

In this case, P,(S) = inf !lYll, where the infimum is taken over all possible factorizations. Proof. We see from 17.3.3 that J , E p,(C(K),L,(K, p)) and P,(J,) 2 1. Hence S = Y J , E ?&(C(K), F ) and P,(X) 5 IIY11. This proves the sufficiency of the above condition. Conversely, let S @,(C(K),P).Then we choose p E W ( K )such that IlSfll 5 G

{

l f ( W ) I r dp(o))”‘

for all f E C ( K ) .

K

Consequently, the operator Yo: Jrf + Sf extends to Y f B(L,(K,p),F ) with ilY!I 5 6.Moreover, we have S = YJ,.

17.3.6. The next statement is also along the lines of the preceding results. Proposition. Let 1 5 r

< 00

and suppose that F has the metric extension property.

.4n operador 8 E 2 ( E , 3)i s wb.ssolv,telyr-summing if and only if there exists a eominutative

diagram

E A 4

S

,P

il Y I

834

Part 4. Basic Examples of Operator Ideals

~2~i.j~ some conipact Hausdorff *spaceK , a probability p E W ( K ) ,A E B(E,C ( K ) ) ,a d Y E g(Lr(K, F ) . In

this case,

P,(S) = inf Ilyil IIAli, where the infimum is taken over all possible factorizations. P r o o f . The sufficiency of the above condition follows immediately froin 17.3.5. I,& S E !&(E, F ) and piit G- := PJS). Then there is a probability ,u E W (UBl)such that

By setting Ax :=f,, where /,(a) := (x,a), we obtain a n operator A E 2 ( E , C ( U p ) ) with jiAil = 1. Let J , denote the canonical map from C(UE,)into L,(U,,, p). Then Yo(Jrf,):= Sx defines an operator Yofrom M(J,A) into F. Hence, we can find an uxtexsion Y E 2(L,(UE,), F ) such that S = YJ,A and IlYIl = IIYoil 5 G-.This completes the proof.

17.3.7. If r = 9, then the preceding criterion is tnic for arbitrary Banach spaces. P r o p o s i t i o n . An. opemtor S E tltere exists LC commutatiue dingram

B(E,F )

i s Gbsolutely 2-summing i f and ordy ij

with some compact Hausdorff space K , a probability ,u E W ( K ) ,A 6 G(E, C ( K ) ) ,c u d y i- 2(L2(K,p),F ) .

TR this case, P,(S) = inf [IYIlIIAli w?here the infimum .is taken over all pas.&le factorizations. Proof. Suppose that S E b z ( E ,F ) and use the saiiie constiriction as in the previous proof. Obviously Y o oxtends continuously to M ( J 2 A ) .P u t Y := YoQ,where Q is the orthogonal surjection from L2(UE,,p ) onto X ( J , A ) . Then we have S = YJzA and IlYll = IlYoil 5 G-.

17.3.5. We now treat n further example. p ) be a memure space. If s E Lr(f2,p), P r o p o s i t i o n . Let 1 5 r < oc, and let (B, then Sf:= sf detines an operator S E ?&(GJB, p), L,(P, p ) ) with P,(S) = llSll = &sllr.

17. Absolutely Summing Operators

235

P r o o f . Given f l , ...,f, E L,(Q, p ) , for every choiceof al,...)L Y , E X there exists a y-null set N ( a i ) with

Hence we can find a y-null set fl such that (*) holds for all rationals al,..., a, E 2 and all w 6 N simultaneously. Then, by continuity, the same estimate is true for all al,...,a, E X . Consequentlv

Finally, we get

17.3.9. As a special case of 17.1.4 or directly from the criterion 17.3.2 we have the

P r o p o s i t i o n . If rl 5 r2, then [prl, P,,] & [p,,,P,J. R e m a r k . The canonical map JrIfrom CEO, 11 into L,I[O, 11 shows that the inclusion l , c ?$,, * is strict for rl < r2 and 1 2 r, 5 00. On the other hand, it will be proved in 21.2.11 that all operator ideals ?J3, with 0 < r < 1 coincide.

17.3.10. Let us recall that 8 denotes the ideal of coiiipletely continuous operators. P r o p o s i t i o n . If 0 < r

< 00,

then?, cB.

Proof. Let 8 E ?$,(E, F ) . By 17.3.2 we have llh'x/\5 0

1l(x, a)]

p(

{US?

lIr

"')

for all x E E .

Hence, given a weak zero sequence (x,,),Lebesgue's theorem of bounded convergence implies lim IIh'xn]1= 0. n

Clearly, the identity map of ll is completely continuous, but not absolutely r-sumnling.

17.3.12. Let us recall that ?Eldenotes the ideal of weakly compact operators. Proposition. I f 1 < r

< 00. then [Q,, P,] = [m, 111.11 o [Vr, P,].

Proof. Let S E ?$,(E, F). As in the proof of 17.3.6 we define the operator Yo by Yo(J,f8) := Sx. It may happen that Yo cannot be extended to all of L,(UEp, p). However, there is an extension Y to Fo := M ( J , A ) .Since 1 < r < M, we see that Fo is reflexive. Moreover, the operator X := J,A transforms E into Fo and is absolutely r-surnming. This completes the proof. 17.3.12. The next result follows from 17.3.9 and 17.3.11.

P r o p o s i t i o n . If 0 < T

c 00,

then !& c m.

230

Part 4. Basic Examples of Operator Ideals

17.4.

c2,, y)-Dominated Operators

17.4.1. Let 0 < p , q 2 00. An operator S E B(E, F ) is called ( p , q)-dominated if it belongs t o the quasi-nornied ideal

[P(P4)’=(,,)I where l / r := IIp

:= [ P ( r . P , q P p(r,P,q)l

+ l/q.

17.4.2. We now generalize the criterion given in 17.3.2. Theorem. Let 0 < p , q < co. An operator S E B(E, F ) is (p,q)-dominatedi f and only i f there exist a constant CT 2 0 us well as probabilities p E W (U E , )and v E W (Up,,) .such tlmt

for nll x E E and b E F’. Here we suppme that UE,and UF#,are equipped with the weak E-topology and F’-topology, respectively. In

this case, D(p,q)(S) = inf

CT.

Proof. The snfficiency of the above condition is evident, since for .zl,..., x, and b,, ..., b, E F’ we have

E

5 C T W ~ ( WX q~()b i ) . Conversely, let S E 9 ( p , q ) ( EF) ) and put CT :== D ( P , q ) ( STake ). [C(UB,)x c(uF?,)]’ equipped with the weak C(U,,) X C(0’,*l)-topology. Then W(U,,) X W(UFr,)is a compact convex subset. For any finite families of elementjs xl, ..., x, E E and functionals bl) ...)b,, 5 F’ the equation

defines a continuous convex function @ on W (UE,) x W (UFTp). Choose a. E UE, and &‘ E Up,, such that and

j3 := wq(bj)=

17. Absolutely Suinming Operators

If a(%) and 6(y,”) denotes the Dirac measure at the point, a. and then we have @(a(ao), a(&))

237

&, respectively,

07

I(SZi, bj)“ - -

=

P

P

ar r

1 r

5 - lr((8Zi3hi))‘ - - (FB)’ 1 r

= - [l,((sxi,

bi))’

- ( m p ( z & ) wQ(bi))‘]

5

Since the collection .F of all functions @ obtained in this way is concave, by E.4.2 vo) 5 0 for all 0 E 3 there are p,, E W(UBt)and yo W ( U p ) such that @b0, simultaneously. I n particular, if @ is generated by the single families (x)and (b), it follows that 1

- l(Sr, b)j‘

r

-

“1

s

,(x,.)pJ dpo(n) - d 4

2,

l{y”, b p dv,(y”)

5 0.

Ufi.,

UE’

Finally, we put 112

:=

{ul

I(%, ~ > Idpo(a))l’p P

and n := {UJ, KY”,

w dvO(~’’)

Then

This completes the proof. 17.4.3. Theorem. Let 1 5 p , q [%(p,q), D ( p , g ) ]

< 00.

[(P:ual, p y 1

Then

p~l-

Proof. Suppose that S E g ( E , F ) can be written as S = Y A with A E P p ( E ,M ) and Y E pt“’(M, F).Given xl,...,x, E and b,, ..., b, E F’,it follows that L((SZi, bi)) = lr((Ari, Y’bi))5 1r(llA~illIIY’bill)

5 Pp(A) wp(zi) pQ(y‘)wQ(bi)* Hence S E %(p,Q)(E, F ) and D(p.Q)(S)5 P y ( Y ) P,(A). - l p ( A x i ) lq(y’bi)

Conversely, let S E % ( p , g ) ( E P ,) and put c := D(,,q)(8).By 1 7 . 4 % there are probabilities p € UB,)and Y E W (UFrp) such that

w(

238

Part 4. Basic Examples of Operator Ideals

for all x E E and b t P'. Let /,(a) := (x, a). Form the subspace M of Lp(UEr, p) spanned by all f, with x E E. By setting Ax :==fi we obtain an operator A f V P ( E ,M ) such that P,(A) 5 1. On the other hand, there exists an operator Y f !2(N,P)with Yf, = Sx for all 5 E. It follows from

IIY'b1/ = SUP {I(/,

Y ' b ) ] :lifzil 5 1) = SUlJ { ~ ( S X b)j: , llfiil 2 1)

that Y' E Vq(P',M') and P4(Y')

s CT.Moreover, X = Y A . This coinpletes the proof.

R e m a r k . Let us mention that

[P@X+ D,,,,,]

=

and

[ V p , Ppl

[P(,,?), D(,,,,I

= [P:ual, py1-

17.4.4. As a special case of 17.1.4 or directly froni the criterion 17.4.2 we have the P r o p o s i t i o n . If p , 5 p 2 and q1 5 q2, then

[P(P,4,),D(P,,q,)l sz

[%J,,r7,),

D(Pz.qs)l*

17.4.5. We now state a counterpart of the preceding inclusion. P r o p o s i t i o n . If 0

< p , q < 00,

then P(p,a) G

P(2,Z).

+

Proof. By 17.4.4 we may suppose that 11. := 1/23 l / q 5 1. Let US recall that d m denotes tho set of all vectors e = (cl, ..., E,) with c i = f 1. If xl, ..., x, E E and b,, ..., b, E P',then we put m

m

1

1

x e := 2' cixi and be := 2 cibi.

Using Khintchine's ineqnality we get

Hence wp(xe)5 Cp2m'PW2(xi). I n the same way we niay check that Wq(be)I Cq2m/4W2(bi).

P ) it follows froni Finally, if S E p(p,q,(E, m

17. Absolutely Slimming Operators

239

17.4.6. By 17.4.4 and 17.4.5 we have the P r o p o s i t i o n . If 2 5 p , g < 00, then P(p,q)= P(2,2).

17.47. Let 1 p 5 to the nornted ideal

00.

An operator S E f ! (E , P )is called p-dominaled if it belongs

[a,, Dpl := [*(l,P,P% R e m a r k . Obviously

17.5.

P(l,P.P*)I*

[a,, Dp] =

D(p,pe)].

Special Absolutely ( r ,p , q)-Summing Operators

17.5.1. We begin this section with the statement that for certain exponents all operators are absolutely ( r , p , g)-sumniing.

[a,

P r o p o s i t i o n . [*(l*l*l),P(1.1.1)I= 11.111Proof. Let xl,..., xm E E and a,, ..., a, E', where E is any Banach space. As in the proof of 17.4.5 we put

zcixi na

2, :=

I)(

and

a, := 2'

1

E~CZ~.

1

Then llzell-I ml(xi) and l\aell 5 wl(ai).It follows from m

2 (xi, ai) = 2-m2 (xe, a,) 6"

1

that

17.5.2. T h e o r e m . On the class of Hilbert spaces [Ip(r,z,z), P(r.z.~)l = [Gr, Srl for all 1 5 7

< 00.

Proof, Suppose that S E &(H, K ) , where H and K are Hilbert spaces. Given ...,x,,, H and yl, ..., y, E K , we put

~ 1 ,

m

X(Ei):= 2 tixi

m

and

Y(qi):= 2 qiyi.

1

1

Obviously X E 2(&,H ) and Y E 2(Z2,K ) . Moreover, IIXll SO, by 15.5.7, we have lr((szi,

~ i ) ) = 1r((SXei, Yei))2

= w2(xi)and

llyll = w&$.

Sr(s) w Z ( ~ i' ()7 ~ 2 ( ~ i ) .

Hence f i E rP(r,,,z)(a, K ) and P(r,z,z)(s) I Sr(fi)* Clearly, the identity map of l2 is not absolutely ( r , 2,2)-summing. Therefore the restriction of !&r,z,z) to the class of all Hilbert spaces is proper. We see from 15.2.2

240

Part 4. Basic Examples of Operator Ideals

that every operator S E !&,,,,,(H, 9)is approximable. Let

8=

qxi*@ yi I

be a Schmidt representation. Suppose that i E g(I).Then for all finite subfamilies ( x i ) and (yi) with i E i we get

(7 IQi

)

/ I l'r

-

4

1 (8% ?/d) 2 P,,.z,z,(4.

Consequently (oi)E & ( I )and l,(ai) 5 P,,,,,,,(S). This proves that ASE Br(H,K ) and S 9 w 5 P(r.z.z)(@. Remark. It can easily be shown that ?&r,z,z,, as an operator ideal on the class of all Banach spaces, is the superior extension of 6,; cf. 15.6.2.

17.6.

Relationships to Other Operator Ideals

17.6.1. Since p2 contains operators which are not approximable, we have QZ$ U2. Moreover, the converse inclusion also fails. Proposition. U24 !&.

Proof. Put o,,:= [ n log (n + 1)l-l and let S E 2(Z2,Zl) be the diagonal operator generated by (o,,). Then, according to 11.11.4 (theorem), we have 00

c c k

m o o

cU n ( S P = c Lp a; c 1

n=l k = n

k-1

00

0; =

=

n-1

c kI$

< 00.

1

Therefore S 6 ?&. On the other hand, it follows from (a,) absolutely 2-summing.

4 1,

that S cannot be

Remark. As observed by H.KONIG[4] we have a somewhat weaker inclusion, c !&, where 21(z,,) denotes the operator ideal generated by the approxinamely mation numbers and the Lorentz sequence ideal l(a,l).

17. Absolutely Summing Operators

241

17.6.2. Kext we state a n immediate consequence of 17.3.7. Theorem. [!&, P2]S [$j, HI.

17.63. P r o p o s i t i o n . [(p,, P2]o [$,HI G

[W,,A,].

P r o o f . Obviously it is enough to show that, for every Hilbert space H , it follows from S E '&(TI, F ) that S E %,(El, P ) and A2(S) 2 P2(S).For this purpose, according to 17.3.7, we consider a factorization

.t

A

Y c

such that / / A ,=( / 1 and /IY/l4 P,(S). We see from 17.5.3 that J2A E !&(H, L2(K,p ) ) implies J2A E U,(H, L,(K, p)) and A,(J,A) 5 1. Therefore Y = Y J J E W 2 ( H ,3') and &(S) 5 P,(S).

17.6.4. The precediiig results yield the Theorem.

[p,,PJ 5 [a,,

A,].

Reinark. Using the identity map from for O < p < 2 .

Zz into Zy we can easily show that

?&$%lP

17.7.

Notes

The concept of a n absolutely 1-summing operator (application senil-integral B droite) was introduced in the fundamental memoir [GRO, chap. I, pp. 134-1641. The main results concerning those operators are presented in the monograph [PIE, pp. 36-44]. I n 1967 the author has developed the theory of absolutely r-summing operators for 1 5 T < w ; cf. A. PIETSCH [ 5 ] . St about the same time B. S. XITJAGIN/A.PSLCZY~SSI [1] found the notion of a n absoliitely (r,p)-summing operator. Finally, the author defined absolutely ( r , p , q)-summing operators in the pre-version of this monograph. The famous factori7ntion theorem for ( p . q)-domi[6]. luted operators was proved by S. HWAPIEB Recommendations for further reading:

[LOT], [SEMI, rxp. 7, 27, 27bis], [SEM,, exp. 2 , 3, 16, 16b1s],[SEM,, esp. 10, 10-121. P. ASSOUAD[l], J. s. COHEN [2], CH. CCJNSTXNTIN[3], V. CRAIU/V. ISTRATESCU 111, J. DIESTEL "4, E. DUB~KSKY/M. s. RAXAXUJ.4N [I], [2], K. FLORET[I], Y. GORJ)EN/D. .J.

242

Part 4. Basic Examples of Operator ldeals

Lmrls/J. 1%.RETHERFORD [ i ] , [ 2 ] , A. GROTHENDIECK [5], G. H. H ~ R D Y /k3. J .LITTLEWOOD [I], A. JOICHI [i], &I. K ~ T O [l], 8. RwIPIE~~/A. PELCZY~~SKI [i], A1. J. KADEC[l], J. TK. LAPRESTE [l], Z. LINDEP\TSTRAUSS/L%. PF: C Z Y ~ S K I [ i ] , B. XAUREY[ 2 ] , B. MAUREY/A.PELC Z Y ~ ~ S K[I], I I<. MIYAZAKI [i], [ 2 ] , W. ORLICZ [ i ] , A. PELCZY~~SKI [Z], A. PIETSCH [3], [4], L8], [I@], A. PIETSCII/H.TR~EBEL [i], M.S. RAMANUJAN [ 2 ] , J. R. RETHEREORD [ Z ] , J. R. RETHEKFORD/C. P. STEGALL [I], J. 1'. RICE [ l ] , D. ROBERT[ I ] , B. ROSEXBERGER [I], S. SIMOTS[l], I. SINGER[i], [?I, 1%;. SLOM-KOWSKI [ i ] ,I. STEPHANI [i], C. SWARTZ 131. P.SAPEAR ~31,~41,I. I. TSEITLI~U [a].

18.

Nuclear Operators

The theory of nuclear operators was created in 1936 when E. J. MURRAYand J. VON XEUMANN investigated the problem “What operators in Hilbert spaces have a welldefined trace?”. They found the ideal Q1 of trace class operators. In the earlier fifties B. GROTHENDIECK and A. F. RUSTON independently extended this concept to operators acting in Banach spaces. I n the following we consider operators admitting a representation

s=

m

C7’d.i

@

yi

1

such that the right-hand series converges in a certain sense. In particular, the condition 00

z:bil llaill IlViIl <

O0

1

yields the classical nuclear operators. Furthermore, we deal wit,h the following operator ideals:

@, q)-compact operators,

%

r-nuclear operators, -4

%J

p-compact operators.

The connection between r-nuclear and absolutely r*-summing operators is given by the formula = Q,..Analogously we have 33; = Pp*. The final section is devoted to the ideal of strictly nuclear operators which has Illany remarkable properties.

18.1.

( r ,p , q)-Nuclear Operators

18.1.1. Let 0 < r 2 00, 1 5 p , q 5 00, and 1 S E B(E, P)is called (r, p , q)-izuclear if m

s = ,y C7jd.i 02Ji 1 1

c*

+ l / r h l / p f l / q . An

operator

244

Part 4. Basic Examples of Operator Ideals

with (ui) E I,,

(ai) E

IQ(E’), and (ye)E mPt(F).In thc case Y = 00 let us suppose that

(4 E cb. We put N ( r , p , q l ( f J ) := inf U G i )

wq4aO wpt(yi)9

where the infimum is taken over all so-called (r,p,q)-nuclear representations described above. The class of all

(Y,

p , q)-nuclear operators is denoted by W(r,p,q). W

Remark. The series 2’ oiai @ yiconvergesin the norm topologyof 2(E2P).There1

fore the operator S is approximable.

K(l,l,l)] = Remark. Obviously [%(1,1,1), 18.1.2. Let 11s := l / r

[a,K].

+ 1/p’ + l/q‘. Then we have the

Theorem. [!R(r,p,e),W(r,p,q)]is an s - w m e d @perustorideal. Proof. We use criterion 6.2.3. (0) It follows from Is = 1 @ 1 that Iz E !Jl(r,p,q, and K(r,p,y)(IcT) 2 1. Moreover, let bo

I&- = 2 U& 0qj 1 Di)

be any ( r , p , q)-nuclear representation. Then 1 = 2 G 1

Consequently,

~ L Y ~ ~ ~ .

18. Nuclear Operators

+ &)b2' W

wp*(Yni)

5 (1 t

N(r,p,q)(fJn)'

1

Consequently W

S

:=

C

2S n =

n=l

O

I,,. *

W

uniani

i=l

11=1

245

0ynj

i8 an ( r , p, q)-nuclear operator with N(r.p,q)(W

5 (1

00

+

2 N(r.p,q)(Sn)s* 1

The case 7 = 00 requires some modifications. We have to ensure that the double sequence (ani)tends to zero. For this purpose, according to 8.6.4 (lemma), we choose a sequence (en) E c, with 0 < en 5 1 and m

m

.L' e 2 N c c e . p , q ) ( S n ) * 5 (1 + E)* L ' N(co,p,qt(SnY. 1 1 Then there are (00,p, q)-nuclear representations 03

fJn

Zonaani 0Yni i=l

==

uch that, for every fixed n,

5 en,

U U n d

wq,(a,t) d wp*(yni)

+ +

[e,'(l

5 [e,'(l

E ) %cu.p,q)~Sn)ls'Q'r

8) N ( m , p , q ) ( ~ n ) Y ' ~ ' -

Now for the double sequences we have lm(bni)

5 1,

Consequently W

00

S := 2'8,

c

u

=

n=l

n=1 i = l

unianiG J yni

is an (ce,p, q)-nuclear operator with

5 (1 + &I2'

~(m,p.qi(f9p

00

21 N(oo,p,g)(Sn)S-

( 2 ) Let S E !Jt(r,p,q)(E, F ) and.&> 0. Consider an ( r , p ,9)-nuclear representation

s=

00

diUi 1

g) yi

246

Part 4. Basic Examples of Operator Ide,tls

-

such that l,(Ot) lVq,(a,)

JVp’(Y2)

5 (1

+

c) y v , p

P)(S)*

I€ T E B(Eo,E ) and R E & ( F ,P o ) ,then ns

RST

=

2 o,T’a, @ Ry, 1

and Ir(0, ) TV,,(T’C(,)

wp,(Ry,) 5 ( 1

+

8) IIRll 3 ( r , p , q ! ( f 7 ) IITII.

s o RSI’ E %(,,,,,,(E, F ) and ;I\-(,,,,,,(RS’T)f lIRll y , , p , , d @

IlTlI.

18.1.3. Analogously to 6.3.3 we now characterize ( r , p , q)-nuclear operators by a factorization property. T h e o r e m . An operator X E e ( E ,F ) i s ( r , p , q)-izuclecir i f and only if there exists commutative diagram

c(

1.

A

+I

such that So E &(I,., 2,) i s a diugonal operator of the form So(:,) = (GJ?) with (uz) I, if 0 < r < 00 a n d (u,) E co if r = 00. Furthermore, -4 5 e ( E . l,,) and Y E i?(l,. F ) .

I n tlbis case,

%.P,,)(S)

= inf

IlYlI

M%)I l 4 >

ushere the infimum is taken over all possible factorizations. Proof, We can use the same method as in 6.3.3. Only the case p = 00 requires some small modifications.

18.1.4. The proof of the following result is also straightforward; cf. 8.6.4. Theorein. The quasi-normed operator ideal [a(,).,

S(,,,,,)]i s miizimrrl.

18.11.5. P r o p o s i t i o n . Let r1 5 r 2 , pl =( p s , a i ~ dq1 5 q?. Furthemore suppose thut l / p l + Pigl - l / r , =( 1/37? l / q , - 1 / r 2 . Then

+

C=

[%(rl,pl,ql)> ~ ( v , . p l . q , ~ I [ % ( r 2 , p 1 , q p ) ’ 1 \ ~ ( r , , p , , q ~*j ]

+

P r o o f . Put I/. := l / r l - ( I / p I/q), 1/13 := I/pl - l/ps, l / q := l/ql 1 ‘qr. Then r 5 r2. Let S E %(,,,PI,ql)(E. F ) . Given F > 0, we choose a representation

s =-

cx)

uza,@ y, 1

~

18. Nuclear Operators

217

such that u, 2 0 and

lrl(ui) wq;(ai)w p ; ( ~ i )2 (1

+

E ) ~ ~ s , , p , . ~ , ) (* f l )

$Ve have W

s = 21 uy(u:"Qcci)@ and lr2(u:'r)

5 lr(fy)

(Oy+Ji)

= Irl(ol)rl'~,

wq;(@QU i ) 5 lq(Cp)W&)

= lrt(op'q\V&$),

5 i P ( p ) wp;(ya)= 1r1(uj)7Jp wP,.(dl'pyi) I

W~;(Y~),

Hence8 is an (r2,p,, q,)-nuclear operator with K(r,.PI,q,)(X)5 (1 f

E)

N(rl,p,,ql)(~S').

18.1.6. We now forinulate, without proof, the evident Theore111.

[%(r.p,g), x(r,p,q)ldUal = W ( r , q , p l >~(r,9,p)Ires*

18.1.7. The preceding result leads to the following Problem. Find all exponents such that the quasi-normed opcrator ideal is regular!

[ % ( r , p , q ) , %,P,*)I

Remark. The regularity is obvious for 11 = 2; cf. 18.1.8.

18.1.8. Using the same technique as in 8.4.9 we get the Proposition 1. The quasi-normed operator ideal [%(r,z,9). S(,,,,,)]is iwjectite. The dual statement is also true.

Proposition 2. The quasi-nornzed operator ideal

[?Jl(r,p,2), S ( r , p , 2is ) ]surjectice.

18.1.9. We now prove a remarkable result which at first glance looks very surprising.

+ l/r > l/p + l/q, then the quasi-nornwrl operutor

Theorem. If 0 < 1 < cx) and 1 ideal [%(r,p,q), X(r,p,q)]is ultraatable. 00

Proof. Consider an N(,,,,,,-hounded farliily of operstcrs S , !Jl(r,p,9)(Ez, F , ) and put := sup ( N ( r , p , q ) ( , Q Li )E: I ] .Given u > q,,we choose ( I , p , 9.)-nuclearreprcsentations W

si = 2 oikark @ Y i k h=l

Such $hat 1,(ark)5 O, wq.(aik) 5 1, wP*f&&) =( 1, and IS fixed. If ok := li?n u2k,then

bil

2 C T , ~1 --.2 0, where i

U

lr(ck)5

ul 2 c2 2

2 0. (1) On the other hand, it follows from wqp(aik) 5 1 that for all finite sequences (&) with B

5 1 we have

and

248

Part 4. Basic ExRmples of Operator Ideals

This proves that Wq*(Uk)

5 1 , where

:= ( U i k ) n .

(2)

yk := ( y i k ) n .

(3)

Analogously we get

5 1, where

Wp’(&)

The proof will be complete if we can show that the ultraproduct S := (Si)uadmits the representation

z OD

S

=

akak@ gk*

1

Since 1

+ l / r > l i p + l / q we can find r, with 0

< r < ro < 00

and

l/so:= l/ro 4-l/p’

+ l/q’ > 1.

7n

By muim5 2 U i k 5 or we have k=l

Hence, given E

{5

> 0,there exists n, such that 1lro

uz}

5 E for all i E I and

n

L no.

k=n+1

Let, x = ( x J U with zi E UE, and put 7a

Uin

:=

uk(z,

a k ) yik.

I;=1

Choose bin E Ups.such tha.t [Ispi- uinli= (Sixi - uin,bjn). It follows from

that

We can find I, E U such that

Moreover,

18. Nuclear Omratom

249

Hence S adinits t.he desired representation OJ

s = 2 aka, @ ?/ka 1

Using (I), (2),and (3) we see t,hstS is an (r,p,q)-nuclear operator with Ntr,p,q)(S)5 0. 18.1.10. For every operator S E 8(El P)we put

%,p.*,(S) :=inf Wi) W q

W Wp’(Yi)

9

where the infimum is taken over all finite representations n

8 = C aiai @ yi. 1

18.1.11. Using the method of 18.1.2 we get the Proposition. N?,,,,, ii? an s-nwm on. &. 18.1.12. We now show that the s-norm N:r,p,qhis regular. Proposition. If S E &(If, P).then Nyr,p,q)(B) = N?r.p,g)(fCFS)

> 0, we consider a finite representation

Proof, Given E

n

KFS = C ciai@ yy 1

such that

I,(%)

W,,(%)

w-&3 5 (1 + E ) N:,p,*)(KFS) *

Let M be the linear span of yy, ..., ?J; E F”. By E.3.1 there exists Jo E g ( M , F ) with llJoli 5 1 + e and KFJoy” = y” whenever y” E M n H ( K F ) . If yi := J&r, then it follows from KFSX = KFJOK~BX= K F

250

Part 4. Basic Examples of Operator Ideals

that

Consequently

respectively, we have where e is a positive constant. Given S E 2(E, P)and E

> 0, we choose an ( r , p , p)-nuclear represent.ation

Q)

S = 2 o,ai @ yi 1

such that

Then there exists n wit,h

2 [(I

+ + 4 N(r,p,q)(4s. ElS

This proves that X:r,p,q)(S) (= N(r,p,q)(S). The converse estimate is trivial. 18.1.14. The following results can be checked using the same technique as in 6.8.4. Lemma 1. Let X E g(E,, F )and 8 E !R(r.p,q)(E, F ) . Then

NPr,p,q)(SX)5 X ( r . p . q ) ( @ IIXII

-

18. Nuclear Oporators

251

Lemma 2. Let S E %(r,p.q)(E, F ) and B E g ( F , Po).Then

-

2 IIBII % , p . q @ )

NP,,,,,,(BS)

18.1.15. Analogously to 10.3.1 we have the following iniprovenient of 18.1.13. Proposition. Let E' or F possesses the metric approximation property. Then

q,,p,q,(s) = N(,.,,,)(S)

for

flzz

sE W E ,F ) .

18.1.16. For spocial exponents the above result holds without any assumption on the underlying Banach spaces. Proposition 1. for all

qr.z.q)(fJ) = %,z,g)(s)

s E 5 ( E ,F )

a

Proof. We know from 18.1.8 (Proposition 1) that the quasi-nornied operator ideal

[!lt(r,2,q), N(r,2,q)] is injective. Hence, without loss of generality we may suppose that F is finite dimensional. Analogously we get Proposition 2. N:r*p,2)(s)= N,,,p>2,(s)

18.2.

for all

s E 5(E,P) *

**-NuclearOperators

18.2.1. Let 1 2 r I_ 00. An operator S E c ( E , F ) is called r-nuclear if it belongs to the normed ideal

[%,, N,1 := [%,,,I),

%,r,1)1.

Moreover, we put Nf(S) := hyr,,,lJS) for all finite operators.

18.2.2. Proposition.

[a,, N,] c [!&, P,].

Proof. Given S E Sr(E,F)and P

> 0, we choose an r-nuclear representation

co

S=

ciai@ yi 1

such that l,(ci) = 1, w,(ai) = 1, and w,.(yi) 5 (1 11~x11 2 (1

+

&)

N,(#)

K

1/7

0;

~(z,ai)lr}

+

8)

R,(S). Then

for all x E E .

This implies, by 17.3.2, that S E !&(I#, F ) and Pr(S) 5 lu,(S). The canonical map J , from C[O, 13 into L,[O, 11 is absolutely r-summing, but noncompact. Therefore 3,+ Q,.

18.2.3. As a suppleinent of the preceding statement we forinulate the Lemma. I/ s E ~ ( z Fz),, then w:(S) = P,.(S).

252

Part 4. Basic F:xsmplos of Operator Ideals

Proof. By 17.3.3 t h m arc numbers pl,...,pm2 0 such that

111

pi = 1 and 1

P u t y i := p;'"Sei. Then for any choice of

rl. ...,

)I,,,

E

X

we have

Heme w,.(?yt)Z Pr(S).It follows from m

= L1 &*ei

3 yi

1

that

B:(s) 5 lr(p:'r)W W ( C i ) wr.(yt) I P,(S). The converse inequality is evident bp 18.2.2.

18.2.4. The following inclusion is an immediate consequeiice of 18.1.5. P r o p o s i t i o n . I f rl 5 r,, t 7 w [!Jtrl, N,,] G [?RrI,XrJ. R e m a r k . Suppose t h a t rl < r2, and let (a,) E I,, \ Irl. Then the corresponding diagonal operator IS belongs to %,a(Z, Z,J. However, it follows froin S $ ?@Tl(Zm, Z,) that 8 fails to be r,-nuclear. Therefore !It,, c Rrt.

18.2.6. We now state a fundamental result which is a special case of 18.4.5. Theorem.

18.3.

[a:,-U]:

=

[p+,€',*I.

p-Compact Operators

18.3.1. Let 1 5 p 2 00. An operator S E B(E,P) is called p - m n w if it belongs to the normed ideal [Rp, K p l

:= m a ? , p . p ' ) ,

~~w,p.pf)l.

Moreover, we put KE(S) := X:m&.p,ps)(S) for all finite operators.

18.3.2. The phrase "p-compact" is derived from the following property. Theorem. A?z operator 8 E B(E,F)is p-wmpa.Ct if and only if there ex&& a mmuWive dingram

18. Nuclear Ouerators 111, this

253

case,

K p ( 4 = inf IIYII IiAll , zchere the

h / ? h 6 U m is

taken over all possible factorizations.

Proof. The necessity of the above condition follows froin 18.1.3. The sufficiency is also evident ; cf. 19.3.2 aud 19.3.7.

18.3.3. We now state a fundnrnental result which is a special case of 18.4.5. Theorem.

18.4.

[R;,KE] = [ap*, D,.].

(p, q)-Compact Operators

18.4.1. Let 1 5 p , q 2 00 and l / p 4-l / q 2 1 . An operator S f e ( E ,P) is called ( p , q)-compct if it belongs to the norined ideal [%J,q),

K ( P . d :=

where 117 := l / p operators.

+ l/q

[%r.p.q),

-

N(r,P.*)I,

1. Moreover', we put K:,,,q,(S):= X!,,p,q@) for all finite

18.4.8. The following inclusion is a special case of 18.1.5. Proposition. If p1

p2 and q1 5 qz, then

K(P,,*,)l 5 [%P*.l*)> K(P,.P*)I * 18.4.3. Leinnio. If 1 < p 5 q' < 00, then [%P1.W

K ( p , q ) ( II :; -+ )7;

5 cpaq*.

Proof. Let

be the representation described in 6.3.4. It follows from Khintxhine's inequality that

where (e) denotes the family of all e This means that wp.(e)5 2nlJ"cp,, ously we get wq,(e)5 Zn/q'cq,.Consequently

K ( p , q ) ( I2;: -+

&*. Analog-

Zi) 5 2-n211'r~-p4e) wpe(e) 5 cp,cpf.

Using the method of 6.3.5 we obtain the Proposition. If 1 < p

q'

< 00, tWen$+,,,,

R(,,,).

18.4.4. Now 18.4.2 and 18.4.3 yield the = a,,,?,. Proposition. If 1 < p , q 5 2 , thela 8(p,p)

18.4.5. Finally, we cheek estimate is required.

ZL

fundamental formula. For this purpose the following

{<

& (/,xi'

r*

bi)jr*

I K7ip,&7) ,Vp*(Xj) W q * ( b J .

-

18. Nuclear Ooerators

18.5.

255

Special ( r , p, q)-Nuclear Operators

18.5.1. Generalizing 6.7.2 we get the Theorem. [!R(r,l,l),R(r,l,l)]i s the smallest r-normed operator ideal. 18.6.2. Furthermore, we have the interesting T h e o r e m . On.the class of Hilbert spaces PX(r,l,l), q,,1,1,1

= [Gr, S'l

for 0

< I 1.

Proof. Let S E !R(,,,,,,(H, K ) , where H and .K zre Hilbert spaces. Given e > 0, we can find a factorization S = YSoAsuch that ljYlj l,(oi) ilAi] 5 (1 E ) N(,,l,l,(X). The diagonal operator So E B(l,, Z1) is generated by a sequence (ui) E &. Obviously we may suppose that oi 2 0. P u t S;(Ei) := (offi). h'ow the following diagram commutes :

+

s

H

+K

+

I

If l/p := l / r - 1, ehen

Si-*E Gp(Za,1,)

and S,(St-') = lT(oi)l-*.

It follows from 15.5.4 (Lemma 1) that

S;'A E G2(H,1,)

and S2(S;d2A)5 ll.(bi)7'2IIAII,

YS;, E G ~ ( zK~), and S 2 ( y S 32 ]iYil lr(oi)*'z. Hence, by 15.5.9 and lir = 112

S,(S)

+ l/p + 112, we have S E G,(H, K ) and

r S,( YS;jz)s,(S;") S 2 ( S 3 ) 5 lIYll Mod l I 4 5 (1

+

E)

N(',l,lm-

This proves that

w,T',l.l)l s [G'(H, K ) , %I.

[%,,l,l)(~,

The converse inclusion follows immediately from 15.5.2. 18.5.3. By a n obvious modifioation of the above proof we get the Theorem. On the class of Hilbert [%(?,1,2),

y r , l , z ) l = 1 6 , S'l

~ p p a c e s

for 0 < r 5 2 .

266

Part 4. Basic Examples of Operator Ideals

18.S.P. The following result is trivial. Theorem. On the du.ss of Hilbert spaces ~ ( r , z , z )= l [Gr,S,l

[%,2,2)3

for

0
Remark. It can be easily shown that % t ( r , 2 , 2 ) , as an operator ideal on the class of all Banach spaces, is the inferior extension of 6,; uf. 15.6.4.

18.6.

Relationships to Other Operator Ideals

18.6.1. Theorem. Let l / ~-, 1 + ljr - ( l / p + l/q) > 0 and l/p a(r,p.q)

+ l j q 2 1. I'hm

2 %a*

Proof. Clearly r 5 ro, q p', and l / q - l / r 5 l/p' - l / r o . So, by 18.1.5, we have ?,U(r,p,q,E ?&~re,p+p~~. Suppose that S E !R,r,p,q)(E, P).Thcn we can find a factorization S = Y&A such that So E i!(lp, I,) is a diagonal operator generated by (ui) E I,e. Applying 11.11.3 we get SoE a,,(Z,, I,) and therefore S E %,,(E, F).This completes the proof.

Remark. It can be seen from 11.11.4 (theorem) that in the above inclusion the exponent r, is the best possible one. 18.6.2. As a suppleiiient of thc preceding result we have the Theorem. Let 0 < r

< 00

(xnd

ljp

Proof. It follows from q 2 p' that

+ l / q 5 1. Then %tr,p,q)

%(r,p,p)

E a,.

E !R(r,p,p) & a,.

18.6.3. Lemma. Let S g ( E , P) and rank (S)= n. Then there exists u finite rep esmtation I&

s = 2 oiaj :3yi 1

szzh t?mt luij 5 11511, lfaill = 1, and l\yiil = 1.

...,y,!

Proof. According to B.i.8 we can find yl, = 1, iibkll = 1, ailti (yL,bk) = t i k . Hence

X ( S )and b,:

...; b, E P' with

11

Sz = 5 ,Sn,hi) y, for all

zE

E.

1

By srtting oi := IIS'bJI and a , :=- o;'S'bi

we obtain the desired representation.

We are now ready to check the Theorem. If 0 < T 5 I, then 8, G %(r,l,l).

F ) , we choose L, E

f'toof. Given S E

[IS- L,,ji If S,

2aZn(S) and

8 ( E , F ) such that

rank (L,) < 2" for n = 0, 1, .

..

:= Ln+, - L,,then

llSnll

jjS

- L,// + //S-

I&,(A')

and rank (S,) < 2nks.

18. Nuclear Operators

257

Take finite representations 'p+l

S n = L' aniani 0~

n i

I-1

llynill 5 1. Obviously

such that lanil 5 IISnll,llanill 5 1, and 00

n=O

m

Z"+P

X 2

IGng!'

5 C 2"+' IISnll' 0

i=1

W

< 23+2T d y 2n-1 az.(S)r

-

0 m

52

< Q3.

3 + 2 u~k (~~ ) r 1

Hence the operator 2"-1

W

s = sn= /y

/y

n=O

i=1

0

is

(T,

bnifl,,

Q yn,

1, 1)-nuclear.

R e m a r k . Since 91?(Z2, Z2)

= ?&(r,l,l)(Zz,

Z2), the above result cannot be improved.

18.6.4. It follows from 11.11.4 (theorem) that ?&(7m, for any r . However, we have the

Theorem. R"

ZI) is not contained in 21T(Zw,Zl)

91,.

Proof. Let T E ?&(E,F ) and S E R(F, G ) . Choose nuclear representations 03

m

1

1

T =z 'a i 0y i and S =

b, @ zk

such that the sequences (a&), (yi), (bk),and (zk) are absolutely d-summable. Define the operators A f O(E, 4) and 2 E 2(Ez,G ) by m

Az

:= ((2,a,)) and

Z(6,) := 2 icrzi. 1

Since m

w

2 c I(!/Ji, i=l k=l

00

bk)i2

W

5i 2 k?/i1 12 IIbk112, c =1 k-1

we know from 15.5.6 that the matrix ((yi, bk))generates a Hilbert-Schmidt opentor E e(Z2,Z2). Finally, it follows from ST = ZMA that ST E U2(E,G).

18.7.

Strictly Nuclear Operators

18.7.1. An operator S E O(E, P)is called strictly nuclear if (an(@)E lo. The class of all strictly nuclear operators is denoted by so. R e m a r k . The sequence ideal lo has been introduced in 13.2.14.

*

17

PiPtsrh- Oncrntnr

258

Part 4. Basic Examples of Operator Jdeals

18.7.2. We now show that in the definition of Rothe approximation numbers can be replaced by arbitrary s-numbers. P r o p o s i t i o n . Ro= n U p= n BP. O
O
Proof. The assertion follows froin 9lP cf. 14.2.13 and 14.2.18.

s 8,and 8,E U,, where l/p = l / q f 312;

18.7.3. Clearly Rocan bc generated by certain ideals of nuclear operators also. P r o p o s i t i o n . lo = n O e e l

=

n %(,,2,2,.

o
Proof. By 18.1.5, 18.6.1, and 18.6.3 we have %(ro.l,l)

where l/ro = l / r

E %(r*2,2) 5 a, s %.l,l),

+ 1.

18.7.4. The phrase “strictly nuclear” is derived froin the following property. T h e o r e m . An operator S E B(E,3’)is strictly nuclear if and only if b3

s = 2 (TicLi 0yi 1

such that (ai) E lo, [la115 1, and liytll 5 1. P r o o f . Let S E Ro(E,F ) . We choose L, E g ( E , F ) such that

5 2nn(S) and rank (L,) < n. for

1 1 8-

n = 1 . 2 , ...

If S,:= L,,, - L,, then \lSnii5 IjS - L,j/

+ /IS- Ln+J5 4a,(S)

and rank (&)

< 2n.

According to 18.6.3 (lemma), take finite representations 2 11

2 c n i a n i 0Yni *=1

Sn =

such that loni]5 I]Xnli, llanill 5 1, and ]jynijl 5 1. Obviously w

2n

b3

m

2 2 lonilp 5n2 2n llXnllp 5 2 1 + 2 P 2 m,(S)p < n=l i=l =l n=l whenever 0 < p

00

< 00. Hence the operator S admits the required represeiitetion

1

Conversely, suppose that

s=

00

oiai @Jyi, 1

-.-

2 0. We see where (ci) E lo, ]laill 5 1, and llyill 5 1. Moreover, let crl 2 0 ,2 from 14.2.13 (lemma) that (i2ci)E lo, as well. Define the operators A E g ( E , Zz),

18. Nuclear Operators

so E % 1 2 , 1 2 ) ,

and y €

259

O(L F ) by W

Ax

:= (i-l(z,a J ) ,So(fi) := (i2aiEi),and Y ( q i ):=

1

i-lyiyi.

Then S = YD0A.Finally, So E Ro(Z2, Z2) implies S E Ro(E,F ) .

18.7.5. The preceding proof also yields the following criterion. Theorem. A n opera.tor S E B(E,F ) .is strictly nuelear i f and o d y i f it admits a factorization

t

/j where So E .t?(Z2, 12) i s n dicryonul operator of the form S,(ti) = (o,ti)with (a,)f I,. Furthermore, A E O(E, Z2) and Y E 2(Z2,F ) . 18.7.6. Prom the definition of ?Xowe get immediately the Theorem. ? . isI & an operutor , ideal. R e m a r k . Obviously the family of quasi-norms (A,) with 0 < p < 00 yields a topology on Rosuch that the algebraic operations are continuous. So we inay say that Ro is a (complete and metrisable) topological operator ideal. However, since Woc !R(r,l,l, for 0 < r 5 1, this topology cannot be generated by a single quasinorm; cf. 6.2.6 and 18.6.1.

18.7.7. Theorem. The qperator ideal ?J is l completely o symmetric, injective, and mrjeetive. Proof. By 18.1.6 and 18.1.8 the assertion follows from =

n

?x(~,~,~).

o
18.7.8. T h e o r e m . The operotor ideal goi s minimal and nmximal. Proof. The minimality is a consequence of 18.7.5. Moreover, we see from 14.3.7 that Rois the intersection of maximal operator ideals; cf. also 18.1.7 (remark) and 18.1.9.

18.7.9. Let us agree that Go denotes the restriction of ?Xo to the class of Hilbert spaces. T h e o r e m . The operator ideal 3, iqs the unique extension of Go to the class of all Banach spaces. Proof. Suppose that U is any operaior ideal such that a ( H , H ) = %,,(H, 11)for all Hilbert spaces H . e I?*

260

Part 4. Basic Examples of Operator Ideals

(1) Let S E U(E,F ) . By 11.4.3 we can choose X, E i?(Z!, E ) and B, E B(F, Zi) such that IlXnll 5 1, llBnll 5 1, and B,SX, = enl, with 2p, := h,(S). Form the Hilbert space H := Z,(Zg). Put m

X :=

00

n-lXnQn and B := 2' n-'J,B,. 1

1

Then BXX E %(H, H ) = Ao(H,H ) . Therefore (a,(BSX)) E , have since Q,(BSX)J , = r 2 p n I nwe

.,I

On the other hand,

h,(S) = 2p,nn(ln) 5 2n2an(BSX). Now it follows from 18.7.2 that S f !Ro(E,F ) . Hence U 5 Ao. (2) Let S E !Ro(E,F ) . Then we consider a factorization 8 = YSoA according to 18.7.5. Now it follows from SoE %o(Z,, 12) = U(Z2,Z2) that S E U(E, F). Therefore

A,

s II.

18.7.10. Theorem. The operator ideal A, i s idempotent. YS,A as described in Proof. Let S E A,(E, 3') and take a factorization S 18.7.5. Here SoE k?(Z,, Z2) is a diagonal operator of tho form Xo(Ei) = (biti) with (oi)E 6 and ci2 0. Put AS'~'~(E~) := (aif2Ei). Then St'* !R0(/,, l,). Hence we have S E (%o)2(E,F ) . This proves that !Ro G (Ao)2. 18.7.11. At this point Q denotes a bounded open subset of the n-dimensional Euclidean space 9%. Let H ( Q ) be the collection of all harmonic funct,ions admitting a continuous extension to It turns out easily that H(Q) becomes a Banach space with respect to the norm lifil := sup ( [ f ( o )Q[ :E O).

a.

Proposition. Let Q and Q, be bounded open subsets of 9*such that $2 the canonical map from H ( O ) into H(.Qo) i s strictly nuclear.

3

a,. Then

Proof. First we show that the canonical map I is absolutely summing. For this purpose we choose a number e > 0 such that

K,(wo) := (w E 9%: IIW - wo/I 5 @) belongs to 9 for all wo E Go.If V,, denotes the volume of the n-dimensional unit ball, then the mean value property implies If(wo)l

1 sQ"F,

1

e n v nR

",(WP)

This proves that

IlIfIio

IJ' lJ'

If(o)ld o 5 - 1 f ( w ) ; do,.

:= sup {lf(wo)l:

0,

f S2,l

5-

if(w)i do.

P

Therefore, the canonical map I is absolutely suninling by 17.3.2. Let 0 < p < 00 and take a natural nuniber n with np 2 2. We now choose bounded open subsets s2, such that Q = Q,,

3

Bj2n--1 3 -..3 9, 3 a, 3 Qo.

18. Nuclear Operators

If 4 denotes the canonical map from H(f2,) into E1(f2k-1), then 12&+ This proves that

I = (Id2n-I)

**.

(1'211)E (Udn G

4221,

261

E 912 by 17.6.4.

E gp.

Consequently I E R,,.

18.8.

Notes

As stated before the concept of a 1-nuclear operator is due to A. GROTHENDIECK [l] and A. F. RUSTON[l], [Z]. I n 1969 the theory of r-nuclear operators for 1 r < 60 has been PIETSCH [l], and P. SAPHAR [Z]. Thr: case r = 2 developed by S. CHEVET [l], A. PERSSON/A. [l]. I n the pre-version of this monograph the author defmed so-called goes back to P. SAPHAR ( i , p , &nuclear operators. If 0 c: r 5 1 and p = q = 1, then one gets n class of operators which wes already treated in [GRO, chap. 11, pp. 3-31]. Another special case is the ideal of those operators admitting it compact factorization through 2,; cf. W. B. JOHNSON [l] and -1.PIETSCH [ll]. Strictly nuclear operators were first studied in [GRO, chap. 11, pp. 6-91.

Recommendations for further reading:

W. BAUHARDT [l],B. S. BRUIIOVSKIJ [l], J. DALORD [l], E. DUBINSKY/M. S. RAMANUJAN [Z], T.GOBDON/D. R. LEWIS/J.R. RETHERFORD [l], [2], C. V. HUTTOK[2], H. JARCHOW [l], V. K. KHOAX[2], J. TH.LAPRESTB [l], K. MIYAZAHI[3], R. OLOFF [l], A. PIETSCH [7], [S], [lo], M. S. RAMANUJAK [l], D. J. RANDTKE [l], 121, [3], J. R. RETHEBFORD [ 2 ] , B. ROSENBERGER [l], 131, T. TERLIOGLU r21, [3], [4], I. 1. TSEITLIN [2], G. R. WALKER [l].

19.

Integral Operators

In this chapter we investigate the ideals of (r, p , q)-integral operators which are defined to be the maximal hulls of the corresponding ideals of ( r , p , p)-nuclear operators. While the (r, p , q)-nuclear operators admit a “discrete” factorizat.ion

E

s

+P

the ( r , p , q)-integral operators are, roughly speaking, characterized by a “continuous” diagram

.I

4.

Y

where 8, is a diagonal operator of the forin Sf := sf with s E L,(S,p). The main tool for proving those factorization theorems is the ultraproduct technique developed by n. DACUNHA-CASTELLE and J. L. KRIVINE. The ideals ?.Itr, 3,,and gr* of r-nuclear, r-integral, and absolutely r*-summing operators, respectively, are related to each other by the following procedures:

%

niaxinial huII mininiak kernel

a%* The situation for the operator ideals S,, diagram.

eP,and Pp. can be described by a eimilar

19. Integral Operators

263

(r, p , q)-Integral Operators

19.1.

+

19.1.1. Let 0 < r g 00, 1 5 p , q 5 00, and 1 -t ljr 2 1/13 l/q. An opelator S E E(E, P)is called (r,p , q)-integral if it belongs to the quasi-normed ideal := [%r.p.q),

[z(r,p,q), 4r.p.q)I

Remark. By 8.7.6 we have

N(r,p,q)Imar-

[ 3 ( l , 1 , ~ i ,1(1,1,1)] =

[a, I].

t9.1.2. As a counterpart of the preceding definition from 8.7.14 we get the Theorem. [%r.p,,,2

N(r*P.q)I = [%,P,9),

I(r#Pd?)p i n .

19.1.3. The next result follou7s immediately from 18.1.5. Proposition. Let r1 l/pl l/ql - l/rl 5 11232

+

5 r4, p1 5 p2, and

q1 5 q2.

+ 1/42 - l / r ~ Then .

Furthermore suppose

c

[3(rt.p,.qx), I ( r l s p , d

2 [+,.p,.q,),

4 r s . p s d*

19.1.4. By 8.7.10, 8.7.12, and 18.1.6 we have the Theorem.

dual

P ( , , p , q ) , I(,,p*q)l

- c - b ( T . Q , P ) > I(r,q.p)l.

19.1.5. We now observe that the definition of ( r ,p , q)-integral operators, for certain exponents, yields nothing essent’iallynew. Theorem. Let 0 < r

< 00

[ 3 ( r , p , q ) . I(r,p,q)I

and 1

+ l / r > l / p + l/q. Then

= [ ! R r * p , q ) , N(r.p.q)lceU*

Proof. The assertion follows from 8.8.6 and 18.1.9. 19.1.6. The factorization theorem stated at this point is an immediate consequence of 18.1.3 and 19.1.5.

+

Theorem. Let 0 < T < m and 1 + l j r > l / p l/q. An operator S E 2 ( E , F ) i s ( r ,p , q)-integral if and only if there exists a cornmutatice diagram

.

,

Al4

so

1,

such that SoE 2(7,,, $1 i s a diagmal operator 9f the form S&i) = (oifi) with (ci) f 1,. Furthermore, A E 2 ( E . lq,) tind Y i 2(lp,F”).

I n this case, I(r,p,q)(s)

= inf ll Yii Moi) ll4l

tihere the intimum i s taken over ull possible factorizations.

264

Part 4. Basic Exemplcs of Operator Ideals

+

19.1.7. Since (r,p,q)-integral operators with 0 < r 5 00 and 1 l i r = 1/p are characterized in later section8 there only remains the following gap.

+ i/y

1. An operator S E O(E, F ) is (w,p , q)-integral Conjecture. Let, l/p -?- l / q : if and only if there exists a conimutative diagram

i

A

Y

+ such that -4 E 2 ( E , Z&) and Y E f?(Zp(I),F“) with some index set I . Hero l(p,ql denotes the embedding map froin Zq@) into I#).

In this case, b , P , , ) ( S ) = inf IIYll 1 141~ where the infiinuin is taken over all possible factorizations.

19.1.8. Now an iinproveinent of 18.1.13 is given. Proposition. Let E or P be finite dimensional. Then n?r.p,q)(S)

== I(r.p,q)N

-

for all S E W E ,P)

Proof (sketch). Tising ultraproduct techniques it can be seen froin 8.8.7 (remark) that every operator S E 9c,,p,,)(E, P)admits a factorization

.i

+ Y

such that A E 2 ( E , Eo), So E S(,,p,q)(Eo, Po), and Y E 2(Fo, F”), as well as IIYII I(r,p,q)(So) IlAIl = I,,,,,,)(S),where Eo and Fo have the metric approximation property. Indeed, we may achieve that Eo = L,,(M, p ) and Fo = L p ( N ,v ) ; special cases are treated in the followingsection and 19.1.6. Suppose that E is finite dimensional. Given E > 0, by 10.3.5 there exists B E V(F,,, Fo) with BSoA = SJ and ilBll 5 1 E. Take the factorization B = JB,, where J is the injection from M ( B ) into F,. Then it follows from llBoll = IlBll and

+

ZJ:r,,,,(BS,A) 2 %,p,q,(BoSoA)= I,r,,,,,(BoSd)

19. Integral Operators

265

that B:r,g,q)(s)

IIYII X 7 r , p , q @ J = IIYII %*p,q)(~so-4 5 lIYl1 4 r , p * q P o S o ~ ) 5 (1 -t&) IIYll I ( * * p , q , ( ~ o l )l 4 = (1 f 8 ) I(*?P&)(B)* S: , p , q ) ( K d )5

Hence K!r,p.q,(Sf 5 I(r,p,q)(S). The converse inequality is trivial. The case, where F is finite dimensional can be treated analogously or by duality. 19.1.9. The following estimates can be checked using the same technique as in 6.8.4.

Lemnia 1. Let X E $(I$,, E ) and S E 3(,,p,q)(I$, a).Then

q,p,q,(sx) d I ( , , p * q ) ( S )IlXllL e l r i n i ~2. Let S E 3i,,,p,q)(E, F ) and B E &(PiF,). Then

y,JJ,,,(~fJ) 5 IlBll I ( , , p . q ) ( S )19.1.10. The following result is an immediate consequence of the preceding lemmas and 18.1.4.

T 11 eo r e in.

13(r.p,q), r(r,p.qJ0 1% il.ill = i%.p.q)3 10, Il.li1 0 I(f,p.q)l= [%r.p,q),

N(r.p,q)19 s(r,p,q~l.

1'3.1.11. Analogously to 10.3.1 we have the following iinprovement of 18.1.15. l'roposition. Let E' or F possess the metric approximatima property. Then N:r,p,q)(S)

I(r,p,q)(s)

1

19.1.12. Finally, we state

a

for all

S E WE7 P).

result which is a consequence of 18.1.4 and 19.1.11.

Theorem. Let E' or F have the metric approximation property. Then !Rt(r,p,q) is the of B(E, F ) in 3 ( f , p , q ) ( E F ), . Furthermore, the qwwi-nomns N(r,p,q) and I ( t . p , q ) coimide a ( r , p , q ) ( EF , ).

I(r,p,q)-doSe& hull

19.2.

r-Integral Operators

19.2.1. Let 1 5 r 5 the norined ideal [3rj

00.

An operator S E B(E, F ) is called r-integral if it belongs to

I,] := [R,, Nrlmax-

19.2.2. As a special case of 19.1.2 we have the Theorem.

[n,,N,] = [S,, l,lmin.

19.2.3. The next lemma is fundamental in the sequel. Lemma. Let (Ki) be a family of compact Hausdorff spaces. Then every ultraproduct (C(Ki))ltcnlz be represented as a Bamck spuce C ( K ) with some compact Hausdorff space K .

266

Part 4. Basic Examples of Operator Ideals ~~

Proof. Since the ultraproduct (C(Ki))uis a commutative B*-algebra, the assertion follows from Gelfmd's representation theorem [DUN, p. 8761. The conclusion can also be checked by using lattice-theoretical methods ;cf. C.2.4. 19.2.4. Lemma. Let K be a compact Hausdorff space. Then, given E > 0, there exists L E g ( C ( K ) ,C ( K ) )such that Kk(L) d 1 and for i = 1 , ...,n. und

fl,

[lfi

...,f n E C ( K ) - Lfillm 5 E

..

Proof. Cover K by open subsets G,, ., G, such that l f i ( s ) - f i ( t ) l 5 E for s, t E (7,. Then there are h,, ...,h, E C ( K )satisfying the following properties (partition of the for all t f K . for all t

4 G,.

m

h&)

=1

for all t E K .

1

Furthermore, fix t , E G,, ..., t, E G, and denote the corresponding Dirac measures by dl, . ., 6,. Then

.

m

L := 2

@ 11,

1

is the required operator. Clearly ~ ~ ( 6= , )1. Moreover, it follows from

;1

1 that

1. Hence wl(hk)5 1. so me have NL(L) 5 1. On the other hand, 00

Thk completes the proof. As an immediate consequence we get the Theorem. Let K be any compact Hausdorff space. Then C ( K ) has the metric nppro;rrimation property.

19.2.5. Lemma. If X E g ( E , C ( K ) ) ,then pm(S) = l[Sll. Proof. Take a finite representation n

s=zaj@fi 1 n

such that

3 ~ ~5aIISll. i ~B y~ 19.2.4

(lemma) there exists L E g ( C ( K ) ,G ( K ) ) with

1

NL(L) 2 1 and llfi - Lfi!lm5 E for i = 1, ..., n. I-Ience

n",(s, I NO,(LS) + NL(B - LS)

+ L' llaill R

2 X o ( L ) W!I

1

llfi

- Will I (1 -L 8 ) IlfJll.

This proves that NO,(S) 5 IlSll. The converse inequdity is trivial.

19. Integral Operators

36‘i

19.2.6. We are now ready to establish the main result of this section. Theorem. Let 1 5 r < 00. An, operator S E B(E, P ) ia r-integral if and only i f me of the following statements is true: ( 1 ) There exist a compact Hausdorff space K and a probability ,it E W ( K )as well as operators A E E(E,C ( K ) )and Y E E(L,(K,, p ) , 3”’) such that

E-

W)

KFS

+P“

+

/j

,IJ,(K,p)

Jr

Here J, denotes the canonical mup from C’(1i) into L,(K, p). In this mse,

I,(@ = inf II YII llAll> where the infimum i s taken orer ull possible factorizations. ( 2 ) There e.&t a probability space (52, p ) as well as operators A md Y E B(L,(Q, p ) , B”’) such that E

KFS

i

I,

B(E,L&2,

p))

’p“

A Lm(Q, P )

f

+

I

W Q , p)

Here I , denotes the embedding map from L,(Q, p) into LJQ, p).

I n this case,

I,(fl) = inf IlPll IIAI!, where the infimum i s taken occi all possible fmtorixations. (3) There exist a meamre space (Q, p ) as well as operators A f 2(E,LT,,(O,p ) ) c u d Y E E(L,(Q, p), PI’)such that KpS‘

268

Part 4. Basic Kxamules of Ouorator Ideals

Here So i s a diagonctl operator of the form S,f In this Ccxse, I,(#) = inf

= sf

with s € L,(Q, p).

llyll llsoll Mil,

where the infimum is taken ower all possible factorizations. Proof. Let S E 3 , ( E , F ) . By 8.8.4 the operator KFS can be written as an ultraJ , where Si = Q$SJ$ with i := ( M , N ) , M < Dim ( E ) , and product N E Cod (P). According to 18.1.3 there are factorizations Si

M

1,

I.,

Sio

+

such that llAill 1, IIYill 5 1, and Nr(Sio)2 (1 c j Xr(Si). By 18.2.2 we have SioE ?&(loo, I,) and P,(Sio)5 (1 E ) I,(#). It follows from 17.2.5 that (Si0)u is absolutely r-summing and Pr((Sgo)u) 5 (1 E ) I,(S). Moreover, by 19.2.3 there exists a metric isomorphism A. from the ultrapower (l,)u onto some Banach space C ( K ) .By 17.3.5 we can find a factorization

+

+

E ) Pr((Si&) 5 (1 such that IIYoll 5 (1 presented in the form (1).

+

+

&(AS).So t,he operator KFS is re-

E ) ~

According to the diagram

we also have a factorization (2), and this again is a special case of (3). Conversely, let us suppose that KFS can be written in the form (3). Then we know from 17.3.8 that So is an absolutely r-summing operator and P,(S,) = IlSoll. Let

19. Integral Operators

269

X E $(go,E ) and B E $ ( F , Po), Since L,(Q, p) is nietrically isomorphic to some Banach space C ( K ) ,by 19.2.5 we have N&(AX) = IIAX/J.Hence, given E > 0, there X,, E O(ZZ,, L,(Q, p ) ) , exists a factorization A X = XoA, such that A, E O(Eo, and IlX,,ll llAoll 5 (1 E ) &411 IlXll. Finally7 18.2.3 yields

Zz),

+

N!(BSX)

= N:(B7K,SX)

2 x(B"PSoXo) ljAoll

IlBll IIYII P,(&O) l l ~ o l IlAOll l 5 (1 4 IPll IIYII l l ~ o l l l 4 IlXll Consequently, E 3,@, F ) and I,(&) 5 IlYIl llfloll ll4. = P,(B"YfloXo) IlAoll 2

+

Remark. The analogous characterization of co-integral operators will be given in 19.3.9. 19.2.7. Theorem. [S,, I r P = [p,,P,]. Proof. Using the fact that Fi"jhas the metric extension property, by 17.3.6 and 19.2.6, we have [!&(IT7

PJ), P,] = [ 3 , ( E ,PJ), I,].

This proves the assertion, since [Vr,P,] is injective. 19.2.8. Restating 17.3.7 we get the Theorem. [S2,12] = [Ip2,Pz]. 19.2.9. I n contrast to the preceding result, we have the Proposition. Let 1 5 T

< bo and r =# 2. Then 9,+ 3,.

Proof. The embedding map from 1, into l2 is absolutely l-summing, but not l-integral.

For the case 1 < r (remark).

< co and r $. 2 we refer to A. PELCZY~~SKI [ 5 ] ; cf. also 22.4.13

19.2.10. The following result is a consequence of 18.2.4. Proposition. If rl 2 re, then [3,,, I r J G [Sr2, ITS].

< r,. then 3,c !&

Remark. The inclusion is strict for rl 19.2.11. Proposition. If 1 5 r

< 00,

Proof.By 17.3.10and 19.2.7 ~ e h a v e 3 , G ? J 3 ~ c ? B . 19.2.12. Proposition. If 1 5 r

< 00,

then 3,c ?B.

Proof. By 17.3.12 and 19.2.7 we have 3,5 8, c Cpu. The assertion also follows from the fact that every r-integral operator obviously factors through a reflexive Banach space.

19.2.13. Finally, we state the fundamental Theorem. [3:, 1 3 = ['PI*, P,.] and [p:, P:]

= [&,

I,*].

Proof. The first equation follows from 9.1.2 (remark) and 18.2.5. The second formula is a consequence of 9.3.1.

270

Part 4. Easic Examples of Operator Ideals

19.2.14. By 19.2.8 and 19.2.13 we have the Theorem. The normed operator ideal [?&,P2]is self-adjoint. 19.2.16. Let 1 5 r 5 00. An operator S E B(E, F ) is called strongly r-integral if it admits a factorization

S

E

C(K)

Jr

+Lr(K, P )

Obviously we can also dcfine a norm for those operators. I n this way we obtain an ultrastable normed operator ideal. Every 2-integral operator is evcn strongly %integral. This is not so for r = 1 and r = 00. Moreover, we conjecture the same negative result for all r 2. Remark. Strongly l-integral operators are extensiveiy treated in [DIV, pp. 165 to 1691.

+

[a,,

19.2.16. Clearly [ST,Il]lnJis the maximal hull of NrlinJ. But it seems very Nr]lnj coincides with the minimal kernel of [Sr,IrlLnj.In other unlikely that [a,, terms, we do not know whether Nr]ln3 = [Q, 11.111 o [3,, Ir]lnJo [8,11.11]2 However, the following formula is true.

[a,,

Proposition.

[al, N,Y = [St, 11.111 o [&, I r P 0 133,11.111.

Proof. Let S E d Y J ( E F , ) . By 19.2.2 we can find a factorization JFS= Yh’J such that -4 E Q(E, E,), So E 9,(E,, F,), and Y E Q(Fo, Finj), as well as llYll Ir(So) IlAll 5 (1$- E ) rVinj(S).Let J , and J y denote the canonical injections from Eoo:= M ( A ) into E , and from F,, := J4(XoA)into Po, respectively. Then A , So, and Y induce operators A,, So,, and Posuch that the following diagram commutes:

1 \-do

Obviously A , E R(E, E,,), So, E 3 y ( E o o Po,), , and Yo E R(F,,, F ) . Moreover, IlAoll 5 IIAll, I?j(Soo) 5 IJS,), and ilY,ll 2 IlYil. This proves that [%, N,li”j G [St, /i.ll] o [S,, I,? o [R,il.i[]. The converse inclusion can be derived from 19.1.10.

27 1

19. Integral Operators

19.3.

p-Factorable Operators

19.3.1. Let 1 5 p 5 cw.An operator S E B(E, 3') is called p-factorable if it belongs to the normed ideal

[I?, L,]

:= [Rp, Kp]mrtx.

19.3.2. As a special case of 19.1.2 we have the

Theorem. [A,, H,] = [I?, LPlmin. 19.3.3. Obviously 19.1.4 yields the

T h e o r e m . [I?,, LpIdua1= [I?,,, lip,]. 19.3.4. The next lemma is fundamental in the sequel.

Lemma. Let 1 2 p < 00 and let ((sZi, pi))be n family of meusure spaces. Then every ,ultraproduct (L,(Qi,pi))u can be represented as a Bamch space Lp(Q,p) zuith some 'vzeasure q a c e (52, p). Proof. Obviously the ultraproduct (L,(Qi, pi))ubecomes a Banach lat.tice with respect t,o the natural ordering. Now the assertion follows from C.2.2, if we shoir. thnt

Ilh -txzi)uIlP = Il(Z1i)UlP

+ Il(x2d11!P whenever (xli)u

A

( x d U = 0.

For this purpose put

p. .__ .-- xli - (zli A zpi)and xg6 := zpi- (xli A x 2 i ) . Then (x:Ju have

+

= [q:li)ll and (x& x:ilip

Il(4i

+

= (xzi)wMoreover, zyi

+ Ilz;Jp.

= llz~illp

4 i ) U I P = IlC4)UliP

A

zii = 0. Consequentiy

we

So

-t lI(4i)dP.

This completes t,he proof.

19.3.5. L e m m a . Let (8,p) be n measure space uizd let 1 5 p < 00. Then, given fl, ...,f n E L,(Q, p ) and E > 0, there emkt8 L E g(Lp(Q,p), Lp(Q,p)) such that K:(L) 5 1 and l l f i - LfijIp5 F for i = 1, ...,n. Proof. Choose simple functions fy, ...,f: E Lp(Q,p) with llfi 5 ~ / 2 Then .

c!jp

me can find disjoint p-measurable subsets GI, ...,Qm such that

..

ft =

II

8&, k= 1

T-hcro

.,h , are the corresponding chara.cteristic functions. TTithout loss of generality we limy suppose that 0 < &&) < 00. Define the functions u,,..., um E p) and 91, ., 21, E Lpt(G,/l.) by Uk := p(sZk)-"' h k and vk := ,/L(Qk)-'''' h k , respectivdy. h,,

&(a,

..

Then m

IJ := 2 Vk 0U k I 111

is the required operator. It follows froin

:1

IVf'kUk

1:

m

=

2' \&IP

I?#

5 1 that

? p(Qk)-l

llhkl/: 5

*

272

Part 4. Basic Examples of Operator Ideals

Hence wp,(uk) 5 1. Analogously we have wp(vk)5 1. Consequently K;,(L) 2 1. On the other hand, Lhk = hk implies .Lfy == f 8 and therefore

As an immediate consequence we get the Theorem. Let (Q, p ) be any inearnre spuce und let 1 5 p < ce. Tlwn Lp(Q:p ) hm the metric approximation property. Remark. The result is also true for p = ca,since L,(O, ,u) can be identified with some Banach space C ( K ) ;cf. 19.2.4. 19.3.6. Using the method of 19.2.5 we can check the Lemma. If S E g ( E , Lp(s2,p)), then K#3) = IlSl!-

19.3.7. We are now ready to establish the main result of this section. Theorem. Let 1 p < 00. An operator S E &(E, F ) is p-factorable { f aad 0’)21y if there exists u commutative d i q m m

such that A E E(E, Lp(Q.p ) ) und Y E !2(Lp(f2.p),F”). Here (Q, p) 6 a suitctble measure space.

I n this case,

Lp(S) = inf IjYIi $4il, where the infimum is taken over all p o s ~ b l efacton’zrntims.

Proof. Let S E EJE, P).By 8.8.4 the operator KFS can be written as an ultrawhere Si= QCSJC with i := ($1, X ) .-M E Dim (F),and N Cod (p). product &(S~),J, According to 18.1.3 there arc factorizations

-

such that llAili 1 and ii17jl!5 (1 + E ) K,(S,). By 19.3.4 the ultrapower (lP)ucan be represented as a Banach space L,(Q, ,u), Hence KFS factors throngh A,(&?,p). Conversely, let us suppose that KFS admits the described factorization. Then by 193.6 we have K i ( B S X ) = K:(BzKFSX)5 IJB”YIJK;(AX) 5 /JBIJ IIYI;l]Ali 1IXiI f o r Y < @(E,, E ) and B E @ ( F , Fo). Consequently S E B J E , F ) arid Lp(S) 5 (/YiIIIAli.

19. Integritl Operators

273

19.3.8. Obviously, if p = 2, we obtain the ideal of Hilbert operators defined a t the point 6.6.1.

Theorem.

[Bz,L,]

= [8, HI.

19.3.9. We now deal with the ideal of co-factorable or oo-integral operators. Theorem. An operator X E B(E,F ) i s co-factorable if trrd only if one of the following statements is true: (1) There exist a eo7npact Hausdorff spnce K as well as operators A E 2 ( E , C ( K ) )crnd Y E f?(C(K),F ” ) such that

I n this case.

L,(S) = inf llYll IIAl!: where the infimum is taken over all possible factorizations. ( 2 ) There ex& a m e w r e space (a. ,u) as well a8 operators A Y E B(L,(O, p), F”) such that

< B(E, L,(O,

p)) and

I n this case,

L a m = inf IlYll

llAIl7

where the infimum is taken over*all possible factorizations.

Proof. The criterion (1) can be checked with the technique used in 193.7. Therefore it remains to show the equivalence of (1) and (2). If KFS admits a factorization (l),then

(Kp)’ Y’’KC(Ky4 = (KF,)’ K y 3 - A = KFS. Hence KFS factors through C(K)”. But, if we identify C(S)’ with some L,(B, p): then C(K)“ and L,(Q, p ) coincide. So there exists also a factorization (2). The converse implioation is evident, since every Banach space L,(O, ,it) can be represented as some C ( K ) . 19.3.10. Analogously t o 19.2.13 we have the fundamental Theorem. 1R

[a;,I$ = [ap*, ] D,*] and [%o*,,

PiPtsch. Onrratnr

D*,]= [Ep., LpJ.

274

Part 4. Basic Examples of Operator Ideals

19.3.11. Let 1 5 p < 00. An operator S E E(E, F ) is called strongly p-factorable if it admits a factorization

Obviously we can also define a norm for those operators. In this wuy we obtain an ultrastable norined operator ideal. Every 2-factorable operator is even strongly %factorable. It seeins to be likely that this is not true for p =+= 2. An operator S E B(E, P)should be called &mgly oo-factorable if it factors throzgh a Banach space C ( K ) . Furthermore, an operator AS E(E, P)is said t o be discretelyp-factorable if it adrnits a factorization

where I is some index set. The normed ideal of those operators is denoted by [&, F,]; cf. 4.7.1 and 4.6.1 for p = 1 and p = M, respectively.

19.4.

(p, q )-Factorable Operators

+

19.4.1. Let 1 5 p , q 5 00 and 1/p l / q 2 1. An operator S E E(E, F) is called ( p , 9)-factorable if it belongs to tlhe normed ideal L(P.,)I

I%Lq),

:= [%,.qb

K(p.q)lmax*

19.4.2. As a special case of 19.1.2 we have the heo or em.

r&,

K(,,,)I = [I?(,.,),

~ ( ~ , ~ d ~ ~ ~ .

19.4.3. The next result follows from 18.4.2. P r o p o s i t i o n . If pl 5 p 2 and qI 2 q2, then

-

E [ ~ ( P 2 d 7 & L(P2,Pp)1

@(Pl.T1b ~ f P , d 7 , ) 1

19.4.4. On the other hand, by 18.4.3 we have the P r o p o s i t i o n . If 1 < p 2 q’ < co,tlhen i?(2,2) f?,,,,). 19.4.5. Now 19.4.3 and 16.4.4 yield the P r o p o s i t i o n . If 1 < p , q 5 2, then I?(,,,) = R e m a r k . Let us recall that & e , z , is the ideal of Hilbert operators.

19. Integral Operators

19.4.6. Theorem. Let l / p

275

+ l / q > 1. An operator S E 2(E,P)i s (23,q)-factorabze i f

n d only i f one of the following statements i s true: ( 1 ) There exist a probability space (Q,p ) as well as operators A €

B(E,Lq>(S, p))

rrnd Y f i?(Lp(Q, p),I?") such fhat

I

A \

Here

denotes the embedding map from Lq,(Q,p ) into L,(Q, p ) .

I n this case,

L ( p , q d 4= inf II Yil l l 4 where the infimum i s taken over all possible factorizabim. ( 2 ) There exist a measure space (Q,p ) as well as operators A Y E i?(Lp(Q, p ) , F") such t h t

2(E,L,.(Q,

p ) ) and

I

.I

where the infimum is taken over all possible factorizations. ( 3 ) There exist measure spaces ( M , p ) and ( N , v) as well as suitable operators A E $(E, L,,(M, p ) ) and Y E 2(LP(N,Y), B"') such that E

K,-8

'P" A

Y

276

Part 4. Basic Examples of Operator Ideals

Here So is n positive operator frmn L,.(M: p ) into L J N , v). I n this case, Ii(P.*d@

= inf

IlYlI l l ~ o lll l 4 >

where the infimum is taken over all possible factorizations. Proof (sketch). The necessity of condition (3) can by checked by ultraproduct techniques. The crucial point is to show that every positive operator from Lqt(M,p ) into Lp(N,v ) factors through a diagonal operator. To pass from (2) to (1) we put Q0 := {co E

lslr d p

0 : s(w) $: 01 and y o ( d ):= A

[aJ 1~1'JIp]-~.

Then (Go,p,,) is the desired probability space. Finally, the sufficiency of condition (1)can be proved by a straightforward generalization of 19.2.6. For further information we refer to [SEM,, exp. 15, 161, J. T. LAPRESTI? [I], and the famous thesis of B. MAUREY [2].

19.6.

Multiplication Theorems

19.6.1. Theorem. [2(p,Q), L(p,q)]0 [@:ya',

p:ya'] 2

isp,I,].

a). Let X f &(Eo,E ) and Proof. Suppose that T E ?@'"'(E, B ) and 8 E f?(p.q)(B, C E &(G,Go).By 19.1.9 we have K&,*)(CS)5 IlClI L(P,47)(4So, given E

> 0, there exists a finit.e represent,ation

19. Integral Operators

277

19.5.2. As a special case of the preceding result we formulate the T h e o r e m . [$, Lp] o

[vy, P p ] 5 [gP, I,].

19.5.3. Passing to the injective hulls from 19.5.1 we get a multiplication formula which is analogous to 20.2.1.

Theorem. [f?&, Lf&,] o [?-@"' > Pdual q' 1C =

19.6.

Ppl.

Division Theorems

19.6.1. T h e o r e m . [ 2 ( p , qL(p,q)] ) , = [3,,I,]

0

[?@tpal,

ptyal]-l.

P r o o f . Suppose that S E 2(E,F ) belongs to SPo (g:?al)-l. Take B E g(P,Po), Lo E ~~,.,,.,(F,,Eo), and X E g(E,, E ) . Given E > 0 , by 17.4.3 we can find a factoriM ) , X o E v;Ya1(M, Eo),and zation Lo = X,Bo such that 13, E !@pt(Po,

P;~"'(xrJ) P,.(Bo) 5 (1

+

qp~,&o).

We have the diagram

and thus, by 19.1.9,

Consequently, according to 18.4.5 (lemma),

The converse inclusion is a consequence of 19.5.1.

278

Part 4. Basic Examples of Operator Ideals

19.6.2. As a special ease of the preceding result we formulate the Theorem. ,!lE

19.7.

L,] = [a,, Ipl 0

PdpUB11-.'

Notes

The concept of a 1-integral operator was introduced in the fundamental memoir [GRO, chap. I, pp. 124-1481. The theory of r-integral operators goes back to A. PERSSON/A. PIETSCE[l], and operators factoring through L, were investigated by J. LINDENSTRdUSS/A. PEECZYI~KI [l] and mainly by S. K W A P I E[S]. ~ An indispendable reference is also the famous thesis of B. MAUREY121. I n any case the reader should consult [SEN4, exp. 16, ISbi*]. Finally, the author defined (r, p, q)-integral operators in the pre-version of this monograph. Recommendations for further reading:

[LAC], [LOT], [SAF,].

D. DACIJWHA-CASTELLE/J. L. KBIVINE [l], T.GORDON/D. R. LEWIS [1], Y. GORDON/ D. R. LEWIS/J.R. RETHERFOBD [l], [Z], J. R. HOLCB[2], J. TH. LAPRESTE[l], D. R. LEWIS [Z]. K. Itbyazig~[3], A. PELCZYNSKI [5], A. RETSCH [9], J. R. RETHERFORD [2], I. I. TSEITLIN c21.

20.

Mixing Operators

In this chapter we investigate the ideal B+s,P, of (s7p)-mixingoperators the significance of which follows from

%

m(8.P)

E @P

forO
m(s,P)

Since 8,E for 1/r + l / s = l / p 5 1, as a corollary we get the famous multiplication formula !& o Fp7 E ?&. Moreover, it turns out that the operator ideals ?lR(a,p) and ?&r,p) almost coincide.

20.1.

(s, p)-Mixing Operators

8 5 00. An operator 8 20.1.1. Let 0 < p is a constant c 2 0 such that

qs,p)(fw

s

q

h

i

E E(E, F ) is called

( 8 , p)-mixing if

there

)

for all finite families of elements q ,..., zm 6 E .

W e put

M(8,p,(S):= inf 0 . The class of all (8, p)-mixing operators is denoted by !!R(s,p). Remark. Obviously [WO?.P)7

1).z,O?,P)l

= [Pp, PPI and

[ ~ ( p , p )%LP)I ,

= 107 l1*l11-

20.1.2. Using standard techniques we get the

[m(s,p), M(s,p)]is a lzormed operator ideal /or p >= 1 and a p-normed

Theorem. operator ideal for 0

20.1.3. The phrase

“(8,

p)-rnixing” is derived from the following property.

Proposition. A n operator S E e ( E , P)is (s, p)-mixing if and only if every weakly p-summable sequence is mapped into an (s, p,)-mixedsumnzable sequence. Proof. Use the method of 17.2.3. 20.1.4. We now state an important criterion.

Theorem. A n operatM S IS 2 0 such that

E(E, F ) is

(8, p)-mnixing if

and only if there exists a

cons&&

..,b,

for aW finite families of elements zl, ...,x , E E and functionnls b I , . In this cme M(s,P)(i3) = inf u.

E

F’.

280

Part 4. Basic Emmplcs of ODerator Ideals

Proof. To show the necessity of the above condition, givon bl, ...,b, 5 B”,

[: I-’

n

define the discrete probability v = 2’ vkdk, where vk := llbkjlS 2 j/bL/js and & de1

notes the Dirac measure at bE := bk/llbkll. If S E %(s,p)(E,F ) and xl,...,xm E E . then by 16.4.3 we have

Conversely, the inetpa1it.y

nieans that

for all discrete probabilities v E W (Up,).Since these probabilities are dense in W (U p > ) with respect to the weak C(U,,)-topology, we have (+) for all v E W ( U F f )So . it follows froiii 16.4.3 that m(s,pl(Szi) 5 mp(zi). This completes the proof.

30.1.6. Theorem. Tlte quasi-normed operator ideal [?&,,,,, Bl,,,,)] is maximal. P r o o f . Suppose that xl, ..., xmE E and bl, ..., b, E F’. Write x:, ..., .& if zl, ...,r,,, are considered as elements of their linear span LW. Obviously we have J$xq = x8and w,(x!) = wp(xJ. On the other hand, put

N := (y E F:(y, b,)

=

- - a

..

= (y,

Then there are b!, .,b: E (FIN)’ with follows from S E aE,;(E, F ) that

bm) = 0 ) .

(9;)’b! = bk

and I,(bp) = &(bk). Now it

20.1.6. The next result is evident. Theorem. The quasi-normed operator ideal

[!Dl(s,,,),fil(8,pJ tk injective.

20.1.7. The following criterion is a generalization of the fundamental characterization given in 17.3.2. T h e o r e m . Let 0 < p 5 s < 00. A n operator S E e ( E , P) i s (s, p)-mi&?q if and there exists a constant G 2 0 such that for v E W(Up)we can fin& p E W(U,,)

only if

20. Mixing Operators

281

tohenever x E E. Here we suppose that UE#ond UFtcrre equipped with the weak E-topology and P-topology, respectively.

In thi.9 caRe,

31(s,p)(S) = inf 6. Proof. If the above condition is satisfied, then

) owP(x,).Hence ASE !Jjl(s,p!(E, F ) and lII(B,p)(S) 5 6. This means that r q s , p , ( S z i5 If s 2 1, then for every Y f W(U,.) we consider the operator J , E ?@,(P, L,(Up<,Y)) assigning to y E P the function f , with f&b) := (v,5). It follows from Ps(JV) == 1 and S E mts,p)(E, F ) that J,S f P p ( E ,LJU,., v)) and P,(J$) 5 31(e9,p)(&’); cf. 20.2.1. Hence, by 17.3.2, there exists ,u E W ( U E t such ) that

for all x E E’.

The case 0 < s < 1 can be treated analogously, since the criterion 17.3.2 is also true for absolutely p-summing operators mapping into a complete quasi-normecl linear space.

20.1.8. As an immediate consequence of the preceding charactcrization we get an interesting multiplication formula. Proposition. Let 0 < p Ern(t,S,,

-9

5 t 5 bo. Then

J h * , l0 [rnn(s,p),%s.p)l G [rn,,,),

%t,P)I

*

Remark. The conclusion can also be obtained froiii the definition of mixec! (s, p)-summable families directly.

20.1.9. Proposition. If p1 5 pz 5 .s, 5 s,, then Cf%,,P,),

%s&l

s

[%S,,P,b

%8>P,,1

*

Proof. Thc assertion follows iminediatcly from 20.1.7. 20.1.10. We now state the main result of this section. Theorem. Let l / r [?@n Pr1

S

+ ljs = l / p 5 1. Then.

[ 9 J 4 s , p ) 7 sI,s,p)l-

Proof. E’irst we claim that, given a prabability p on a compact Hausdorff space K , the canonical map Jr from C ( K )int,oL,(R, p ) is (8, p)-mixing and Mf8,p)(Jr)= 1. Let

282 fl,

Part 4. Basic Examples of Operator Ideals

...,fm

r’/s

E C ( K ) and h E L , ( K ,p) with llhllr. 5 1. It follows from p / r

+ rf/p‘= 1, and l / r + l/s + 1/21‘ = 1 that J If#’/‘

I(Jrfi,h)I

IfJ”a

+ p / s = 1,

lhlr’/”Ihr’Ip‘ d p

K

then we get

and m

m

Z I(gi, h)ls 5 J Z I f i l P K

1

lhl”

d wp(fi)P-

1

Hence m(s,p)(Jrfi)S U t i )

w,(gi)

5 wp(fi)*

This proves that J , E ?Jll(s,p)(C(K), L J K , p ) ) and M(s,p)(Jr)5 1. Finally, the assertion follows from 17.3.6, since the normed operator ideal [%ll(8,p),M,,,,,] is injective. Remark. It was proved by G. PISIER [2] that the assumption p 2 1 is essential.

20.1.11. As an immediate consequence of 16.4.5 we have the Theorem. Let l / r

+ l/s = 1/p. Then

[ m ( s , p ) , M(s,p)l

s [%r.p),

P(r.p)l.

Remark. It follows from 22.3.5 (remark) and 22.6.3 that 9Jlm(s,p) $: ?@(r,p) if 0 < p < s < 2 and l/r l/s = 1/21.

+

20.1.12. We now show that Theorem. Let 1 5 p (Q(r.p)

m(s,p) and

almost coincide.

5 so < s 5 00 and l/r + 11s = 1/p. Then

?Jll(s,,p)*

Proof. Suppose that S E ?&r,p)(E,F ) and x,, ..., x, E E. Given any v E W ( U p ) , we consider the operator J , E ?&o(F, Lso(UPr, v)) assigning to y E F the function f, with f,(b) := (y, b}. Clearly P,JJ,) = 1. Choose ql, ...,qm E L8;(UF,, v) such that l/Jvt!3x& = (JJzj, qi} and IIgills; = 1. We now defini: operators X E B(E,”.,E ) and G e(Ls0(UF7 y), ZE) by m

Eixi and

X(Ei):=

af := ((f, gi)).

1

Then iiXi1 = wp(xi)and llGll = 1. Moreover, let T E,:Z(J!

Z2) be the diagonal operator

20. Mixing Operators

generated by

(ti)

M(P,l)(T)

283

17. Applying 20.1.10 we have

5p

p m = l,.(ti)

-

Then it follows from 20.2.1 (remark) that P(Sfl,l)(f=T)5 PV,P)(S)IIXII &,l)(T) 2 P(r,p)(S) WP(Zi) Wti)

9

where 11s := l/p - l/r. By 18.2.3 and 22.6.5 we get

NZ;(SXT) =( .Wp(Zi) lP.(ti) with some constant

(r

2 0. It follows from 18.4.5 (lemma) and

that

Hence

So, by 16.4.3, we get for all Y E W(UFn). m ( s , , p ) ( w5 G W p ( & )

*

This completes the proof.

20.1.13. P r o p o s i t i o n . Suppose thut F is an intermediate space of (Fo,F,) posses-Sing J-type 0. Let 0 c p S so, s1 5 00 and 11s := (1 - @/so O/sl. If X E E(E,P A ) , then rS E ?lR(s,,p)(E, Fo) and S € !D2(sl,p)(E, PI)imply S E !?JI(s,p)(E,F ) . MoreWer,

+

M ( 8 , p ) ( 8E: -+ P) 5 M(a,,p)(S: E+

M ( s l , p ) ( E~ : + F1)'.

Proof. P u t l / r := l/p - l/s, l / r o := 1/p - l/aO, and l/rl := l/p - l/sl. Then 1/r = (1 - O)/ro O/rl. Let zl, ...,2, E E with wp(si)5 1. Given t' > 0, we have Szj = toi~,,i and Sxi = tliyli such that

+

lre(roj) lr,(tli)

+ 2 (1 +

S (1

5 1,

8)

H(s,,p)(S:E'--j, F a )

and

E)

M(sl,p,(S:E + P I )

and lvsl(yli) 5 1 -

ws,(~oi)

Furthermore, we may suppose t'hat zOi> 0, and zli > 0. P u t yfi := tTISzi. Then I,(ti)

5 l r , ( ~ o i ) l - ~L l ( t l i ) ' .

On the other hand, it follows from

I(yi, b)i that

= I(yoi, b)ll-e l(yIi, b)jo

for all 6 i F'

t i:=T

and

284

Part 4. Basic Examples of Operator Ideals

Consequently m(8,p)(szi!

2

mS(?/i)

- (1 4E ) 81(se,,)(x I : E -> $'o)'-'

~I(s1,p)(8 : E -+ P,)'.

This proves the assertion.

20.1.14. Let us define the space ideal 48.p)

:= 'pact

(m(S.P)).

20.1.15. We now prove that M(s,p) does not depend on the parameter p . Theorem.

M(s.g) for 0

:

< p < p < s < 00.

Proof. By 20.1.9 we haw M(s,p) M(s,q). Conversoly, let E E M(s,q). Then

M ( s , q ) ( I M5) M(,,,)(IE) for all M E Dim ( E ) . Noreover, 20.1.8 yields

and by 20.1.13 we get M ( q , p ) ( I I w )I )r(s,p)(I~~)'-@

nl(p,p)(Iaf)b

where 8 is determined by l/q = (1 - 0 ) l s qs.p)vMY

+ 0/p. Consequently

I %?,g)(~M) 5 M ( s , q ) ( G

Finally, it follows from the rnaximality of

[m(s,p), M,,,,)]

that E E M(s,p).

20.1.16. We now give an interesting characterization. Proposition. Let 1 < s (= 2. A Banuch space E belongs to

if and only if

B(%> E , = ' p s ' ( c O , E ) . W

Proof. If (zi) m,(E), then X(Ei):= _2: E*zi defines an operator X

B(c0, E ) . SO

1

we get a metric isomorphisin between m,(E) a i d 2(co,E ) . Moreover, using the same method as in 18.2.3 we can see that in this may ttt(8,1)(K)is mapped onto !&(~, 3). Consequently, the statements w,(E) = m(s,l,(E)and 2(co, E ) = @,.(c0, E ) are equivalent for cvery BanacIi space E. This proves thc assertion.

50.1.17. Theorrm. The operator idenl !l.Rm(s,p)i.9 proper if txnd only if s s >p.

>2

~

l

d

Proof. It will be shown in 23.3.5 that tho identity map of 72 belongs to !&2,p, with 0 < p 5 2. So !l.R(s,p) 2 !J.R(2,p, is non-proper for 0 < p 5 s 5 2. Moreover, we have ?Jll(p,p) = 0 for all p > 0. On the other hand, if s > 2 and s > p , wc Bee from 20.1.11 that !J.R(s,p) & ?&r,p) with 1/23- l / r == l/s < 112 and p 5 r < co. SO W ( s , p ) is proper by 17.9.7.

286

Part 4. Basic Examples of Operator Idcnls

20.2.3. Proposition. Let 1 5 p 5 s 5 Ern(p.,,% q p , . s , , 1 0

P S ,

00.

Then

I81 2 [3,,I,].

Proof. Suppose that T E 3 , ( E , F ) aiid S E 9.11(pr,sl~(B’, G). If X C g ( X 0 , E ) and C E g(G,Go),then it follows from 19.1.9 (lemma 1) and 20.2.2 that

N,(CSTX) 5 3X(pp,s,)(CS) R,(TX) 5 llC!j N(,~,,+S) I J T ) IlXll. Hence ST E SP(E,G ) and IP(ST)2 III(pr,,r)(S) Is(T). Remark. Using the same method we get mp;.P%

%wd 0 [%.P.,?),

for 1 2 po 5 p 5 co and l/p - l/r

I(r.p,q)l

=

E ~%o,Po,q)>

470.Po,q)l

lipo - l/ro.

20.2.4. We now state the main result of this chapter. Theorem. Let l/r

[@nPrI

+ 11s = l/p 5 1. Then

2 [ r n ( s , p ) , M(s,p)I and

“Pn PrI E [ W p c . s , ) , %pr,st)l*

Hence (l),(2), and (3) follow immediately from the preceding propositions. Suppose that T E V,(E, B’) and S E !Rr(F,G ) . Given E

F

> 0 , we take a factorization

S

+

such that I1211 &(So) /JBII5 (1 E ) N,(S). Since I , has the nietric extension property, we know from 19.2.7 that BT € 3,(E, I , ) and I,(BT) = P,(BT). Hence by (3) we get SoBT E 3,(E, I,.) and IP(SJ3T) 5 €‘,(So) IlBIl P,(T). Clearly we may achieve that Z E @(Zr, a). Then it follows from 19.1.10 that ST = ZSOBT f ?Rp(E,F ) and XP(ST)2 llZll I,(S,BT) 5 (1 + E ) Sr(8) P,(T). Finally, (5) is an immediate consequence of (4). Remark. It was shown by G . PISIER[2] tliitt (1) docs not hold in the case where l/r l/s = l/p > 1 in general.

+

287

20. Mixing Operators

20.2.6. The results of the preceding theorem are represented in the following multiplication table

20.3.

Division Theorems

20.3.1. As an improvement of 20.2.1 we have the Theorem. Let 1 5 s [ r n ( * , p ) , i%.p)l

co. Then =

[bs,

0 [@P> P P I .

..

Proof. Suppose that S E B(E,F ) belongs to ?j3T1o ?&,. Let xl, ., x , E E and b,, ., b,, E P'. Define the operator B E B(P7I:) by By := ((y, bJ). Then P,(B) s l s ( b k ) and therefore

..

d PP(B8)W p ( Z i ) d P,' Hence, according to 20.1.4, we have 8 E This proves that

[bs,P81Y 0 t @ p ,

Ppl

SR(,,,)(E,

0 P p ( 4 Wp(XI)

UbJ*

F ) and M ( s , p ) ( 8 )I p,'

0 Pp(8)-

E C f R S * P ) , %.p)l.

The conversc inclusion follows from 20.2.1. 20.3.2. The folloming formula can be checked by analogous considerations. Theorem. Let 1 5 p 5 s 5 00. Then

[m(p*,,'), N(p,.s~)= l E%p, ;v,l

0 [%a7

N P-

'30.3.3. Theoren?. Let 1 5 p 5 s 5 60. Then [m(pT,s,)7&p,,s,J

20.4.

=P p ,

Ipl 0 [a,, I,Y.

Rotos

Although the theory of ( 8 , p)-mixing operators is here preseated for the first time, most results can be iinplicitely found in the thesis of B. MAUREY[2]. See also [SEM,, exp. 13, 17, 211 and [SEX5, exp. 121. The i n c h i o n 8, c ?lRm(s,p) ~ s i t hl / r l / 8 = l / p 5 1 wm established bjA. PIETSCH [5]. The multiplication table is taken from 9.PERSSOK/A. PIETSCE [l], and the division theorems are due t o J. Punr, [l].

+

Recommendations for further reading:

-

3.MAU~XEY/G. PISIEI~ [I], G. PISIER [2], $1. P. ROSEXTII-~L [41, r71.

21.

Type and Cotype

In the following we deal with the ideals and C&p) of operators possessing (8, p)-type and (27 p)-cotype, respectively. The theory of these operators was created by B. MAUREYin. 1972. The basic formulas are !$38

!$3p f o r O < p < 8 5 2

OZ$;;

and

!& for 9 5 p < w . Moreover, we have z(s,p, = 2 for 0 < s < p < 1 which proves that the operator ideal f@, with 0 < r < 1 does not depend on the parameter r. o 'ppG

From our point of view the operator ideals investigated in this chapter are toolb to deal with absolutely summing, nuclear, and integral operators. However, operators of (s,p)-type and (2,p)-cotype also play a significant role in the structure theory of Banach spaces. On the other hand, using these concepts J. HOFFMANN-JQRQENSES and G.PISIER have characterized those Banach spaces in which the law of large numbers or the central limit theorein hold.

21.1.

Stable Laws

31.1.1. Let 0 < 8 I;2. Then by Bochner's theorem there exists a probability ys on the real line and on the complex plane such that exp (-1718) and

J exp (i.iz) d,u,(a)

= 9

exp (-Itis) = J exp (i Re (617))dp,(cu), y:

respectively; cf. [KAW, p. 3971. Let us mention that in the literature y, is frequentcly called a stable law. If .s = 2 and s = 1, then these probabilities have a Lebesgue density given by 1 exp (-lx12/4)

VG

1

and 4 1

+

241

+

(real line) IKl2)

and

1 exp (-IaI2/4)

47L

and

1

(complex plane), j42)3/2

respectively. 21.1.2. Put sf := s if 0 < s .= 2 and s+ := 00 if s = 2. Then there exist the absolute moments

21. Type and Cotype

< p < s+.

for all exponents p with 0

and

[r CSP

=

s

We have

r ~ r+)]'" ) r

289

(complex plane) ;

(9)

cf. [KAW, p. 4301. I n particular,

c2p

=;

8

[r ("-

p)]'",

-

cZ2= 2 . cZ1= 1.z

(complex plane).

31.1.3. We now prove a fundamental formula, where u ,: denotes the m-fold product of p5. L e m m a . If 0

< p < s+, then

Proof. Observe that the coordinate functionals f l , . . .. f m , defined by f t ( a ):= a, for a = ( a t ) ,are independent randoin variables on the probability space (Zm, p:) with corresponding Fourier transforms f z ( t ) = exp (- l t l s ) . Let x = ( E , ) be any m

wctor. If f := 1

tifr,then

Clearly the randoin variable g with g(cw) := cw 11x11, defined on the probability spaccb (Z, p,) has the same Fourier transform. So f and g are equidistributed. Consequently

an6

coincide. 1 g Pietech, Oprrutor

290

Part 4. Basic Examples of Operator Ideals

21.1.4. In the following we denote by ax and bx the coordinatewise product of b = ( p i ) with 5 = (li),respectively. Leainia. Let 0 < s, t 5 3 and 0 < p < min (s+,t+). Then

a, = (ai) and

cS;l

{1

1 Ilbxll,P dpT(b)}lIp

I!uxllf d,uP(a) = yc ~ l {

3-m

for all x E Zm-

xm

Proof. By the preceding lemma we have

and

Proof. By the preceding lemma we have

On the other ha.nd, it follows from 22.3.1 that

{ J l!axllf dp,”(c~.)

c ~ ’

x

m

Remark. If s = t = 2, then the above formula remains true for 0 < p f 2. In case 2 S p < 00 we get t,he converse estimate.

21.1.6. Let 0 < s 2 2 as welloas 0 < p x1, ...,xmE E , we put

< s+.

For every finite family (xi),whore

21. Type and &type

291

21.1.7. We DOW state an important result of J. H O F F N A " - J ~ R G E N S E X [l];cf. dso [SEM,, exp. 61. Proposition. Let 0 < p t(s,q)(xi)

< q < s+. Then there e&ts

a constant bspq> 0 smh tha

5 bspqt(s,p)(Xi).

21.1.8. Let 0

< p < 00. For every finite family (zi), where xl,..., xm i E , we put

21.1.9. Por completeness we mention an estimate which is analogous to 21.1.7; cf. [LIT, vol. 111 and [KAH, p. 151. Proposition. Let 0 < p < q < 00. Then there e&ts a constant bpq > 0 such that pq(xi) S bMrp(xi). 21.1.10. Pinally, a result of G . PISIER[l]is stated; cf. also [SEM,, exp. 31. Proposition. Let 1 < s 5 2 crnd 0 < p < s+. Then there exists a conatant btp > 0 such that rp(xi)5 b:pt(s,p)(xi).

21.2.

Operators of (s, p)-Type

21.2.1. Let 0 < s 5 2 and 0 < p < s+. An operator S E B(E,F ) is said to be of (s,p)-type if there exists a constant (r 2 0 such that t(,,,(S%)

5 4(%)

.

for all finite families of elements xl, .., x, E E. We put T(8,p@) := inf (r. The class of these operators is denoted by Z(s,p,. Remark. It follows froin 21.1.7 that Z(s,p)does not depend on the parameter p .

21.2.2. Using standard techniques we get the Theorem. [ Z ( s , p ) , T ( s , pis ) ] a normed operator ideal for p 2 1 and a p-normed operator ideal for 0 < p < 1. 21.2.3. Theorem. The quasi-normed operator ideal [ Z ( s , p )T,,,,,] , is mciximal. Proof. Let S E zr$(E,F ) . Observe that Cod ( P ) , the set of finite codinicnsional subspaces of F , is directed. We have

...,x,

for al,..., amE Y and xl, xl, .., x,. Then

.

1 I)*

E E. Let iM E Dim (E) be the linear span of

292

Part 4. Basic Examples of Operator Ideals

So, by Fatou’s theorem, we get

3 )and ‘J!(8,p)(4 5 Tr$(fi). Hence S € Z(s,p)(E,

21.2.4. Theorem. The quasi-nomzed operator ideal [Z(s,p,,T(,,,,] i s injective and surjective. Proof. The injectivity is evident. To check the surjectivity let S E Z;FA,(S, F ) . Given xl,..., x, E E and E > 0, we choose xy, ..., :x E ESurwith xi = Qpz! and ll4ll I (1 4 IIXill. Then

+

t(s,p)(&)

= t(s.p)(sQ,&)

I T(s,p)(fiQ~) l a ( 8 ) 5 (1

+

E)

TSti,(S)&(xi) *

Hence S E Z(,,,)(E,P)and T(s,p)(S) 5 TYZ’,,(S). This proves the assertion.

21.2.6. We now state the basic result of this section. Theorem. Proof. If xl,

Tt$;l G [!)J M(s,p)l. l(,,,), ...,x, E E and bl, ...,b, E F‘, then by 21.1.3 we have

= t(S,P)(fi’bk) l q x i )

5 T(S,p)(s’)1s(bk)W p ( X i ) ‘

Hence the conclusion follows from criterion 20.1.4.

21.2.6. Proposition. Let 0 < p < so < 5 5 2. An g e r d o r S E B(E,F ) is of (so, p)-type if there exists a constant cr 2 0 such that

(*I

r,(Szi) d 4(4 for all finite families of elements xl, ...,x,,,E E . I n this case

T(s,,p)(45 css0c;; P r o o f . By (*) we have

inf 6.

21. Type and Cotype

Observe that ,uE is invtiriant under all transformations follows from 31.1.5 that,

2 GS,C;-d IS&!)

a

--f

ecc

293

with e € 8". So it

*

'l'his conipletes the proof. Rcinark. Operators satisfying (*) with 0 < s bc of Raileinacher (s, p)-Lype; cf. G. PISIER [l].

52

and 0

0;)

are said to

21 2.7. Using the same method as above we get the

P r o p o s i t i o n . Let 0 < p 5 2. =In o p r n t o r S E 2(E,F ) i.s exists ri constant CT 2 0 . w C h that

of

( 2 , p ) - t y p e if there

(*I

rp(S.%) i a t , ) for all finitc fauzilies of elentpnts q ,. . .. x,,,c

E.

In this case, T(z,p)(S)5 c22cg1inf u . Reinark. I n the case where s

=

2 thc condition (*) is also necessary; cf. 21.1.10.

51.2.8. By 21.1.10 and 21.2.6 we have the

< p < s2 5 s1 5 2, then Z ( s , , p ) Z(8z,P). T h e o r e m . If 0 < p < s < 1, then = 2.

P r o p o s i t i o n . If 0 21.2.9.

Proof. Let E be any Banacli space. Then r p ( x i )5 l,(xi) for xl,. .., x, € E . Hence the identity map of E h:~s(s,p)-type by 21.2.6. So 2 & Z(y,p).

21.2.10. As a consequence of 21.2.5 and 21.2.0 we formulate the T h e o r e m . If 0

< p < s < 1,

tl~en9R(S,p)= 2.

21.2.11. Now 20.2.1 yields the impressive T h e o r e m . If 0

vP

< 1) < < 1 , f?Len

= 'p8,

21.2.12. A Banach space is paid to be of (s,p)-.typeif it belongs to the space ideal T(8.P)

:= Space

(2(,,,,).

Banach spaces of 1Z~tdeniach~r (s, p ) - t y p e are defined analogously. R e i n a r k . Let us recall that the parameter p is indeed superfluous; cf. 21.2.1. (remark).

294

Part 4. Basic Examples of Operator Ideals

21.3.

Operators of Ultra (s,p)-Type

-

21.3.1. Let 0 < s 5 2 and 0 < p < s+. An operator S E B(E,F ) is said to be of ultra (s, p)-type if there exists a constant (T 2 0 such that t(S$)(SXi)

5 uws(xi)

...,xm E E.

for all finite families of elements zl,

We put

U(s,p)(S) := inf (r. The class of these operators is denoted by U(s,p). Remark. It follows from 21.1.7 that U(s,p)does not depend on the parameter p .

21.3.2. Using standard techniques we get the Theorem. [u(s,p)7U(s,p)] is n normed opemator ideal for p 2 1 o p e r a t o p i d e a l f o r O < p < 1.

aid

CL

p-nurmed

21.3.3. Next we state an cvident Theorem. [ % ( S , P ) , T(S.,)I 0 [bs, PSI

$z[U(S.Pb U,s,,,l.

21.3.4. We now prove a basic result of this section. Theorem. [ b p , Ppl 5 [ U ( S , P ) , U ( S , P ) I . Proof. Suppose that S E V p ( E ,J’). By 17.3.2 there exists a probabilit,yp E W ( U E p ) such that

...,x,,,E E , we have

Hence, for xl,

P)and Uf8,p)(S)5 P,(S). This proves that 8 E 21(s,p)(E, 21.3.6. In order to establish the following theorem we need the Lemma. Let 1 5 p < 00 and suppose that the operator A E B(I,“, 1;) is gelaerated hy the matrix (aik).Then

21. Type and Cotype

295

Proof. 3 y Schmidt's factorization theorem it is enough to deal with the following special cases. (1) Let A E o(lr,IF) such that A(&) =; ( m k t k ) and Iyk 2 0. &aKly f& := (OLk) G t I l be writ.ten in the form a = C A,e, where 1, 2 0 and 2 1, = IIAli. Since we have 8"

8" t(z,I])(qpk)

= t ( 2 , p ) ( q ) for

all e = ( E ~ )with

5

t(Z,p)('%Xk:k)

ck =

= \ ( A I\

&tp,p)(&kzk)

8"

&I, it follows that

t(2,p)(&)

*

(2) Observe that pm is invariant under all orthogonal transformations A E B(ZT, ZT!.

80 we have

(3) To treat the case m t(?.p)(Xl,

* *> 5 m 7

+ n another trivial fact, namely = t(2,p)(%

0)

-

*

*,

7

is required.

We are now ready to check the Theorem. Let 1 5 p [Z(2,p), T(2.P)I 0

< 00. Then

[P*, P21dUB1 c= [u(2,p), U(0,p)l.

.

Proof. Suppose that T E @y1(E7 F ) and 8 E 2(2,p,(F, G ) . Given xl, .., x , C E, m

by setting X&) := 2 && we get an operator X E f?(Zg,E') with IlXll

= w2(xk).

1

According to 19.1.9 there exists a finite representation 18

TX =

1jai

@ ZJ~

1

such that

Iz(A$) Wz(ai)

5 (1 f

W,(?/i)

8)

~ ~ ~ , i , ~ ?5) ((l Tfx )&) p?a(T)

wdXk).

n

Define A E B(ZF, Zz) by the equation A := and /]All= w2(ai).Now it follows from n

8TXk

= 8Txt?k = i=l

that.

OLikAiSyi

cri @ e,. Then a i k := (Ltek, ei) = (ek7 ai)

296

Part 4. Basic Examplcs of Operator Ideals

21.4.

Operators of (2, p)-Cotype

21.4.1. Let 0 < s 5 2 and 0 < p < s+. An operator S E B(E,3’) is said to be of (.g. p)-cotype if there exists a constant (r 2 0 such that 18(sxt)

5 d(8,p)(2t)

for all finite families of elenients q,..., x,,~f E . We put C(8,p)(8)

:= inf

0.

a(s,p). R e m a r k . It follows from 21.1.7 that a(s,p) does not depend on the parameter p. Moreover, we have a(8,p) = f? for 0 < s < 2 ; cf. [SEJI,, cxp. 71. The class of these: operators is denoted by

21.4.2. Using standard techniques we get the Theorem.

[a(2,p), C(2,p)].is a normed operator ideal.

21.4.3. The next result is also stated without proof. Theorem. The normed operator ideal [6(2,p), C(2,p)] is nmxirtzal.

21.4.4. Theorem. The nornzed qerator ideal

[&(2,p),

jectizt e .

C(z,p)]is injective, but not

~UT-

Proof. The injectivity is evident. Since, by 22.3.4, the identity map of all Banacli = 2. spaces & ( I )is of (2, p)-cotype, surjectivity would iniply the false formula 6(z,p)

21.4.5. The following result is trivial. T h e o r e m . [ E ( Z , P ) , C(2,,)1

0 EU(O,P), U ( 2 , P ) l

G [V2>p21.

31.4.6. Now it follows a famous Theorem. Let 1

5 p < 00. I’hen

[ a ( z . p ) >C(2,P)I 0 [%.P)>

T(2,P)I

s [f?2,

1421.

Proof. By 21.3.5 and 21.4.5 we have

C(,,,)]

0 [ 2 ( 2 . P ) , T ( 2 , P ) I 0 [V2,

s

P21dua1

EV2, p21.

So 19.6.2 yields the assertion. 31.4.7. As a n imniediate consequence of 21.3.4 and 21.4.5 we get the main result of this sectioii. Theorem. Let 2 s p E6(2,P)> C ( 2 , P ) I 0

< m. Then

[PP, PPI s

p21.

[P2?

21.4.8. Next the adjoint version of the preceding forniuln is given. Theorem. Let 1 < r, =( 2 . Tlwu [ 3 2 , 1 2 1 0 W(2.pf)’ C(2,P*)l

EP

p ,

&I

*

21. Type and Cot.ype

297

Proof. Suppose that T E C(z,p)(E, F ) and S E &(F, a).Moreover, let C E g(G, Go), Lo E !&(Go, Eo),and X E S(Eo,E). Then, by 18.4.5 (lemma) and 19.1.9, we h a w

!trace (8TXLoC)j = /trace (TXLoCS)I5 P2(TXL0)M;(CX) -

C(2,p)(T) llxll BP*(Lo)llcil I

2 W .

ST E Sp(E,G ) and I,(ST) 5 12(S)C,,,,*)(l').

Hence, it follows from 19.2.13 that

31.4.9. We now state an immediate consequence of 20.3.1 and 21.4.8 as well as of 19.2.7 and 19.2.8. Theorcin. If 1 < p 5 2 , then W(2,P*),

C(,>,*)I c=

[?JJ$Z,P),

M(2,p)l.

does not depend on p. So we have indeed Itemark. Let us recall that C(z,p) %l&z,q)for 0 < p < 00 and 1 < q 5 2. However, it seeins to be unknown whether this inclusion also remains true for 0 < q 5 1.

21.6.10. Pinally, we establish a supplement of 21.2.5 and the preceding result. Proposition. If 1 < p

< 00, then ; ; :% :

6(2,p*,. Proof. Suppose that S c Z$'$i(E, li') and xl,..., x , E E . Choose b,, ..., b,,' t 3'' such that (&xi, bi) = llSxi112 and l]biil = [j8zill. Since the coordinate functionals f l with f i ( a ):= mi form an orthogonal family in L z ( X m,up), , we have tn

I~(SZ~)' = (SZ~,b,) 1

= CG'

j"

(2

1

m

2' akS'bk

& t ~ i ,

1

, x m

I C2,%,*C&,,*)(Xi)

dpy(cx)

t ( 2 , P W k )

- C 2 p % p * C 2 q , , p ) ( ~ ' ) I,(bk)

4 2 , P * W

*

Therefore, since 12(bi)= l2(5z,),it follows that 12(8Zi)

1dua.l r c z p c z pr;c;* __ q2,pJs) t(Z.P.)(4

*

Hciicu X E a ( 2 , p * ) ( F E ), .

Remark. We will prove in22.3.4 that theidentity map of El is of (2,p*)-cotype. =I=6(2,p*). Therefore On the other hand, this map cannot belong to

21.4.11. A Banach space is said to be of (s,p)-cot?ype if it belongs to the space ideal C(8.P)

:= Space

(a,,,,)) *

Remark. Let us recall that the parameter p is indeed superfluous; of. 21.4.1. (remark).

298

Part 4. Basic Examples of Operator Ideals

21.5.

Notes

From the very beginning the scqnence of Rademacher functions and Khintchine’s inequality have played an important role in functional analysis. It was the idea of J. BRETAGNOILE/ 1). DACUNHA-CASTELLE/J. L. KRIVINEEl] to extend this machinery by using other sequences of independent random variables. The concepts of Banach spaces with super- and subquadratic averages have been introduced in a paper of E. DUBINSKY/A. PEE,CZYI$SKI/H. P. ROSENTHAL [l].The final definition of operators having (8.p)-type and (s,p)-cotype was made by B. MAUREY [2]. Important contributions are also due t o T. FIGIEL/G.FISIEE[i], B. MAUREY/G.PISIER [I], 121, and G . PISIER [I], [3], as well as to J. HOFFMANX-JBRGENSEN [I] and s. K W A P I E [71. ~~ Recommendations for further reading:

[KtzC], [KAH], [SEM,, Denx journhes p-radonifiantes], [SEM,, exp. 5-7, 8, 15, 17, 221, [SEM,, exp. 3, 6, 8, 24, 251, [SEM,, exp. 3, 4, 6, 71, [LIT, vol. 111. A. BADRIKIAN [I], s. CHEVET [2], J. HOFFMANN-J0RGENSEN/G.PISTER [I], M. J. KADEC [I], M. J. KADEC/A.PEI,GZY&SKI [i], W. LINDEIA.I’IETSCR [i], J. LINDENSTRAUSS [l], B. MAUREY [I], E. BI. NIAISHIN [l], A. PELCZYT~SKI [6], J. PUIIL[l], J. R. RETHERFORD [el, H. P. ROSENTEAL

[a], [7].

22.

Operators in L,-Spaces

Given an operator ideal on the class of Banach spaces we niny ask whether its Hilbert space part coincides with some operator ideal 6,.The answer is affirmative for most of the known ideals of absolutely summing, nuclear, and factorable operators. However, there is also a counterexample due to G. BENNETT. Next we introduce so-called L,-spaces which are closely related to those defined by J. LINDENSTRAUSS and A. PELCZY&SKI. The main purpose of this chapter is t o investigate diagonal operators from I, into 1,. Using the concept of a limit order we give almost necessary and sufficient conditions for those operators to be absolutely r-summing, r-nuclear, p-factorable etc. According t o an important theorem of H. KONIGthese results can be carried over to weakly singular integral operators.

22.1.

Operators in Hilbert Spaces

22.1.1. It can easily be seen that, for 0 < p absolute moments

< 00, there exist the n-dimensional

We have

and

Observe that c y . = c Z p ;ef. 23.1.2. P u t a,, := c!$/cg). Then 1

alp 2 a2p2

1

... 2 0

and lim n1/2anp = c+&'. It

22.1.2, Let 02" be the normalized rotation-invariant measure on the n-dimensional unit sphere S:. L e m m a . If 0 < p

< co,liken

800

Part 4. l3i~sicExamples of Operator ldcds

Proof. Thc above forinula can be checked by using polar coordinates. It also follows from 21.1.3 and

22.1.3. Lemma. If 0 < p

< 00, then. P p ( k Zg ->

Z;)

=z

a;;,'.

Proof. Using 17.3.2 and the preceding lemma we get, P J I : 1% -+ 1;) 5 Q;;,'. To establish the converse inequa1it.y we suppose that

where p is some probability on the closed unit ball

\ Ilxii:

1=

CZG;(~)

U;.Then

5 op j J ~ ( xa>lp , da;(z) d,u(n) u; s;

S;

J iiailg d,u(u) 5 o p n ~ ~ .

GPUE~

U;

so

2 c and by 17.3.2 we have P,(I : 1;

'19.1.4. Lemma. I f 1

D ( p , q ) ( IIf:

-+

)!Z 2 a;;.

< p 5 q' < 00, the%

-> )Z;

= 0;;~;;

and

L(p*,q,)(I: 1; 3 Z;) = ru;Ctbptinq.

Proof. By 17.4.3 we have

5 P#: z;

D ( p , q ) ( I2;: + )z; If the operator A,, E i?(Z!, then llAlapii= anP.Let f l .

L,(S;,

--f

If) P y y I : z'; -+ )z; 5 a;;.;;,'.

o;)) is defined by

=

C ei @ f i . 1

Since ( f i , f k ) = n-lcik, we get the &gram

Hence

Anpz:= f,, whero /$(a):= (2, a),

..., f, denote the coordinate funotionals fi(cr)

n

An,

(1) := ai. Clearly

22. Operators in Lp-Spaces

301

Finally, it follows from n = trace ( I ) 5 D ( p , q ) ( I I!:

3

I;) Ii(P.,qt)(I: 1; -+ Z;) 5 n

that we have equality in (1) and (2). 32.1.6. Lemma. If 1

Lf,"b,(l:z;

3

p

2;)

q'

< 00, then

= unq,u;I;:.

Proof. Let the operators Aw9and A,, be defined as in22.1.4. Then we have t'he diagram T

Since .;;Anp is a metric injection, it follows that

;z

g (p,q)(I: nj -+2;)

(*I

5 unq*a;;pl*

By 19.5.3 we have ;;ZC

= P,(I :

;z

3

z;) 5 L$,\)(Z :z;

--f

2;) P y ( I :z;

3

2;) 2 a;;.

Therefore equality holds in (*). 22.1.6. Lemma. If 0 < I, < s 5 2, then

M ( 8 s P ) (2;I : 3 Zg)

= T ( s , p ) ( IZt: ->

)2;

= u,~u;;,'.

Proof. Let xl,...,x, E ;2 and write xi = (fil,

...,E i n ) . Then

302

Part 4. Basic Examples of Operator Ideals

Hence, by 21.2.5, we have

B€(8,p)(I: 12" + 1;) 5 T::$(I: Z: + Z;)

= T ( 8 , p ) ( I1;: + )Z:

5 (x,,u;;P~.

(*I

On the other hand, 20.2.1 implies

P J I : 13" -+ ZP) 5 3 q P ) ( I I;: -+ 1;) P8(I:22" -+ z;). Therefore u;;

5 i?I(s,p)(I: 1; 3 )z; a;;.

So equality holds in (*). Remark. It turns out that TC2,JI:I:

22.1.7. Lemma. If 2 2 p

<

00,

--f

Zi) = 1 for 2 5 p < W.

then

C(,,pl(I: 1; i Z;) = aspa;:. Proof. Let xl,...)xm 1: and write xi = (Ell,

.,., 2iJ.

As in the preceding proof

we have

Now it follows froin 2 5 p that

Hence

C(2,p)(I: 1; 3 Z;) S anpa;;. On the other hand, 21.4.7 implies

Pz(1: 1; + )z;

s c(2,p)(l: 1; + )z;

P J I : 1;

--f

z;).

Therefore an, 5

c(2,p)(l: z; 3 )z;

unp.

Remark. It turns out that C(2,p)(I: lf -+ 1;) = 1 for 0

< p 5 2.

22.1.8. Theorem. On the elms of Hilbert spuces

3, = G2

for 0 < r

< 00.

Proof. Since the quasi-norined operator ideals [91(8,p), NYI,,,,)] and [ 6 ( ~C ()2,, p ) ] are rnaximal, it follows from 22.1.6 and 22.1.7 that the identity map of every Hilbert for 2 5 p < 00. Hence, s 2 2 and to space belongs to 9Jl(s,p) for 0 < p

by 20.2.1 and 21.4.7, the operator ideal pr does not depend on the parameter r. Finally, ?&= G2 yields the assertion; cf. 17.5.3.

Remark. I n order to prove that !& = 6, for 1 S r < w we could also use 19.5.3 and 22.1.5 as well as the fact that all operator ideals on the class of Hilbert spaces are

symmetric.

22. Operators in Lp-Spaces

303

22.1.9. Theorem. On the class of Hdbert spaces

R1=31=6,. Proof. It has already been proved in 15.5.3 that R, = 6,. Since GIis maximal, breaking a fly on the wheel, we get 3,= !Ryx= 6,;""" = G1.

22.1.10. T h e o r e m . On the class of Hilbert spaces

R, = 3,= G2

for 1

< r 5 00.

Proof. Clearly, Pr= G2 iniplies .3;, = (!&*)* = 6; = G2. Since B2 is minimal on R P= Gmin- G2;see also 22.4.3. t.he class of Hilbert spaces, we get !Rr = !

23.1.11. Next we state a n immediate consequence of 22.1.4; cf. also 19.4.4. Theorern. On the class of Hilbert spaces

B(p,r)= B for 1 < p 5 q' < 00. 22.1.12. Theorem. On the elass of Hilbert spaces ?$(,,p,

= 6,

0

for

< p 5 2 and

112 - l/q = l/p - l/r

< 112.

Noreover,

P(r,p) =B

for 0 < p 5 2 and

l / p - l/r 2 1/2.

Proof. It follows from 22.1.6 that the identity map of every Hilbert space is (2,p)-mixing for 0 < p 5 2. Hence, if l / p - l / r < 112, by 17.2.6 and 20.2.1 (remark) as well as by 17.5.3 we get !&r,p) = ?4$,,2) = Gq.On the other hand, if l/p - 1/r 2 112, it follows that i? = !4#c,,2, 5 !&,.p).

22.1.13. The remaining case has been treated by G. BENNETT [2] and G. BENNETT/ V. GOODMANIC. M. N E W [l]. If B(,,,) denotes the Lorentz operator ideal defined in 14.2.20, then they proved the Theorem. On the class of Hilbert spaces for 2

P ( r , p ) = G(2r/p,r)

22.2.

< P < r < 00-

L&3paces

22.2.1. Let 1 5 p 2 co and put

.

L, := Space (Bp)

In what follows Banach spaces belonging to the ideal L, are denoted by L,.

22.2.2. We now state a n iniportant characterization. Theorem. Let 1 s p 5 00. A Banach space Pbelongs to Lp i f and 07dy if E" is iamwrphic to a complemented subspace of some Banack space L p ( Q , p ) defined on a suitable measure space (Q, p). P r o o f . The assertion follows from 19.3.7 and 19.3.9.

304

Part 4. Basic Examples of Operator Ideals

22.2.3. T h e o r e m . Let 1 < p < 00. A Bamch space E belongs to L, if and only if E is isomorphic to n complemented subspace of some Banach space Lp(C?> p ) defined on CI suitable measure space (Q, p). Proof. We see from the preceding theorem that all L,-spaces with 1 are reflexive.

< p < ca

22.2.4. Clearly 19.3.3 yields the T h e o r e m . (Lp)dua’ = L,,,

32.2.5. The next result is a consequence of 19.4.4. Theorem. If 1

< p < m, then L, C= Lp.

22.2.6. The following theorem implies that the Banach space ideals L, and q, p $: 2, and q 2. non-comparable for p

+

+

T h e o r e m . L, n L, = F for 1 5 p

L, are

< 00,

F for 1 < q 5 cw,

L, n L,

=

L,nL,

=L,

for l < p , q < O O , p + q g .

Proo-f. Let E’ E L, n L,. Then 22.4.2 yields 2 ( E , E ) = !&JEl, E ) for 1 5 p 5 2 and B(E, E ) = ?&(E, E ) for 2 < p < r < 00. Since all operator ideals ?&are proper. we see that E is finite dimensional. Hence L, n ,L = F. The formula L, n L, = F with 1 < q assertion is proved in [LIN, p. 2011.

22.3.

5 00 follows by duality. The remaining

Banach Spaces of (s, p)-Type and (2, p)-Cotypo

23.3.1. We begin wit.h a n elementary L e m m a . Let ( i l f , , ~a)d (N,v) be a n y measure spaces. Suppose that the scnlnr function f is ( p x v)-?tzeasurableon ( M x N ) . If 0 < p < q < 00, then

P r o o f . Assume that the right-hand integral is finite. Then by f ( a ) := If(&, .)I” we define a function f on M taking values in L,,,(N, v) almost everywhere. Now the required inequality follows from

and

22. Operators in Lp-Spoces

32.3.2. P r o p o s i t i o n . If 1 5 q 5 2 and 0

4E

305

< p < s < q, then

T(8.P).

Proof. Let ( 0 , ~be) any measure space. If fi, , . . : f m E L,(Q,p), then by E.5.2 we have

Hence

This means that i o q ( f i ) 5 l&fi). Since rp(fi)5 rq(fi)it follows from 21.2.6 that Lq(Q,p) has (8,p)-type. Moreover, '(S,P)(ILq(Q,p)) < =c

(*)

qSc-1. qp

22.3.3. Proposition. If 2 5 q

< 00

and 0

< p < 00,

then

L, E T ( 2 , P ) . Proof. Let (Q, p ) be any measure space. If f l , ...,f m E Lq(Q,p), then by 21.1.3 and 22.3.1 we get

Hence bq(Q, p ) has (2,q)-type and therefore (2,p)-type with 0 < p

< P 5 P. If 1 5 q 5 2 a d 0 < p

(*I

T(2,P,(Lq(Q,,,) 5 C2*Gpl for 0

22.3.4. Proposition.

< m. Moreover,

co,then

4 E C~2.P). Proof. Let (Q, p ) be any measure space. If f l , and 22.3.1 we get

20 Pietach. Operator

..., f,,,

E Lo@, p ) , then by 21.1.3

306

Part 4. Basic Examples of Operator Ideals

Hence LJQ, p) has (Z,q)-cotype and therefore (2, p)-cotype with 0 Moreover,

< p < co.

s

C~Z.P)(IL,(S),,)) e2pcG1 for !2IP < 22.3.5. Summarizing the-preceding results we have the fundamental T h e o r em. Lq E

M(2,p)

for 1

(*)

5 q 5 2 , 0 < < 2,

Lq E M ( 8 . p ) for 2 S q

< s',

0
< < 2.

P r o o f . The first inclusion follows from 21.4.9 and 22.3.4. If 1 < q 5 2, we may also use 21.2.5 and 22.3.3. Moreover, 21.2.5 and 22.3.2 imply L, E M(s,p) for 2 5 q

< s',

0
< s < 2.

R e m a r k . Let us mention that the preceding result cannot be improved. Namely, by 20.1.17 we have M(s,p)= F for 2 < s < m and 0 < p < 8. On the other hand, according to 22.4.7 and 22.4.11 (remark),

P s ( I : 1; for 0 < p

+-

Zt) 5 nlls

and PJI:

17 +- Zy) =: (n log n)l's

< s < 2. Hence

lim N ~ s , p l (1',I :+ I,".) = 00. n

This means that I,.

M(s,p).

22.3.6. We now give a, finite version of the preceding theorem which follows iminediately from the formulas (*) in 22.3.2, 22.3.3, and 22.3.4.

L e in ma. M ( z , p ) ( I : 1: + 1;) 5 $,+$

for 1 < q 5 2, 0 < p

5 c z p , c ~ l for 1 5 q 5 2, x ( s , p ) ( I : 1; --f )z; 5 c*lsc7; for 2 s q < s',

N ( 2 , p ) ( I : 1;

--f

1:)

< 2, 1 < p < 2, 0 < p < s < 2.

R e m a r k . It wouldbevery interesting t o know precise estiniates of N(2,1)(1: $7 +Zy).

+

22.3.7. Let us recall that ?lRm(s,p) and ?J3(r,p) with 1/r 11s = l/p 5 1 almost coincide. We now show that these operator ideals actually have the same limit order. T h e o r e m . If l/r

+ 11s = l/p 5 1, then

4rn(.SJl),u,v) = 4 v ( r , p ) , u, 4 P r o o f . Since IJn(oo,p) = !&p,p) and ?l.R(p,p)= !&oo,p), in what follows we may suppose t h a t 1 5 p < s < co. Let p < so < s. By 20.1.11 and 20.1.12 we have r n ( S , P ) c= v ( r , p ) 5 m ( s , , P P Hence i t ( m ( s , p ) , Ul

-

v) 2 J$p(,,,), u, 71) 1 4 m ( s o , p ) >21, v)

Pinally, t,he assertion follows from (*!

22. Operators in L,-Spaces

307

I n order to verify this estimate we put 8 := sols. By 20.1.8 and 20.1.13 we get

M ( s , s o ) ( I2;: +-):Z 5 X(m,80)(I: Zz -+ Z;)'-'

5 Pso(l:z;

M(so,so)(I: Z; +-

2 nl-5

3 2;)l-e

and

B€(8*p)(I: 2; --f 13) 5 ilx(80.P)(I: z; -+ I;) M(s,so)(f:z;

-+ I:).

Consequently

M(so,p)(I: 3 );z

5

implies = < ,) p,tn+(s-so)iso. &,p)(I: -+I

So (*) follows by applying 14.4.3. R e m a r k . The limit order of

22.4.

will be investigated in Section 22.6.

Absolutely +Summing and ?--IntegralOperators

22.4.1. It has been proved in 17.3.9 that the operator ideals ?&strictly increase with the parameter r 2 1. For special Banach spaces E and F , however, the components &(E, F ) are constant on certain intervals. Theorem. Let It and P be arbitrary B a m c h qaces. Then

5 u 5 2,

0 < r 5 2,

!&(Lt',F ) = ?&(L,, 3 ) for

3

!&(L,, F ) =!&(La, a)

for

2 5 u 2 00, 0

v 2 ( E ,L,) = v r ( E ,L,)

for

15v

5 2,

< r < s < u', 2 5 r < 03'.

P r o o f . The assertion follows from 20.2.1 and 22.3.5 as well as from 21.4.7 and 22.3.4. Remark. If 2 < v 5 00, then there cannot exist any interval A such that V r ( E ,L,) is constant for r E A and all Banach spaces E ; cf. 22.4.12 and 22.4.13,

22.4.2. We now state the adjoint result. T h e o r e m . Let E and F be arbitrary Banach spaces. Thew

3 , ( E , L,) = 3 , ( E , Lo) for

15 v

5 2,

3 w ( E 7 L w ) = 3 r ( E , L vfor ) 2 5 ~ 5 0 9 ,v < r s 0 3 )

S2(L,,F ) = 3,(L,, P) for

1 5 ' 1 ~5 2 ,

1 < r 5 2.

Proof. The assertion follows from 20.2.3 l n d 22.3.5 as well as froin 21.4.8 and 22.3.4. R e m a r k . I n particular, we have i?(Lm)A,) = p2(L,, Lo) for 1 5 ZI 5 2 and C(L,, L,) = !&(L,, L,) for 2 < v < r < 0 0 ; cf. 20.1.16. 20"

308

Part 4. Basic Examples of Operator Ideals

22.4.3. The next theorem is a n immediate consequence of 22.4.1 and 22.4.2. Theorem. Let 1 5 u, v 5 2. Then pr(Lu,L,)with 0 < r with 1 < r 5 co do not depend on the parameter r.

< 00

and 3,(Lu, L,)

22.4.4. We now prove the famous Grothendieck T h e o r e m . O(L,,L,)= 'pr(L,, L,) for 0 < r < m. Proof. By 22.4.1 we have p,(L,, L,) = !&(I,,, L2) for 0 < r 5 2. so it remains to check B(L,, L,) = @,(L1, h,).Let S E B(L,, L,) and (xn)E m,(Ll). Define the operator X E B(&, L,) by

L,) Since the dual of L, is a n Lar-space, it follows froin 22.4.2 that X E (Ippl(Z,, and therefore S X E Ip:m1(Z2, L,). Hence, by 19.5.2, we get S X E !&(&, L,). This proves that (Sx,)= (&'Ken)is absolutely 2-surnmable. So S E !&(Ll, L,). R e m a r k . The first proof of this theorem understandable for average mathematicians was given by J. LINDENSTRAUSS/A. P E L C Z Y [l]. ~ KSee I also [LIT, vol. 1. p. 681.

22.4.6. The so-called Grothendieck constant cG is defined to be the infimum of all numbers e 2 0 such that Pl(S) e /lSl\whenever S E I!@, Z2). Although nobody needs the exact value of Grothendieck's constant, everybody likes to know it. Our method yields cG 5 M[,,,)(I: 1, + 2), lb&,11(1: 2, + 21). R e m a r k . I n the following we list some upper bounds of Grothendieck's constant which have been obtained until now :

A. GROTRENDIECK [4]

< 2,302 A. GROTHENDIECK [4] < 4,604

R. E. RIETZ[ 11 < 2,261

J. L. KRIVINE[2] < 1,782

(real case)

S. KAIJSFJ~ [l] <1,807

G. PISIER[5] < 1,527

(complex case)

22.4.6. We now prove the adjoint version of 14.4.4. Lemma.

On the other hand, we have This proves the assertion.

310

Part 4. Basic Examples of Operator Ideals

for 2susc0,2 Since n 5 P2(I::1 -+ 1;) P 2 ( IZ; We now have the

~

v

~

w

.

-+ I : ) , identity holds in (*).

Proposition.

This yields t,he diagramm :

I 2

22.4.10. The following trivial statement is an important tool in computing further limit orders. Lemma. Let [U, A] be a normed operator ideal. Then

A(%*, u,v) Proof. If

E

+ I.(%,

v, u) I 1.

> 0, then

n 5 A*(I: 1;

--f

l:) A(I: 1; -+ 1;) 5 e*~nl(l['.u,v)+&enl(~,v,~)+"

where e and e* are some constants. Remark. For most of the examples identity holds in the above inequality. HOWever, there are normed operator ideals [%, A] for which A(%*, u,v) A(%, v, u) = 2 ; cf. H. MONIG [2].

+

32.4.11. Lemma. Let 1 5 s

PP(k1,". -+

z); g

< 2 and 0 < p < 8. If E > 0. f l w i

Q?zl/s+&,

where e is sorne constant. Proof. Define a,, by l/so:= l/s + E . Clearly p .< so < s, if Then 22.3.6 implies P,(I : 1; -+ )z;

5

,,)(I: ,;z

--f

z;,)

P s 0 ( l ,:;z ->

E

is sufficiently small.

);z 5 c,,oc,-aP,o(I:79 + 1:).

22. Operators in Lp-Spacee

Now the assertion follows from

cf. [SEK, exp. 311. Protract,ed computations yield

P,(I :;z -+ 2,")

",(n log n)"S;

G . BAUMB.4CH/W. IAIXDE [I]. 22.4.12. P r o p o s i t i o n . If 1 5 r 5 2 and 2 5 u 5 00, then Cf.

15 v

5 u', u'2 v 5 0 0 , 5 u 5 co, 1 2 v 5 r ,

for 2 5 u 5 r ' , l/u' for 2 5 u 5 r ' ,

1jv

4cpn u , v) =

l/v

for

7'

Moreover,

A(!&,

u,?I)

+ A(%,,

u, u ) = 1.

Proof. It follows from 14.4.6 and 22.4.11 that

+

A(?&,

u,v) 5 I,(!&, u, u') I@, u',v) 5 l/u'

A(?&,

u,w ) 5 A(&

and u, u')

+ A@*,

for u' 5 v

v', v) 5 1/v for v I ,u'.

On the other hand, according t o 18.2.2 and 22.4.7,

A(@,,

u,V ) 5 L(Bj U, 00)

Y W T ,

u, v) 5

+

jb(@r,

60,

r)

+ A(E>r , v)

yields

l / v for r 2 v , l / r for r

5 v.

Consequently

4%

u,a ) 5

I

l/v for 2 5 u 5 Y', 1sv 5 u', l/u' for 2 5 u 5 r ' , u I 5 v 5 m ,

i

f o r r ' ~ u ~ c le s, z i g r . l/r

for r ' g u s c o , r s v s w .

Analogously it follows that

l(%n,*, b, U ) 5

1lu l/r'

1 u, for r' 5 w.

for r'

Moreover, &(I: Z; + Zi) = 1 and

A(%,,

u,

u)5 n(c,v, 1)

+ A(%n,,

1, 2)

Q

+ A(2,2 , u)

311

312

Part 4. Basic Examples of Operator Ideals

imply

A(?&,,

v, u)5 l/v‘

for 2 S u 5

00.

Summarizing the above estimates we have

A(!%*,

0, u)5

l/v’ €or 2 5 u 5 r ’ , 1 5 v 5 u‘, l/u €or 2 s u g r ‘ , u ’ ~ v ~ m , l/v’ for r ’ s u 5 o c , , l s v s r ,

I I

llr’

for r ’ s u 5 o a , r ~ v ~ o c , .

Applying 22.4.10 we see that identity holds in (1)and (2). Hence

A(!@,, u,v)

+ A(%,.,

v, u ) = 1.

The remaining case, where 1 5 u 5 2, can be treated by 22.4.1 and 22.4.2. Then we get I.(!@,, u, v) = A(!&, u, v) and A(?&, v, u)= A(%*, v, u). This yields t,he diagrams :

1

f

22.4.13. Proposition. If 2 < r

!

< oc, and 2

l/u‘ for 1 S u s r ‘ ,

3.(!@,, u, 4 =

where

e

:= 1/‘

e

A(!&, u,v)

5 m, then 2 ~ ~ 5 0 0 ,

l/v

forr‘sug2, 2 5 v 5 r r , for 2 S u S m , 2 5 ~ 2 1 ,

Ijr

for r’ 5 u

I

00,

l/u) ( l / v - l/r) + (l/r’ -112 - 1/r

Moreover,

v

+ A(%r,,

21,

u)= 1.

r

S v 5 cyi,

22. operators in Lp-Spaces

313

Proof. First of all we mention that the most complicated case, where r' < u < 2 and 2 < v < r, has been recently treated by E. D. GLUSKIN[2]. See also E. D. GLUSKIN/A. PIETSCH/J. PUHL [l], and A. PIETSCH [23]. In the following we only deal with the remaining values of u and v. Since we have 5 A(%,, u,w) the estimates from above are immediate consequences of 22.4.12. Hence l/u' for 1 5 u 2 r', 2 5 v 5 00,

A(!&u,v)

i

W & , U , VIjv ) ~for l/r

2 5 ~ 5 0 0 2, s v S r , forr'susce,rsvsco.

(1)

As in the preceding proof we get

CLld

A(%,.,

v, u) 5

l/v'

for 2 5 u 5

l/u

for l s u s r ' ,

00.

Consequently 2svVco,

l/w' for 2 5 u 5 0 0 , 2 5 w 5 Y , l/r' for r ' ~ u s c or,s v o m . According to 22.4.10 identity holds in (1) and (2). Hence

+

I.(?&, u,v) v, u)= 1. The remaining case, where 1 5 v 5 2, can be treated by 22.4.1 and 22.4.2. Then w, u ) = v, a). we get A(?&, u,v) = A(?&, u,w) and This yields the diagrams: Ib(%,,

r

r' I

3 14

Part 4. Basic Examples of Operator Ideals

Here Q' := 1/p -

(1/u - l/r') (1/ r - l/v) l/r - 112

R e m a r k . The level curves in Gluskin's squares are hyperbolas. R e m a r k . Since I.(?&, u , v) < A(&, u, v) whenever r' < u < 2 and 2 < v < P, we see that ?&=# Srfor 2 < r < 00, and, of course, also for 1 < r < 2 ; cf. 19.2.9.

22.5.

p-Dominated and p-Factorable Operators

22.5.1. Theorem. If 1 5 u 2 p 2 v 2

00,

then B(L,, L,) = B,(L,, L,).

Proof. Let S E B(L,,, L,) and 1 5 p 5 2. If X from 22.4.1 t,hat X E VY'(E, L,,).Now 19.5.2 yields

W ( E ,La), then it follows

HX E @Pal(E, L,) 5 &(E,L,) G 3 J E , 4). Using the formula Bp = S p o (@y we )get 1S E B,(L,, L,). The case 2 2 p 2 00 can be treated by passing to the dual operators. R e m a r k . The above result means that

P r o o f . The estimates from above are consequences of

On the other hand,

we get the estimates from below. Now by 9.1.8 and the preceding lemma we have the Proposition. l/v-l!2 for 1 5 ~ 2 2 ,1 5 ~ 2 2 : for I S u U 2 , ~ ~ v ~ c A(i?2, u,a) = l l v - l l u for 2 5 ~ 5 0 0 1, 5 ~ 5 2 , ( 1 / 2 - 1 / u f 0 r 2 ~ u ~ 02 05 v, I C w ; .

c

,

22. Operators in Lp-Speces

315

Moreover, I.(&, u, v)

+ qa,. v, u)= 1 .

u

i

///'/I

v

1 2

A(&, u, v)

0

for 1 5 ~ 5 2 ,

112) - 11.

for 2 5 u 5 p ,

=, d

for 2 g u s p , for Z S u U p ,

0 l/v-l/u

i

where 0-:=

f o r p s ; ' ~ 5 m 1. 5 1 ) S p .

l , p - l i u for p 5 u 5

00,

p 5 z) 5

L i .

(112 - 1/20 ( 1 / u - l / p ) 1j2 - l i p

Mort over,

qep,u,v) + f ( P p ' 11,, u ) = 1. Proof. First of all we mention that the most complicated case, where 2 < u < Y aiid 2 < v < p , has been recently treated by E. D. GLUSKIN/A.PIETSCH~J. PUHL [I]In the following we only deal with the remaining values of 91 and T. The estimates from above follow from

A(!&,

91.

v) 5 A&?,,

and l.(i?,,u,23)

u,c)

On the other hand, !&,, o 2, o

5 ?.(a,z i , p ) f i ( 2 . p .c ) .

Qpl & 3 implies e

A ( ! p y , q, u)+ Ace,. u.

27)

+ A(@,,

Using the known limit orders of @ !,, estimates from below.

and

0,

p)

2 1.

vy,by

it

suitable choice of q . w e get the

316

Part 4. Basic Examples of Operator Ideals

This yields the diagrams

? Here g'

:= 1 -

(l/u

- l/p') (1/2 - l/v) V P - 112

22.5.4. The above result can be carried over to the case where 1 < p j ( eP'I U,P ' )

=:

2(EP, of, u') and A&*,

< 2 by

a,v) = A D p , v', w').

Since 2 , = 3- and 3, = PI,we also know the limit orders of 2, and of 2, and 3,. 82.6.

as well as

Absolutely (P, p)-Summing Operators

22.6.1. We now prove ib classical result of W. ORLICZ El]. Proposition. If 1 5 q 5 2, then L? E P(2,ii. Proof. By 20.1.11 and 22.3.5 we have L, E M(z,l)2 P(p,l). However, there is also a direct proof. Let (Q, p ) be any measure space. If fl,

So LJO, p ) has the Orlicz property.

...,f,,, E L,(Q, p), then

22. Operators in L,-Spaces

22.6.2. P r o p o s i t i o n . If 2 5 q

317

< 00, then Lq E P(q,l).

P r o o f . Let, (Q, p ) be any measure space. If f l ,

...,f m E L,(Q, p), then

So L,(O, p) belongs to P(,,l). 22.6.3. As a counterpart of 22.3.5 we now have the Theorem.

Lq E P(r,p) for 1 5 q 5 2 . 0 < p < 2 , l/p - l/r = 1/2, L, P ( r , p , for 2 2 q < w . 0 <: p < q’, l / p - l j r = 112’. P r o o f . Let 1 5 q 5 2. I f 1 5 p 5 2 , then 17.2.6 implies Lq E P(z,l) 5 P(r,p) for 1/p - l / r = 112. If 0 < p < 1, then L, E M(l,p, by 22.3.5. Tlicrefore it follows froiii 20.2.1 (remark) and Lq E P(s,l) that Lq g P(r,p, whenever I / p - 1/r = 112. The case 2 5 q

< 00

can be treated analogously.

22.6.4. To prove tho deepest result of this section we need the following

a@&,

S u b l e m m a . Let S E F ) and 1 infimum of all constar& 5 2 0 such t h d

< p < r < 00.

Then P(r,p)(S) equals the

J for all finite families of functionals bl,

...)b,

5 P’.

Proof. First we consider any constant for which (*)isfulfilled. Given xl, ...,xm E Zk, = I I ~ X ~ I \ +Writing ~. me choose bl, . .., bm E F‘ such that ( t k k , bk) = li#zkllr and xk = ( f 1 k , * * * ) Erik) we get in

n

1,(SXx.)‘ = 2’ (XrC,, b k ) = c’ k=l

5 al.p(bk)W.p(xk)

m

(Sei, b k )

ti&

i = l &=l

= (Tlr(8z&)‘-’

wp(zk)

-

This proves that P(r,p@?)5 a. I n order t o check the converse statement we take arbitrary bl, ...,b, E 3’’. If (Sei, bk) + 0, then define f i k by

318

Part 4. Basic Examples of Operator Jdods

Otherwise put

$ik

= 0. We get

and therefore

..., tnk).Since

where %k = (Elk;

it follows that

-4

I-1

I

f l(Sei, bk)lP’

k=I

UP’

5

m

l(8xk, bk) 5 lr(&xk)lr*(bk) 5 P(r,p)(S) 1,&). k-1

We are now able to establish the fundamental Lemma. Let S E i!(lL, P)and 1 < p Pro(#)

< r < r, < 00.

Then

5 grorp(r,p)(&)>

where g,,, is sonze constant.

Proof. Put. q := up’, s := ur‘ and so := ur& where 0 < u < 2/p’. Obviously 0 < u < so < s < q < 2. If b,, .,a, E F’, using 21.1.5 and the preceding sublemma we have

..

So the sublemma yields

22. Operators in L,-Spaces

319

In the real case we get

The same limit appears in the complex case. 12.6.3. Theorem. If 1 5 r

< ro < m, then p(r,l)(Lm, F ) Qr0(Lm, F). such that 1 < p1 < rl < ro and l/pl - l/rl = 1 - 1.

Proof. Choose p , and r, Then & p(rl,pl). On the other hand, the preceding lemma as well as 17.1.3 and 19.2.5 imply

v(r,l)

%l,p,)(L

F ) G V r o ( LF >)

Remark. The above theorem means that ?&r,l) o 2, E

32.6.6. Proposition. If 2 5 r

4 q 3 ( r , 1 ) >u,21)

=

< bo, then

lo

I/w-l/u for l z u z r , 1 $ w 5 u , for 1 5 u u r , u 4 v s m , llv - l/u for r 2 u 5 0 0 ; 1 5 v 5 r , l\r-l/u

for

r S u Z m ,rSvSca.

Proof. By 22.6.1 and 22.6.2 we have

A(?&r,l), q, a) = 0 for 1 5 Q 2 r . Now it follows from 4@(T,l),

+ WL u>v)

u,v) I 4V(T,,,,u,u)

that

This proves the estimate from above.

:{20

Part 4. Basic Examples of Operator Ideals

By I,(ei) 5 P(rrl)(I: -+ I:) wl(ei), we have nl/r5 P(r,l)(I:1; -+ ZEj n'!U. Consequently I(!&r,l), u,v) 2 1/r - l/u. Moreover, u,v) 2 3.(8,u,v). This yields the desired estimates from below. We now give the corresponding diagram :

29.6.7. The following result is due to G. BENNETT [l]and B. CARL[l].

+ l / u - l/v.

Proposition. Let 1 5 u 5 v 5 2 and l / r = 112 I from 1, into I, is absolutely (r, 1)-summing.

Then the embed-

ding map

Remark. The spccial cases u = 1, u 6.5.4 and 22.6.1, respectively.

=

2 and 1 5 u = v

52

were treated in

22.6.8. Finally, we give some diagrams which are due to B. CARL/B. MAUREY/

J. PUHL[ 11 : 'I 2

L+L

r 2

22.7.

Embedding Maps of Sobolev Spaces

22.7.1. Let l2 be a bounded open subset of the n-dimensional Euclidean space 9" having n sufficiently smooth boundary. For 2 2 0 we write 1 = 6 [A],where [A] is the greatest integer not exceeding I and 0 2 8 < 1. If 3. = 0, 1,2, ..., then the Sobolev space W#I) consists of all functions f defined

+

22. Operiltors in L,-Spaces

321

on Q such that the weak derivatives Df of order la1 2 I. exist and belong to L,(Q). I n the case where R 0, 1,2, it is assumed that, for all a wit,h1 . 1 2 [I.], we have

+

...

and

Then W#2) is called a 8obolev-8Zobodetzkij waee and a Holder space, respectively. Let us mention that W$2) becomes a Banach space with a suitable norm. The theory of these spaces is fully presented in [TRI]. 33.7.2. As a basic result we state Sobolev's

Theorem. Let A/n > l / u - l / v 2 0. Then W:(SZ) c L@). 22.7.3. For every operator ideal U the Sobolev limit order u,(a, II, v) is defined to be the infimum of all A > 0 such that the embedding map from Wt(i2) into LJQ) belongs to a. 22.7.4. Sobolev's limit order is closely related to the ordinary one. So, in order to obtain informations about embedding maps of function spaces it is enough to consider diagonal operators in sequence spaces. The non-trivial proof of the following L KONIU[l]. result is given by E T h e o r e m . Let

[a,A] be u quasi-normed operator ideal. l'hen

o,(%, u, v)/n = A(%,

26,

v)

+ l / u - I/@,

where 12 i s the dimension of Q.

22.7.5. Suppose that the bounded kernel K is infinitely differentiable on Q the eventual exception of the diagonal. Then, for certain u and v,

K , :f ( t ) -+

g(8) :=

SZ with

J IS - tl"(a-l)K(g,i?) f ( t ) dt

Q

with a > 0 defines a so-called weakly & t &ar

iittegrcrl operator from Lu(0)into

WQ). Special examples are the Riemann-Liou.viL?eoperators of fractional integrdion given bv

B, : f ( t ) + g ( s ) := -

J'

rb)0

where 0 5 s, t

(8 -

tp-1 f ( t ) at,

5 1.

22.7.6. The significance of Sobolev's limit order follows from t h e next T h e o r e m . Let U be an operatorideal. If a

> ua(%, u, v), then K , c 21(Lu(Q),L*(fl)). P

R e m a r k . The above condition is almost necessary and sufficient, since, conversely, a < u p , ~ ~ (u, U ,4 implies Ra 6? U(L,[O, 11, LJO, 11). 21 Pietsoh, bpcrat,or

332

Part 4. Basic Examples of Operator Ideals

22.8.

Notes

There is an extensive literature dealing with operators in L,-spaces. First of all we have to mention the pioneering papers of A. GROTHENDIECK [4] and J. LINDENSTRAUSS/A. PELCZYRSKI [l]. The coincidence of Hilbert-Schmidt operators and absolutely r-summing operators in [4] and A. PIETSCH [5]. Further results conHilbert spaces was proved by A. PILCZY~SKI cerning absolutely (T, p, q)-summing and ( T , p, 9)-nuclear operators in Hilbert spaces can be [2], G. BENXETT/V. GOOD MAN^. M. NEWMAN [l], B. CARL [I]. [2], [&I, found in G. EENNETT D. J. H. GARLIBG [3], Y. GORDON/D. J. LEWIS [3] and A. PIETSCH/H.TRIEBEL[l]. The main theorems about absolutely r-summing operators from L, into L, are taken from S. KWAPIE~~ [5], [6] and I?. SAPHAR [3]. See also G. EENHETT [l], B. CARL [2], [3], D. J. H. GARLING[5], A. PIETSCR [16], and A. E. TONG[l]. The definition of a limit order was given by A. PIETSCH [14]. We also refer to a famous paper of H. KONIQEl]. Recommendations for further reading:

[LIN], [LOT]. G. BAUMEACH/W. LINDE[l], G. BENNETT [4],B. CARLIB.MAUREY/J.PUHL[I], C. CLARK [I], J. A. COCJIRAN[I], Y. GORDON/D. R. LEWIS/J.R. RETHERFORD [l],[2], D. J. H. GARLIXG/ Y. GORDON[l], E. D. GLUSKIN [2], J. R. HOLUB[3], [5], J. S. HOWLAND 111, H. KONIG[2], [3], [4], J. L. KRWINE[2], S. K W A P I E[l], ~ [2], D. J. LEWIS121, J.LINDENSTRAU~~/H.P.RO~E THAL [I], 13. MAUREY[2], A. PEECZY~~SKI/W. SZLENR[l], G. PISIER [5], N. POPA[l], J. R. [ 2 ] , R. E. RIETZ[l], R. ROGGE [l], [2], H. P. ROSEXTHAL 141,[6], N. TOMCZAKRETHERFORD JAEQERMANX [l], A. E. TOXG[2], L. SCHWARTZ [l].

LUST Theory

23.

This cha.ptor is devoted to special kinds of absolutely summing, nuclear, and intsgral operators which play an important role in the structure theory of Banach spaces. The abbreviation LUST is derived from the phrase “local unconditional structure”.

23.1.

Absolutely z-Summing Operators

23.1.1. An operator X E B(E, F ) is called absolutely z-summing if these is a constant c 2 0 such that

for all finite families of elements zl, We put

...,xm E E and functionals bl, ...,bm E P’.

P,(S) := inf 6. The class of all absolutely t-sunzming operators is denoted by ?&.

23.1.2. The proof of the following result is straightforward. Theorem.

I?&,

P,] i s a normed operator ideal.

23.1.3. Analogously to 17.1.3 we have the Theorem. The normed operator ideal [PI, P,] is maximal.

23.1.4. Theorem. The normed operator ideal [?&,PI] is completely symmetnk. Proof. Obviously we have PP”’(fJ) = P,(X) for all elementary operators. Hence the assertion follows from the niaximality of and 8.7.12.

v,

23.1.5. The ideal of absolutely z-summing operators is rather small. Proposition. I f r 2 1, then [g,, P,] S [ ? P ( r , p , q ) , P ( r , p , q ) l * P r o o f . Let 8 E I;P,(E,I?). Since l/r‘

21 *

,

+ lip + l / q 2 1, we have

324

P u t 4. I3auir Examples of Operator Ideals

23.1.6. T h e o r e m . An operator S E B(E, F ) i s absolutely t-summing i f unCE mdy i f there exists a constant u 2 0 a d a probability ,u E W (UBp X UF,,) such that

I(%, a} (y", b)l

@'x, b)l 5 u

dp(u, y")

for all z C E and b E F'.

us. x UP..

Here we suppose that UBtand UFttare equipped with the weak E-topology curd 3"-toplogy, respectively. In th& case,

P,(S) = inf u. P r o o f . The sufficiency of the above conditioii is evident; cf. 17.3.2 and 17.4.2. Conversely, let S E ?&(E, P) and put u := P,(S). Take C(UETx Up*)' equipped with the weak C(UEpx UF*r)-topobgy. Then W(UE,x UFrr)is a compact convex subset. For any finite fanlilies of elements xl,. .., x,,, E E andfunctionals b,, . .., b,,, E F' the equation m

~ ( p:= ) 2 ~ ( S Xbi>l ,, - u 1

WE'

J

1(xi, a>iy", bi)l dp(u, y")

x Ute,

defines a continuous convex function @ on FV(Gfi,x UF.v).Choose ':y E UFtesuch that

a,

6 U,. and

If 6(uo,y,") denotes the Dirac measure at (h,y,"), then we have m Q(d(a0,

Y;))

X [I(Sxi, bi)i -

=

i(xi, uo) (YC,hi)\] 5 0 .

1

Since the collection 9 of all functions obtained in this way is concave, by E.4.2 there is po E W(UB.x UF-) such that @(yo)5 0 for all @ E 9 simultaneously. In particular, if @ is generated by the single fainilies (x)and ( b ) , it follows that

J

I(Sx. b)i -

UEfx

I(x, a>(Y", b)l dpo(a, 2/") 5 0 . t,

This completes the proof.

23.2.

a-Nuclear Operators

23.2.1. An operator S .

B(E,F ) is called u-sl;uclear if

N

S=

a i @ yi 1

with a,, %, ... E E' and y,, y2, in the operator norm.

...E F

such that the family

(ai @

yi) is summable

23. LUST Theory

325

We put W S ) := inf sup

K

I1

z.‘ I(z, u L >(yE,6)l: llzd 5 1, Wll 5 1

,

ahere the infimum is taken over all so-called o-nuclear representations. The class of all o-nuclear operators is denoted by 8,.

53.2.2. The proof of the following result is straightforward.

[a,,Nu]i s a normed gperator ideal. A sequence ( e , ) with e, + o is called a hyperorthogonal

Theorein.

23.2.3.

qpace U if its linear span is dense and if I(,\ 1L = 1 , 2 . ...

basis of the Banach

1 : l II: l

5 lrrlimplies 2’ Eiei 5

for

R e m a r k . Every hyperorthogonal basis is unconditional. Conversely, a Banach space with a n unconditional basis can be renormed such that the basis becomes hyperorthogonal; cf. [STS, pp. 558-5601.

23.2.4. The standard examples are given in the trivial P r o p o s i t i o n . The unit sequence basis of l p with 1 5 p

< co a i d

co i s hyper-

orthogonal. 23.2.5. T h e o r e m . An operator S E 2 ( E ,F ) is a-nuclear if and onhy i f there exids corn mutative diagrn m

.such that A E Q(E,U ) and Y ortliogonal basis.

Q( U , 3’). Here U i s s m e Banach space having a hyper-

Jn this mse, N U ( W

= inf

IIYII IIAll7

where the infimum i s taken over all possible factorizations. Proof. The sufficiency of the above condition is obvious, since 03

S = 2 A’fi @ Ye,, 1

where (ei) denotes the hyperorthogonal basis and coordinate functionals.

(Ii)is the sequence of corresponding

To check its necessity we take a o-nuclear representation m

S = 3 ai @ y i 1

326

Part 4. Basic Examplca of Operator Ideals

such that

Obscrve that 05

an := SUP

2,' I(%, ~ {nil

i( ~ ) ib)l: ,

l l ~ ld l 1, llall 5 1

tends to zero; cf. [PIE, p. 251. Using the method of 8.6.4 we can find that 1 2 el 2 ez 2 ... 2 0 and

+

05

zeb2 I(%,

U i ) (Yi,

(en)E co such

a)] I (1 4') WS) IIXI! Ilhll-

1

Let U be the collection of a11scalar sequences t = (ti)for which ( tigi*yi) is suniniable. Then U becomes a Banach space with the norm i'tl: := ~ v ~ ( t ~ g ; ~and y ~ ) the , uriit sequences form a hyperorthogonal basis. Define A E 2(E,U ) and Y E 2 ( U , F ) by m

A x := ((z, ai)) and Y(r7i):=

qiyi. 1

As usual we pnt p n ( E i , .-*, t n ,

tnti,

a * . )

:= ( E l ,

. . a ,

En,

0,..*).

Then

5 en+l lltll-

+

Hence 11-4 - PnAI:5 en+l(l E ) X&!3) ~ and /IT' - YE',,' 5 p3,Ll. 80 A E %(e, G) and Y E @ ( U ,F ) . In particular, we have //Ail 5 (1 E ) ? S,(S) and iiYl[ 5 1. Since S = Y A , the assertion is established.

+

23.2.6. As an immediate consequence of the preceding factorization propertoy we get tile

Theorein. T h e normed operutor ideal [?&, &] is ?ni?iitn.tl. 23.2.5. The next result is also obvious. Theorem. The normcd operdor ideal [Ru,Nu]is sytnmetric. R c m a r k . I t seems very likely that [nu, X u ] is L o t regular. 23.2.8. The ideal of a-nuclear opcrators is rather 1~1.,i 2 . TVe have the eyident i'roposition.

[R(r,p,q), N(,.,p.q)l S [a,, XI.

23. LUST Theory

327

23.2.9. For every operator S E g ( E , P)we put

[

?J:(X) := inf sup

c:

2 I(x, a$ (yj,a)!:

11

ilxli 5 I, llbIl 5 1 ,

;There the infimum is taken over all finite representations n

s = 2 a*0yi. 1

23.2.10. Proposition. rU: is a nmin on &. 23.2.11. Using the technique of 18.1.12 we can check the regularity of 1;.

Proposition. If S E g ( E , F ) , then N:(S) = X:(KpX). 23.2.12. Analogously to 18.1.14 we have the following estimates. L e m m a 1. L e t X

&(Eo,E ) and X E ?Jt,(E, P).Then

N : ( m 5 Nu(& IlXIl. Lc innin 2. Let S E %JE,F ) and B E &(F,Fo). TherL PI':(BS) 5 IIBII YAW ' 23.2.13. With the method of 18.4.5 we get the Thzorem. [%:, N,*]= [qZ, P,].

23.3.

o-Integral Operators

23.3.1. An operator

X E B(E, F ) is called o-integral if

it belongs to the nornied ideal

1%

I,] := [%", 8,1mas. 23.3.2. As a counterpart of the preceding definition from 8.7.15 we get the Theorem. [%, Nu]= [3,,Iulmin. 23.3.3. The following result is analogous to 19.2.5 and 19.3.6. Lemma. L e t V be an order cornpZete Bunach lattice. I f X E $(E, P), then pli:(S) = IiSll. Proof. First we check the assertion for those operators admitting a finite representation n

So = 2 aoi @ h4 1

such that hi A hi = o if i

+ j . Let 2 c E and g E

I.&, aoi) (hi, g) = I(%,

aoi>(7~,,

g)l

and

Then n

zI(%, 1

n

Uoi)

(hi, g)l = 2 ' &(XL.,not>(hi, 1

s>

V'. Choose A1,

l&i

=

1.

...,1, E X

with

338

Part 4. Basic Examples of Operetor Ideals

Now let S E g ( E , F ) be arbitrary. Take some finite representation in

S =

ak Q f k 1

rn

2' IIaki; 5 1. Siniulating approximation by simple functions, given E > 0, we 1 n can find f o k = tikhisuch that Ilfk - fokll 5 8 and hi A hi = o if i + j ; [LAC, p- 'i]

with

i=l

and [SAF,, p. 1371. P u t nr

80

:= 3 a k

8f o k .

1

Then m

n

So =

G~@ hi with

aoi :=

i=l

&a&.

k=l

Hence m

11s

-

5 N:(S - 80)5

1

ll%ll llfk - f0dl

We now get

N!(S) 5 q ( S 0 ) 4-N:(s - So) 5

llfloll

+

&

E.

I IlSll

+ 2.7.

This proves that N:(S) 5 /lSil. The converse inequality is evident.

23.3.4. Theorem. An operator S E B(E, F ) i s o-integral if and only if there exists a commutatiee diagram

su,ch that A E e ( E , V ) and Y E e ( V ,F'). Here V i s some (order complete) Banach lnttice.

In this ease,

Ids) = inf llyil i1-41, where the infimum .is take78 over all pmsible faetorizations. Proof. Obviously every Banach space with a hyperorthogonal basis is a Banach lattice in its natural ordering. Now it follows by ultraproduct techniques that KFS factors t*hrougha Banach lattice 7, if X E S,(E, F ) . Since KFS = YA implies KFS = (Kp)' Y"h',A, we may replace V by its order complete bidual V". Conversely, let us suppose that KFS = YA with A E 2(E,V ) and Y E B(V, P') such that the Banach lattive 7' is order complete. Let X E g(Eo,E ) and E $(p,PO). By 23.3.3 we have x!(BSx) = N:(pKFSX) I IIB"Yi1 * ( A X ) 5 IPII IIYII IIAIl IIXll. Hence S E 3,(E,

F)and I,(S) 5 IIYII 114 .11.

23. LUST Theory

329

23.3.6. A4nalogo~isly to 19.1.9 we have the

Lemma 1. Let X E g(Eo,E ) and S E 3 J E , P).Tlien

N:(SX) 5 I,(S) I]X]l. Proof. Given E > 0, we choose a factorization hrFS= Y A such that A E B(E, V ) , Y E jJ( T’, F”),and I/ YIl JIAll5 (1 + E ) 1,(8), where P is an order complete Bttnach lattice. Then 23.2.11 and 23.3.3 yield

s:(Sx; = Y:(K*,Sx)5 ilKFl-:!3 ! ( A X ) 5 IIYII IIAXIJI (1

+

E)

I@) IlXll.

This proves the assertion. For completeness we also formulate the dual estimate.

Lemma 2. Let S E 3 , ( E , F )and B E g ( F , Po).Then

qWS) s IlBll U W 23.3.6. Finally, we mention an iiiiportsnt formula which follows from 33.2.13. Theorem. [b:, P:]

= [3,, I,] and

[a,’,1 3 = [?&, P,].

23.3.5. -4 Banach space is said to have LUST, that means local mconclitional strucP.GORDOWand D. R. LEWIS), if it beloiigs to the space ideal

ture (in the sence of

I, := Space (3,). Remark. By 23.2.3 and 23.2.5 every Banach space with an unconditional basis possesses local unconditional structure. Moreover, we know from 23.3.4 that the sarne property even holds for all Banach lattices.

23.4.

Multiplication Theorems

23.4.1. Theorem.

[PI,P,] o [So,I,] s [B1,L,].

Proof. First suppose that T E g(E, F) and S E !@,(P, G ) . Given

E

> 0, choose a

23. LUST Theory

23.5.

331

Notes

The reader is warned that our defiaition of Banach spaces with local unconditional structure is a slight modification of t.hat given by E. DUBINSKY/~. PELCZYASKI/H. P. ROSENTHAL [i]. Nost results of this chapter can be implicitely found in P.GORDOK/D. R. LEWIS[l], [2], [3]. Recommendations for further reading:

[SEM,, exp. 24, 251

T. FIQIEL/W.B. JOHNSON/L. TZAFRIRI [l], T. FIGIEL/G.PISIER [i], W. B. JOHNSON [3], W. 3.JOHNSOX/L. TZAFRIRI [l], D. R. LEWIS[3], G. PISIER [6], J. R. RETHERFORD [2]. See also t,he bibliographical notes of Chapter 28.

Part V:

Applications

24.

Decomposing Operators

The famous Dunford-Pcttis-Phillips theorem states that every weakly compact operator X from Ll(Q,p) into any Banach space E can be represented in the foriii Xf =

J x(0)f ( w ) 440)

for all

f E L,(Q,y )'

0

where x is a p-measurable E-valued function defined on the probability space (Q, p). A Banach space E is said to have the Radon-Nikodym property if this result remains true for all operators. I n this chapter we deal with thc ideal 1 ' ) of so-called Radon-Nikodym operators is that of decomposing and give its main properties. The dual ideal 0 = '1)'"' operators. We also introduce the ideal Q, of p-decomposing operators with 0 < p 5 oc. It turns out that Q, = p y and 8, = Q y for 1 < p < 03. Using the concepts described above we can improve some multiplication theorems for absolutely summing and integral operators. In particular, Grothendieck's formula ?lBo 3 = 92 is established.

24.1.

Measurable Vector Functions

24.1.1. In what follows ( Q , y ) always denotes any probability space. By an Ecalued function x we mean a map from SZ into an arbitrary Banach space E. 24.1.2. An E-valued function x is said t o be p-sinaple if n

Xih,(Cu),

x(w) = I

...,xn E E and ...,Qn of 9.

where q, subsets Q1,

h,, ...,h, are characteristic functions of y-measurable

24.1.3. An E-valued function 2 is called y-measurable if there exists a sequence of p-simplo E-vtllud functiocs a, such that x(to) = l k xn(o)ahnost everywhere. n

24.1.4. Let us mention the trivial Lemma. For every p-measurable E-valued function y-measurable, rts well.

d:

the scalar funclion

I@(.)\\i s

24.1.5. Now an important characterization of p-measurable vector functions is stated. For proofs mo refer t o [DIU, p. 421, [DUN, p. 1491, and [IOX, p. 721. Theorem. An E-valued function x i s ,u-nreasurable i f and only if the f o l l o w i q ccnditions are satisfied: iL (1)There exists a separable subspace A4 such that x(w) E iW almost everywhere. (2) All scalar functions (x(.), u) with a E E' are p-measurable.

336

Part 5. Applications

41.1.6. Lemma. Let x be cc fi-measurable E-valued function. Suppose that for all a E UE, we have I(z(o),a ) / p almost everywhere. Then 112(0~5 )!1p wlinost everywhere.

Proof. First we observe that alniost all values of x belong to some separablc subspace M of E. Let ( x n ) be a countable dense subset of M . Furthermore, pick a, 5 UE#with (xn,ajL)= Ilz,J/. Then = 1,2,

llzll = sup (i(x, a,)! :

...)

for all x E X.

Finally, we can find a p-null set A' such that I(z(o), a,)l

2 p for all w E Q \ A'

and

12

= 1, 2,

...

This yields Ilz(m)ll (= e almost everywhere. 2 1.1.7. I n what follows we do not distinguish between p-measurable E-valued functions which coincide alniost everywhere. The collection of these equivalence classes is denoted by L,(E, Q, ,u). Let

Ilzllo:= inf ( p )= 0: p(w E 12: Ilx(w)ll > e) 5 e j . Using standard techniques we get the T h e o r e m . Lo(E,0, p ) is a complete metric Zineur r p c c with the P-norm l~.llo.

21.1.8. If z is any p-simple E-valued function of the form 7k

a(o)= 2 x i k i ( o ) , 1

the11

1z ( w ) dp(c0) 2 Zip(Qi) :=

?I

1

is well-defined: cf. 24.1.2.

24.1.9. An E-valued function x is called p-integrable if there exists a sequence of p-simple E-valued functions x, such that lim IIz - zn/io = 0 and II

hm.nn

j

//Z,,~(O>)

n

-

x,(o)ll d p ( o ) = 0 .

Then the Bochner integral is well-defined by

J z ( w ) d,u(w) := lim J x,,(to) d,u(to). n

'I

D

94.1.10. By [DIU, p. 451 we have the Lemma. An E-valued functim t is p-inteyrcrble if md only if it is p-tueu8urable and dP(0) < 60-

J Il=(c.)l 12

24. Decomposing Operators

24.2.

337

Radon-Nikodym Operators

24.2.1. An operator X E f?(L,(Q,p), E) is called r@ht decomposable if there exists a ,u-measurable E-valued function x such that

X f = J x(w)f ( w )++J)

for all f E L,(Q, p ) .

R

Then x is said to be the kernel of X . 24.2.2. The following lemma can be derived froin 24.1.6.

Lemma. The kernel x of every right decomposable operator X E 2(Ll(12,p), E ) i s unique almost everywhere. Horeover,

llXIl

= ess-sup ( l ~ x ( w ) w ~ lE: Q ) .

24.2.3. We next state the famous Dunford-Pettis theorem a proof of which can be found in [BOU,, chap. VI, p. 461, [DIE, p. 2251, and [DUN, p. 5031. Theorem. Let E be a B a M space such that E' is separable. Then every operator X E s(L,(Q,p ) , E') is right decomposable. 36.2.4. We are now able to check the deep Theorem. Let E be a reflexive Banach p e . Then every operator 3 E f?(Ll(Q,p), E') is right decomposable.

Proof. We know from 3.2.9 that L,(Q, p ) has the Dunford-Pettis property. So every operator X E f$(L,(O, p ) , E ) is completely continuous. Observe that the embedding map I from L , ( Q , p ) into L,(Q,p) is weakly compact. Thus, by 3.1.3, the product X I must be compact. Consequently 1M := lM(X1)is a separable subspace of E . Since L,(Q, p) is dense in L,(Q, p), we also have M ( X ) = M . So X acts from Ll(Q, p ) into M = MI', and we can apply 24.2.3.

24.2.5. As an immediate consequence of 2.4.3 and of the preceding result we get the Dunford-Pettis-Phillips Theorem. Every operator X E !Xl(L,(Q,p), E ) is right decomposable. Remark. A direct proof is given in [ION, p. 891. See also [DIU, p. 751.

f-1.2.6. Let S 2 ( E , F ) . Then S is called a Radon-Nikodynz operntor if S X is right decomposable whenever X E B(L,(Q,p), E ) . The class of all Radon-Nikodym operators is denoted by g. 24.2.1. Theorem. 9 .is a closed operator d e a l .

Proof. It can be easily seen that m

of operators S, E g ( E , P)with are kernels yn such that

s , X f = J y,(w) n

22 Pietsch. Operator

9 is an operator ideal. We now take a sequence

llSnli< OQ. Given X E 2(L1(D,p), E),then there

1

* f(w) dp(w)

for all

f E L,W, P I .

338

Part 5. Applications

By 24.2.2 we have I]yn(w)]] 5 l]SnXilalmost everywhere. Since ca

m

2' IlgAw)!I 5 Z IIfL!' !!XiI, 1 1 by setting Qi,

Y(0) :=

z

Yn(c.)

1

we get a p-measurable P-valued function which is defined almost everywhere. So we m

have found a kernel y of the operator SX, where S:=ZS,,. 1 S E g ( E , F).Hence the operator ideal 9 is closed.

This proves that

22.2.8. Theorem. The operator ideal 'I) is injective. Proof. Let S E %(E,P).Then, given X E B(Ll(s2,p), E ) , there exists a kernel y such that

S X = ~ J g ( w ) /(oj) d,u(w) for all f E L,(SZ,p ) . R

-

Put B := M ( S X ) . Applying 24.1.6 to the FIN-valued function Q;y y(w) E d l ( 8 X ) almost everywhere. This proves the injectivity of 9.

we see that

24.2.9. A Banach space belonging to Y := Space ('I)) is said to have the RadoaNikodym property. Remark. The thcory of these Banach spaces is presented in [DIE] and [DIU].

242.10. Proposit ion. The B a m c h space 1, possesses the Radon-Nikodym property. Proof. The assertion follows from 24.2.3, since 1, is a separable dual Banach apace.

24.2.11. Proposition. The Banuch spaces Ll[O, 11 and co fail to have the R&ANikodym property. IZ

Proof. Let I be the countable set of all indices (k,n ) with k Porin the intervals

= 1, 2,

...

= 1,

...,2" and

and denote the corresponding characteristic fiinctions by hkn. Then the equation Xf:= ( f , hkn)defines an operator S E B(L,[O, 11, co(I)).Let us assume that L,[O,11 E Y or co(I)E Y. Then X is right decomposable, and we can find a co(l)-valuedliernci X. Obviously z(t)= (hkR(t))almost everywhere. On the other hand, we always have (hkn(t)) 6 since card ((k,n):hkn(t)= 1)= No for all t E (0, 11. This contradicticjn implies L,[O, 11 ( Y and c o ( l ) Y. Clearly, %(I) and co are isoiuorphic.

24.2.12. Proposition. !ll3 c 'I). Proof. The inclusion follows from 24.2.5. Moreover, we have lI E Y \ W.

24. Decomposing Operators

339

24.2.13. Proposition. QJ c 8. Proof. Suppose that some S E QJ(E, F ) is not unconditionally summing. Then by 1.7.3 there exists X E B(co, E) for which SX is an injection. Now the injectivity of QJ implies that the identity map of c,, belongs to B. This is a contradiction.

24.2.14. Proposition. The operator deals ?Band QJ are incomparable. Proof. Let X be t,he operator constructed in 24.2.11. Then X E B \ QJ.On the for ll all illfinite dimensional reflexive Banach spaces E . other hand, we have Ix E B \ ?

24.3.

Vector Measures

be any a-algebra given on a set 9. A map m from $' 3 into a Banach 24.3.1. Let space E is called a vector measure if

for every sequence of disjoint subsets B,, B,,

..- E $' 3.

34.3.2. The variation po of a vector measure m is defined by

The supremum is taken over all finite families of disjoint subsets B,, belonging to %.

...,B, of B

24.3.3. The proof of the following lemma can be found in [DIN, pp. 32-35]. Lemma. The variation of every vector meamre is a smlar measure which is, however, 7zeces8arilyfinite.

not

24.3.4. We now consider a vector measure m and a probability p, both defined on the same a-algebra 8.Then wz is called p-continuous if p ( B ) = 0 implies m ( B ) = o for B E 8.The E-valued measure m is said to be p-differentiable if there exists a p-integrP+bleE-valued function x such that

m ( ~=)J z(w)d,u(o) for all B

c B.

B

Then x is called the Radon-Nikodym dericative of m. 24.3.5. Let m he any E-valued measure. Then the P-valued ineasurc Stti, where 8 E B(E,F ) , is defined by ( S m ) (B):= X[rn(B)]for B 8.

21.3.6. Every p-differentiable E-valued measure m is p-continuous and has finite variation. The fact that the converse stakment is false gwve rise to define RadonNikodyin operators; cf. [DIU, p.501. Theorem. An operator X E 2 ( E , F ) is a a d o n - N i k o d y m operator i f and only i f it ?naps every p-continuous E-valued memure m with finite variation h t o a p-differcntiable F-valued measure Xm.

.

22*

340

Part 5. Applications

Proof. Given X E B(Ll(Q,p), E ) , then m ( B ):= X(hB) for B E % defines a y-continuous E-valued measure m with finite variation. Let us now assume that S possesses the property mentioned above. Then there exists a derivative y of am. This means that = Jy(w)

SX(ILB)

~ B ( w dy(w) )

for all B E 8.

0

Consequently we get

SXf =

J Y(0)f ( w ) 4 4 w )

D

for all p-simple functions f. By continuity the same formula holds for all f E L,(Q, p). This proves that 8 E g ( E , F ) . To check the converse implication we consider a p-continuous E-valued measure m with finite variation po. Then there exists an operator X E B(Ll(12,h),E ) uniquely defined by the condition X(hg) = m ( B ) for all B E %. If 8 E g ( E , F ) , then we can find a p-measurable F-valued kernel yosuch t,hat

8x1= J ? l o b ) fb)dpo(w)

for all f E LdQ, Po)

-

Q

Since the p-continuity is transmitted from m to po, according to the classical RadonNikodym theorem, there is a y-integrable scalar function g 2 0 such that ,tAo(~)=

J g(w) d p ( w )

for all B E B .

I3

Consequently, by putting y := gy,, we get Xm(B) = S X ( ~ B ) = J yo(@) +,(to) B

=

1y(o) dp(cr>).

B

s o y is the derivative of Sm. This proves that every Radon-Nikodym operator has the required property.

24.4.

Decomposing Operators

24.4.1. An operator A 6 B(E, L,(S, p)) is called left decomposable if there exists a y-measurable E-valued function a such that

Ax

= (2,

a(.)) for all x E E .

Then a is said to be the kernel of A . 24.4.2. An operator S 6 B(E, F ) is called decomposing if BS is left decomposable whenever B E f?(F,L&2, p)). The class of all decomposing operators is denoted by a.

24. Decomposing Operators

341

TZ = 9"'. P r o o f . Let S E Q(E,P).If Y E B(L,(sZ, p ) , F'), then Y'K,S is left decomposable.

24.4.3. Theorem.

For x E E and f E &(Q, p ) we have (2,S'Yf) = !Y'Kp!h, f

J (x, ~ ( w )f ) (

)=

~d )p ( ~ ) ,

Q

where a is the corrcsponding kernel. The above equation yields

S ' Y ~= J a ( o ) f ( o ) d,p(w) for a11 f E L,(Q. p). Q

This proves that x't'

g(F',El). Hence D

gaual.

Conversely, let X E (I)*""(E,F ) . If B E g ( P ,L,(O, p)),then S'B'KL1is left decomposable. For x E E and f f L,(D, p ) we have

I(%,N o ) )

{B~X f ) ,= (x,S'B'K,,f) =

f(0))

~Aw),

s1

where a is the corresponding kernel. The above equation yields BSx = (2,a(.))for all x E E. This proves t8hatS E Q(E, P).Hence gd*' Q. 24.4.4. As a counterpart of the preceding result we formulate the

P r o p o s i t i o n . Qdual c 9. ( ~ ) d u a l ) d u n5 l Oreg = g. In order to = (Ddual implies Qdual Proof. Clearly show that Qdw1 + 9 we consider the related space ideals. By 24.2.10 we have I , E Y. On the other hand, the assumption I , E ad"'implies 1'; E Y. Since 1'; and LJO, 11" are isomorphic [SED, p. 4801, the injectivity of Y yields Ll[O, 11 E Y, which is a contradiction by 24.2.11.

24.5.

p-Decomposing Operators

243.1. Let' 0 < p m. An operator A f B(E,L J 9 , p ) ) is called left decomposable if there exists a p-measurable El-valued function a such that

A x = (2,a(.)) for all x E E . Then a is said to be the kernel of A . 24.5.2. Lemma. The kernel i.s unique almost eoeryuhere.

of

every left decomposable operutor A E B(E;L,(Q, p ) )

Proof. Obviously it is enough to show that. m y kernel a of the zero operator vanishes almost everywhere. First we observe that almost all values of a belong to some separable subspace M of E'. Let (a,) be a countable dense subset of M and choose x, E U, such that 2 i(xn, a,)[ 2 Ilu,ll. Then llallO

:= sup (I(x,, .}!:

n = 1, 2,

...I

342

Part 5. Applications

defines an equivalent norm on M . More precisely, we have 2 lla/lo2 /lull 2 jjaljOfor all a E M . Take a p-null set N such that (zn,~ ( w ) = ) 0 for all

OJ

E s;! \ N

and n

=

1,2, ...

Hence l l ~ ( a ) )= //~ 0 almost everywhere. This proves the assertion. 24.5.3. Let 0 < p 5 00. An operator S E B(E,F ) i3 called p-decomposing if BS is left. decomposable whenever B f E(P,Lp(s;!, p)).

The class of all p-deconiposing operators is denoted by

Qp.

Remark. Obviously, we have Q = Q,. 24.5.4. We formulate without proof the trivial

Theorem. Q p is an operator ideal. 24.5.5. Using the method of 24.2.8 we get the

Theorem. The operator ideal Q p is surjective. 24.5.6. Theorem. If 1

p 5 00, then the operutor ideal Q, i s regular.

Proof. The regularity of 0, follows from 4.5.6 and 24.4.3. Let S E Q'dig(E,P)and 1 < p < 00. Then, given B c B(F, Lp(Q,p)), by 1.5.5 therr exists BzE B(F", L,(D, p ) ) such that B = B7KF.Consequently BS = BT(KFS) is left decomposable and therefore S E Qp(E,F ) . This proves that Q, = QFg. To treat the case p = 1 we need the fact that there is a canonical surjection Q from L,(D, p)" onto L,(B, p ) which can be defined by the help of Lebesgue's decomposition; cf. [BOU,, chap. V, p. 611. Then the left decomposibility of BS folloas from BS = (QB")(K,S). Remark. It seeins to be unknowii whether or not the operator ideal 0,is also regular for 0 < p < 1. 24.5.7. Theorem.

Qp

s FpdpU8l.

Proof. Let S E Qp(E,F),Then for every 6 E E(F, Lp[O,11) there exists a unique Lebesgue-measurable E'-valued kernel a such that BSrr = (x,a(.))for all x E E. Clearly B +u defines a linear map K froiii B(F, Lp[O,11) into Lo(E',[0, 11). We noTs consider a sequence of operators B,, B,, E E(F, L,[O, 11) converging to some operator B f E(F, Lp[O,11). Furthermore, let us assume that the sequence of corresponding kernels a,, C I ~ ., .. tends t o some vector function a E Lo@',[0, 13). Then

...

= Lp-lim(2,un(.)) = Lp-limBnSx = BSx. Lo-lim (r,an(.)) n

(I

11

On the other hand, by hypothesis we have Lo-lini(2,an(.))= (x,a ( . ) ) . n

Hence u is the kernel of BS. This proves that the map K is closed arid therefore continuous. Choose 6 > 0 such that \\I311 5 6 iniplies I/allo5 1/2. We claim that

24. Decomposing Operators

343

for all finite families of functionals b,, ...,b, E F‘. In order to verify (*) we inay assume that wp(bi)= S and I[S‘bi[l> 0. Put l i

II~‘bilP

+ + IIfJ’bmll”

:=

IlS’blllp and form the intervals

..*

If h,, ...,h, are the corresponding characteristic functions, t,hen bi @ hi

B := defines an operator B

B(P,L,[0, 11) with ijB[j= 6. Since

m

1

is the kernel of BS, it follows that /\ullo5 lj2. On the other hand, we have that i:n(t)ll= lp(8’bi)for all t E (0,11. So lp(8’bi)> 1/3 would iinply I!alio> 1/2, which is a contradiction. Therefore (*) holds, and t,his means that S’ E !J&,(F’,El). Consequently

a, 5 qy.

31.5.8. To prove the main result of this section we need thc following Lemma. Let 1 p < 00 and A E p y ( E , L,(Q, ,LA)).Then there emkts f i L,(Q, p ) such that \Ax1 5 f

for all

2

E

U,.

I n other terms, A maps the closed unit ball U , into an order bounded subwt of Lp(!2)p). Proof. Take any finite p-measurable decomposition (Ql,

..., Q m ) of B such that

p(Qi) > 0, and denote the family of corresponding characteristic functions by (hl, .,hm).Defineu,, u, E Lp(Q,p) and cl, v,,, E L,.(Q, y ) by ui := ,u(Qi)-’%i and vi := , U ( B ~ ) - ~ ’Then %~.

...,

..

...,

m

L := 1z’j 0ui 1

is an operator in LJQ,,u);of. 19.3.5. It follows from wP(vi) 5 1 that lp(A’vi)5 Ppd‘(A). SO w

fL

:=

2 [IA’viIiui 1

fulfils llfLllp 5 P;mi(A).-Since

344

Part 5. Applications

..

Consequently max (JLAzl:,I, ., ILAs,l) 5 fJ for all finite families of elements x,, ..., x,,E U,. This yields llniax (\LAX,!, ..., ~LAx,l)ll,2 P y ( A ) . Observe that, LAz tends to Ax, if (9,; ...,9,) ranges over all finite p-measurable decompositions of 9 ;cf. 19.3.5. So, passing to the limit, we have

..

Since the family of all functions max (IAxll, ., IAx,]) is directed upwards, according to [BOU,, chap. IV, p. 1371, there exists a supreinum f E Lp(9,p). Clearly f has the required property.

We are now able to establish the fundamental Theorem. If 1 < p

< M, tkcn. Q p

=

,y

and

Qiml= Qp.

Proof. Let S E Q,(E, 3') and X E O(E', Lp(9,p)). According to 17.3.11 without loss of generality we may suppose that B' is reflexive. By the preceding lemma there exists a positive function f E L p ( 9 ,p ) such that JXS'b/5 f for all zi E UFt.Hence Yob := XS'b/f defines an operator Y o B(F', L,(O, p)). It follows froin IF E 9 that IF, E Q. Thus we can find a p-measurable F-valued kernel yo such that P,b = (yo(,),b) for all b 6 li". Hence XS'b = (g(.),b) for all b E F', where y := /yo. Therefore S' is p-decomposing. This proves that Q, & Qy. As proved in 24.5.7 we also have Q p & So we get

vy.

!ppE Q y l & ( v y ) d u a l s vp and dual Q,dual c =(Qp

dual

c

=DpSPyl.

24.5.9. As a counterpart to the preceding theorem we have the Proposition. Q,

+ Py'.

Proof. The canonical map J , from C[O, 13 into L,[O, 11 is 1-integral. Hence J , E 3p1 5 gp$".We now assume that J1E Q1. Then J1has a Lebesgue-measurable C[O, 11'-valued kernel j,. Form the polynomials pn(t):= tn for n = 0, 1, ..., and choose a Lebesguc-null set N such Ohat

pt2(t) = (p?87j1(t)) for au. E

Lo, 1'

\N*

Sccording to Weierstrass's approximation theorem the above equat.ion holds for all functions f E C[O, 13. This means that j,(t) = 8(t) for all t E [0, 13 \ N .

However, there does not, exist any separable subspace M of C[O, 11' contailling almost all Dirac measures 8(t). This contradiction proves that J , 6 Q,. 24.5.10. Finally, we state the trivial Proposition. I/ p1 5 p2,then Qpl E Q,,. Remark. Using a famous theorem of E. M. NIKISHIN [i] it can be shown that the deal Q, with 0 < p < 1 does not depend on the parameter p .

24. Decomposing Operators

24.6.

345

Multiplication Theorems

24.6.1. I n order t o prove the basic result of this section we need a preliminary Lemma. Let (0, p ) be any probability space and 1 5 r < 00. Then for every left decomposable operator iZ E f?(E,L,(.Q, p ) ) the product 1,A is r-nuclear such that S,(I,A) 5 114 .11. Here I , denotes the embedding map from L&2, p ) into &(0, ,u). Proof. First we treat the special case in which the kernel a of A has a countable a(o)= aj}.Without loss of generality we may assume image (ai).P u t Qi := ( w E 0: that p(Sj)> 0. Let f i := ,u(Oi)-l/%i,where hi denotes the charact,eristio function of R,. Ther! 'x1

1,A

=

2,' ,u(R~)~!' C C ~S J fi. 1

It follows from

lr(p(Ot)l'r)= 1 , wm(ai)=r llAll, and wrt(fi)= 1 , that I,A is an r-nuclear operator with N,(I,A) 5 1 1 8 1 1 . We now come t o the general case. Since almost all values of the kernel a belong to some separable subspace H of E', we can find B sequence of p-measurable Elvalued functions a, possessing a countable image such that ess-sup (Ila(co) - an(co)i/: w 0) l/n. P u t A,z := (x,an(.))for all

5

E E. Then we have A = /l.ll-lim A,. Moreover, I

N,(I,A, - IrAn)5 llAm- A,li implies that (I,A,) is a n X,-Cauchy sequence. Since I,A is the only possible limit, we get I,A E a t , ( E ,L,(Q, p)) and

N,(I,A) = lim Nt,(I,A,) 5 lim liAnlI = IIAl]. n

n

We are now able to establish the fundamental Theorem. Let 1 2 r P r ,

< 00. Then

I,] 0 [a.Il-lil S

[a,,K P g .

P r o o f . Let T E Q(E,F ) and S E &(F, G ) . Given E > 0, by 19.2.6 we can find a factorization K,S = ZI,B such that IiZIl /IB// (1 c) Ir(8). Since BT is left decomposable, the preceding lemma yields KGST = Z ( I , B T ) E %,(E, G") and

+

Nr(KGfiF) 5 11211 Br(IrBT) 5 llZllllsrll 2 (1

+ &)

lITll.

This proves that S1' E ?J2Fg(E,(2) and lKf""(ST)5 I,(&')l!2'l.

24.6.2. As a consequence we get the famous Grothendieck Theorem.

[m, 11.111 o [Z, I] = [n,N].

Proof. By 2.4.3 every S E m ( F , G ) factors through some reflexive Banacli , and 2 E %(Go,G ) space Go. More precisely, given E > 0, we can find SoE f ? ( P Go) such that S = 28, and liZI[llSoll5 (1 E ) IISII. Now 24.2.12 yields Sh E gad((&, P'). So, by 24.4.3, we hare 8; E Q((2&3"). Let T E 3 ( E , F). Then T' E Z(F', E') and

+

:%46

Part 5. Applications

therefore T‘S;

%(GA, E’) as well as N(1”SA) 5 I(T’) IiSblI. Csing the reflexivity

of Go we get ST = Z(S,,T) E R(E, G) and

N(ST) 5

llzll N(SoT) i IIZlI IISoIl I(T) 2 (1

This proves that [!& 8.6.4.

11.11]

o [3,I]

[a,N].

+

E)

IiSII I(T).

The converse inclusion is evident by

R e m a r k . Let us niention that a n operator S E B(E,F ) belongs to 9 if and only if S-Y is nuclear for all strongly integral operators X E B(Eo,E ) ; cf. [DIU, p.1751.

24.6.3. We next state soine k i d of Grothendieck’s formula for r-integral operators. P r o p o s i t i o n . Let 1 4

T

< m. Then

[R,11.111 0 P r , 1 7 1 = [%,> 5 7 1 . Proof. Suppose that T E 3,(E, F ) and S E R(F, G). According t o 19.2.6 there exists a factorization KFT : YI,A such that !lYll llAij 5 (1 t F ) I,(!Z’). We have S7 %(F”, G). Since L,(Q,,u)has the approximation property, the operator S7Y is e v m approxiinable. So 19.1.10 yields ST = (S”Y) (IJ) E %,(E, G) a n d

S,(LSY)5

llsq ilAlj 5 (1 + 8) I’SIj I,(T).

r ,

I his proves that

[R,ll.lll 0 P,>I,] E [anKI. The converse inclnsion follows from 19.2.2. R e m a r k . Clearly

[m,ii.ii]

o [3,,I,]

4[a,,S,] for t

< r < 00.

24.6.4. We are now ready to improve 20.2.4. T h e o r e m . Let l / r

+ l/s = 1,‘p5 1 cxnd 1 <

T,

s < m. Then

i):oof. By 17.3.11 every operator T E P,(E, 3’)admits A factorization T = Y T , such that T , < ‘&(E, P o ) , I? < 2 ( F o , F ) , and liYil Ps(To)5 (1 -1 t ) P,(T), where F , IS >: rafltxive Banach space. If X :3,(B’, G), then 24.6.1 iinplies K,SY E %,(Po,G“) a d X,(K,SY) 5 I,(S) IIY~~. Thrrefoie KGST == (K,SY) To E ?Rp(E,G”) and

Np(K,ST) 5 S,(K,SY) P,(?‘o) 5 (1

-c1

p:ores

i’z-:tig

+ F) I,(#) Ps(7’).

p).

to t!ie iiijective Iinlls froin (3) we get (1).

Let 7’ S s ( E , P ) and S I3 Y 3 ( F ,G). Then XT E 3FJ(F”, G) and 1;’ (&IT) = VJ(S). By 13.2.6 there is a factorization K,T = Y1,A with IlYll IlAll 5 (1 F ) Is(T). Since L6(l2,p ) is reflexivc, the operator Y is decomposing. So the injective version

+

24. Decomposing Operators

347

of 24.6.1 (theorem), namely

[Sr,Irlinj 0 [Q, 11.111 G [!It,. hTrlin', yields STY %F(Ls(s2,p), G ) and S:"j(S7Y)5 p ( S )llYIl. By 19.2.16 we can find a factorization S"Y = ZSo, where doE 3y(Ls(s2, p), Go),Z E A(Go,G),and

llZll I:"j(S0)5 (1 + e ) I:"j(S)IjY11. Altogether we get the diagrmi :

~,(-Q,

r ) T & ( Q ,r ) T . a o

Then S0IJ E S J E , Go) and Ip(#,,18A)5 I:"J(SO) llA/l. Finally, it follows from the preceding proposit'ion that ST = Z(SoIsA)E ?JIp(E,G) and

N,(ST) 5 IlZll &(SOLA) 5 (1

5 (1

+

C)Z

+ ).

1 3 4 IlYll

ll4

I:"?(#) 18(T).

This proves (2).

24.6.6. As a special case of the prtwding result we formulate the Theorem. [?&,PJ2 E [!It,N].

24.7.

Notes

Therc is an extensive list of papers dealing with representations of operators in function J. PETTIS[l], I. AT. GELFAXD spaces. We only mention the classical a o r k of N. DUNFORD/B. 111, and R. S. PHILLPS [i]. A full presentation and further references may be found in [DUN, pp. 489-5111. The theory of Radon-Nikodym operators has been developed by W.TJNDE [3] and 0. J. REINOV[i]. For further informattions the reader is referred t o i~ survoy paper of J. DIEsTa/J. J. Urn, [l] and the monographs [DIE] and [DIU]. Tii? coricept of a p-decompocf. [BAD, exp. I?$ Pitrther contributions sable operator was introduced by A. BAUBIKIAX; d r C due to S. KWAPIER [6], P. SAPHAR133, and L. SCHWAETZ [2], [3]. Th- most striking multiplication theorem of this chapter first appeared in [GRO. chap. 3. p. 1321. Other important results are taken f r o i A. PERSSOX 121. Becoxurnendations for furtlirr reading:

[DIN], [ION], [SEn/r,, annex I], [SElil,. csp. Oj. [SEA&, exp. -1-6.51, [SEi\I,. esp. 11, [swal,

[TAR]. S. BOCHNER [l], J. DIESTEL[i], V. K. K O ~ T K O[if, V [2], W. LINDE[B], 131, J. VON KEU[I], A. PIETSCH [4], [9], J. W. RICE[l], R. ROGGE [ I ] , C. SWARTZ [Z],G. I. TABGONSKI [I], E.THOMAS [I], [2], [3], J . WEIDNANN[l], J. WLOKA[l], [a], T. K. TVom 111, [2], [3].

26.

Radoriifying Operators

‘The concept of a cylindrical probability on special linear topological Hausdorff in 1933. Further progress is due to I. ill. spaces was created by A. N. KOLMOGOROV GELFANDand his fanioiis school. However, the final version of this important theory was given by L. SCHWARTZ. In general, a cylindrical probability defined on a Banach space may fail to be o-additive. So the question arises “What operators transform certain cylindrical probabilities in such a way that their images have better properties?” These considerations lead to the concept of a radonifying operator.

In the following we give a sketch of Schwarts’s theory which is presented in his “SBrnin&ir~s’~ from 1969 on. As a climax we discuss the Wiener probability appearing in the theory of Brownian motion. This probability is defined to be the image of some normalized cylindrical Gauss probability with respect to the integration operator. 26.1.

Probabilities on Topological Hausdorff Spaces

26.1.1. Let T be it topological Hausdorff space. Then @(T) denotes the Borel a-algebra genemted by the open subsets. Moreover,R(T)is thc collection of all compa ct subsets. 26.1.2. In what follows we always deal with so-called Borel probabdities which are defined on

a(5”).

A Borel probability ,u is said to be reguZur if p ( B ) = sup (,u(K):K E B, K E R(T)] whenever B @(T).A regular Borel probability is siniply called a Radon probubiZity.

25.1.3. The non-trivial direction of the following criterion is proved in [BAU, p. 2011. Proposition. A Borel probability defined OTLa complete metric y a c e .is regular if a d anly if i t .is macentrated on some separable subspace.

26.1.4. Obviously, every Radon probability ,u is uniquely determined by its restriction po to $(“). We now state an important characterization; [BOU3, chap. IX, pp. 42-47]. Proposition. A m a p po from 9(T)into &?+can be extended to a Radon probability if and o d y if the following conditions are satisfied: (1) Let H , hr E $ ( T ) such that 13 K . Then p o ( H ) po(K). (2) W e have po(K, u K,) 5 po(K,) ,uo(K,)for K,, K 2 E ft(T).Moreover, equcclity holds if K , and K , are disjoint. ( 3 ) Given KOE ji( T )and E > 0, there exists an open subset U containing KOsuch that p o ( R ) 5 p o ( K o ) E for all K E R(T)with K 5 U .

+

+

(4)sup { p o ( K ) :K E R(T)) = 1.

25. Radonifying Operators

349

Cylindrical Sets

25.2.

25.2.1. Let us recall that Cod ( E ) denotes the collection of all finite codiniensional subspaces N of the Banach space E. Given N,, N2 E Cod ( E ) such that N , N2, then x ( N l ) -+ x(N,) defines a canonical map Q from E / N l onto E{N2, and we have the commutative diagram

Hence, the family of all quotients EIN with N E Cod ( E ) constitutes a projective spectrum of finite dimensional Banach spaces. 25.2.2. If N E Cod (E),then &(E) denotes the a-algebra which is the inverse image of %(E/N) with respect to the canonical map Q:. Subsets of the form

2 := (x E E : Q ~ E xB ] , where B E B ( E / N ) , are called cylindrical. We put

8(E):= uN 3N@)* 25.2.3. Proposition. 8 ( E ) is an nlgebra. Proof. Let Z, a

...,2, E 8 ( E ) .

Then Zi E QNr(E), where Ni E Cod ( E ) . Since

N := n N i E Cod ( E ) ,it follows that 1

n

U Zi E 8n.(E) L 8 ( E ) . 1

The remaining properties of an algebra are trivial. Remark. Let us mention that 8 ( E ) is a a-algebra in the finite dimensional case only. 25.2.4. We now give some examples of cylindrical subsets.

Proposition. If K E R(E) and N E Cod ( E ) ,then K

+ N E S(E).

Proof. Since B :=-Q$(K) is compact, we have B E B ( E / N ) .Hence

K

+N =

(Z

E E: Q ~ EXB JE &v(E) S 3 ( E ) .

25.2.6. Proposition. A subset 2 of E is cylindrical if and only if it cnn be written in the form

z = :.( E E : ((5,a,>,...,(x,a,)) E 81, where a,,

...,a, E E' and B E

%(Sfl).

350

Part 5. Applications

25.3.

Cylindrical Probabilities

25.3.1. A map from 8 ( E )into 9+is called a cylindrical probability if its restriction to the a-algebra &(E) is a probability for all N C Cod ( E ) . 25.3.2. We now describe an important method to produce cylindrical probabilities. For this purpose an arbitrary probability apace (a. u ) and a linear map X from E' into Lo(S,p ) are required. Given any cylindrical subset = (Z E E : ((2,al},

we put

zx

= { w E Q : (f1(w),

...,(5,an}) E B } , ...,fn(u4) E B),

where f r := X a j . Since f l , ...,f n are actually equivalence classes, it should be mentioned that 2 , is well-defined up to a p-null set. So by setting p#) := p&) we get a map px from 8 ( E ) into 9+.

25.3.3. The following result is trivial. Proposition. px i s a cylindricctl probability. 25.3.4. For detailed proofs of the converse statements we refer to [BAD, exp. 31 and [SWA, pp. 256-2581.

Proposition. Every cylindrical probability c can be generated by some lirtear X from E' into Lo(O,p ) such that 5 = pX. Here (Q, p ) i s a suitable probability space. Proof (sketch). Let ( a i )with i E I be a Hamel basis of E'. If 0 denotes the I-th Cartesian power of X , then A x := ((2,aj))defines a map A from E into 8. For any finite index set i E g ( I ) we consider the canonical surjection Qi from 8 onto Q,, tlhei-th Cartesian power of X . Put Ai := QiA. Then the family of all images pi:=Aic with i E B(I) coi;stitutes a projective spectrum of probabilities. So, according to Kolmogorov's extension theorern, we can find a probability ,LC on D such that pl = Qlp for all i E S ( I ) ;cf. [RAU, pp. 157-1162] and [PEH, pp. 5-17]. Finally, the desired linear map X from E' into Lo@,p ) is given by I

iliai+x d i f i , Z

where f i denotes the i-th coordinate functionon 0. Obviously, if 2 = ( X E E : A i x E B ] , then ZX = ( w E Q: Q,co 23). Therefore pu,(B)= [ ( Z ) and pc,(B)= p(&) imply ( ( Z )= px(Z) for all Z 6 a ( E ) .This proves that 5' = px. 25.3.5. A cylindrical probability is called Badmian if, given K E B(E)such that 5(K N ) 2 1 - E for all N E Cod ( E ) .

+

E

> 0, there

exists

25.3.6. Proposition. Every cylindrical Radon probability t can be uniquely extended to a Radon probability f . Proof (sketch). In a first step we put

(,(R):= inf ( ( ( K+ N ) : N E Cod (E)]for K E L(E). Since tosatisfies the conditions stated iii 215.1.4,there exists a Radon probability f such that f ( K )= (,(K). Finally, we can show that 4 is tho required extension of C.

25. Radonifying Operators

Remark. Conversely, given any Radon probability, then its restriction to is of course a cylindrical Radon probability.

351

3(E)

26.3.7. We now give a fundamental characterization of cylindrical Radon probabilities which is the key to all what follows; of. [BAD, esp. 121 and [SWA, p. 2961.

Proposition. Let X be a linear map from E' into L,(Q, p). Then ,ux.is a cylindrical Radon probability i f and only i f there exists a p-measurable P-valued function x defined OTL Q such that

X a = @(.), a) for all a E E'.

In this case, the extension px equals the image x ( p ) . P r oof (sketch). Suppose that px is a cylindrical Radon probability. First we show that X becomes continuous if E' is equipped with the locally convex topology generated by the system of semi-norms

pK(a):= sup ( ) ( xa)): , x E K ) , where K E S?(E').

+

Tor this purpose, given E > 0, we choose K E R(E) such that pX(K N ) 2 1 - E wheneverN Cod (E).Takeanya E E'withpK(a) 5 Eandwritef := X a . Nowweput

N := ( x E E : ( 2 , a ) = 01, 2 := (xE E : I(%, a)l 2, := (a E Q: If(a)l

>E),

>E).

+

Then ( K N ) n 2 = 0 implies p(Zx)= px(Z)2 E . Hence IjXa)jo5 E. This proves the continuity of X . According to 26.1.3 we may assume that E is separable. Then there exists a countable subset (ai)of E dense in the locally convex topology defined above; of. [KOT, pp. 261 and 2661. Put f i := Xai. Let Xmdenote the countable Cartesian power of X . By setting A x := ((2,a,)) andf(0) := (fi(o)f we obtain a px-measurable linear map A from E into X" and a p-measurable X"-valued function f on Q. I n order to show that A($=) = f(p),the canonical surjection Q, from X" onto Xnis required. Put A , := QnA andf,, := Q,$f. If B b(X"),then we write

z:= (x€ E:A,x € B ) , z, := (0€ . R : f n ( 0 ) € B ) , 2, := {u E Y " :&,u

E B).

It follows from Z = A-l(Z0) and 2,

=f-l(Z0)

that

[ A ( i i x ) l W o= ~ i u X ( Z ) = cd(Zx)= [f(01)l(Zo)Since the Bore16-algebra of X mis generated by the subsets Z,, we have shown k k t A(,&) andf(p) coincide. Choosea sequence of subsets h', E a(#)with liin fiX(Km)= 1 ni and put,

Then

352

Part 5. Applications m

NOW,for every o 6 Qo, we select a(o)E A-l(f(o))n

u K,.

This yields an E-valued

1

function x defined almost everywhere on Q, Moreover, ( ( ~ ( w )n,i ) ) = A ( z ( o ) )= f(o)>= ( f i ( ( o ) )

wlienevor

GO

E Q,,

irnplies Xui = (a(.), ai) for i = 1,4, ...Using the c0ntinuit.y of X we get X a = ( ~ ( . ) , n ) for all a. E E’. Since the scalar functions
25.3.8. For every cylindrical probability 5 the weak p-th moment is defined by

where 0 < p

<

00.

The case p

= bo

requires the usual modifications.

Remark. The above integrals make sense if, for fixed a F E‘, we restrict [ to the a-algebra &(E), where N := (x: E E : (2,a) = 0).

35.3.9. Lemma. T h e cylindrical probability px h a s

ci

fifhite weak 1)-th moment i f and

d y if X is a continuous linear map from E’ into LJS, p). I n this case wp(p..)= }lXll-

Proof. Let a E E‘ and put f := Xa. Then 6 E : l(x, a)I

p&

5 p)

1

,u(o E Q: If(w)l

5 e ) for

g

2 0.

This yields

J

I(Z, a i i p

dpX(x) =

B

J If(w)ip

~

0

)

.

n

Consequently

s

(1IX4lp: lbll 11 = IIXII’ 26.3.10. For every cylindrical probability the absolute p-th moment is defined by W p ( P X ) = SUP

where 0 < I, < cm.The case p = bo requires the usual modifications. Remark. The above integrals make sense if, for fixed N 5 Cod ( E ) ,we restrict 5 to the a-algebra &(@.

25.4.

p-Radonifying Operators

25.4.1. We begin this section with two elementary statements.

Proposition 1. Let S E G(E, P ) . Then every eylindrieal probctbility i on E i3 mapped into a cylindrical probability Xc on P which is defined by [ S z ](2):== c(S-‘(z)) for z E 3 ( P ) . Proposition 2. Let S E e ( E , 3’)and let X be a n y linear map from E’ into bo(9, p). 1 hen Spx = pxg.

I 1

28. Redonifying Operators

353

P r o o f . Given b,, ...,b, E F’ and B E S ( S n )we , put

U := {X t E : ((2.S’b,),

...,(x,S‘b,))

5 B)

and

v := (y E 3’: ((y, a,>,

.I

(y,b,)) E B ) .

Clearly U = X-l(V). By setting f Z:= XS’bi we get

ux = vx,, = ( 0 E Q:

(f,(W),

...,f&))

E BJ.

Consequently

[Spxl ( V ) = Px(U) == CL(U1) = P(V,S*) = rLLrS’(TT). This completes the proof. 25.4.2. Let 0 < p 5 00. An operator S E 2(E,F ) is called p-radonifying if every cylindrical probability with finite weak p-th nionient is mapped into a cylindrical Radon probability Sc.

The class of all p-radonifying operators is denoted by !Rp.

25.4.3. As an immediate consequence of 25.3.7 we have the following criterion. Proposition. A n operator X E 2(E, F) i s p-rudonif?yinq if and only if, given S E e(E’,LP@,p)), there exists a p-measuruble F-valued fumtion y such that XS’b = ( g ( . ) ,b) for a8 b 6 F’. 25.4.4. The next result can be easily checked.

Theorem. I , is an, operator ideal. 25.4.6. Analogously to 24.2.8 we obtain the

Theorem. The operator ideal

?Itpi s injective.

85.4.6. Comparing Definition 24.5.3 and Proposition 25.4.3 we have the Theorem. !Rp E Qy.

P5.4.7. We now prove the main result of this section. Theorem. If 1 < p < 00, then I , =Q y and = Qp. Proof. Let, S f QF(E, F ) and X E O(E’, L,(Q, p)). According to 17.3.11 and 24.5.8 we may suppose that F is reflexive. Since XS’ is left decon~posable,there exists a p-measurable F-valued kernel y such that XS’b = {y(.), b) for all b E F‘. Hence X E I J E , F ) by 25.4.3. This proves that a dpus1 5 !Rp.

!Ry

The converse inclusion is stated in 25.4.6. Finally, the second forniula can be deduced from the first one, 25.4.8. As an immediate consequence of 24.5.8 and the preceding result we get the

Theorem. If 1 < p

< 00, then !RP= rP,.

Remark. Looking carefully at thegroofs of 24.5.8 and 25.4.7 we see that every p-radonifying operator, where 1 < p < 00, maps all cylindrical probabilities with 2 5 Ptetsch, Operator

354

Part 5. Applications

finite weak p-th moment into cylindrical Radon probabilities with finite absolute p-th moment; cf. 17.2.3. 25.4.9. Next a counterpart of the preceding theorem is stated. Proposition. Rl

+ ?&.

Proof. The canonical map J1from C[O, 11 into L,[O, 13 is absolutely 1-summing. We now assume that J , E R,. Then J1 E Denote by Q the natural surjection from C[O, 11' onto LJO, 13 transforming every measure into its Lebesgue-continuous component; [DUN, p. 1321. It follows that J , = &JiJ, E a,, which is a contradiction by 24.5.9.

a:".

25.4.10. Theorem. 'I) o Q1

?XI.

Proof. Let T E @,(E, F ) and 8 e cf)(F,a). Given X E e(E', L,(Q,p ) ) and E > 0, by 24.5.8 (lemma) there exists f e L,(Q, p ) such that f 2 E and IXT'bl 5 f for all b E Up..By setting Tog := f g we define an operator To E B(L,(Q, p), Ll(Q,p)) for which M(To)is dense in L,(Q, p). Since T is weakly compact, by 1.5.5 we may consider the operator T" e O(E", P ) with T" = KpT". If g e L,(Q, p), then we choose b e U p such that IIT"X'g1I = (TnX'g,b). Now

IITXX'gll = (T"X'9, b) = (9, XT'b) 5

J lgfl dp = IITogll1.

s)

Consequently there exists A

e g(L,(Q,p),B') such that we have the diagram:

Since SA is right decomposable, we can find a p-measurable Q-valued kernel %. If x := fxo, then

ST"X'g

=

Ix ( w )

g ( o ) dp(w) for all g E L,(S, p )

.

s)

Hence XT'S'c = ( x ( . ) , c) for all c e G'. This proves that ST is 1-radonifying.

28.4.11. For completeness we state a final result. Theorem. Xm= ?B. Proof. First observe that the identity map of any reflexive Banach spaoe E belongs to 'I). Therefore IE8E Q. So every operator X e C(E', La@, p ) ) is left decomposable, where the corresponding kernel takes its values in E . This proves that I B E Rm.Xow 2.4.3 implies 111) 5 !)Iw. Let S be the Dirac probability on the singleton Q = (wO). Given x'' E E", then the map z" @ 1 from E' into Lo(Q,6 ) generates the cylindrical probability 6(r")on E .

25. Radonifying Operators

355

Obviously the weak oo-th moment is finite, Let 8 E R,(E, F).Then there exists a &measurable F-valued function y such that

(X"x", b) = (z",S'b) = (2'' 0 1) S b = (y(w,), b ) for all b E F ' . Hence Sl'x''

25.5.

= KFy(wo), and Sis weakly compact by

1.5.4. This proves that ?&E , 8.

Gaussian Cylindricid Probabilities

26.6.1. The normalized n-dimensional Gauss probability y" is defined by

for all B E B(9P). Obviously y" is invariant under orthogonal transformations. 26.5.2. Let H be a real Hilbert space. Then every cylindrical subset 2 can be written in the form 2 = {Z E H :

((5, %I),

..., (2,Z.))

E B},

where (q, ..., z,,)is an orthonormal family and B E S(W"). By setting yH(Z):=y"(B) we get a well-defined cylindrical probability yH which is called the wmlked cylindrical Gaw8 probability on H . Remark. Let us mention that yH is a-additive in the finite dimensional case only; cf. 25.5.9. 26.5.3. A cylindrical probability y on a real Banach space E is called G a u d a n if there exists an operator X E B(H, E ) such that y = X(yH).Here H is a suitable Hilbert space. 26.6.4. Proposition. For every cylindrical Gauss probability all weak p-th m m e n t s with 0 < p < co are finite.

Proof. Let y = X ( y H ) . If a E E'suchthat llX'all= 1, then (10a ) y = (1 @ X ' a ) yH is the normalized 1-dimensionalGauss probability. Hence

and therefore wp(y)5 g, IIXII.

25.6.

Wiener Probability

26.6.1. Let 0 < ;I< 1. Then CIIO,13 denotes t.he Banach space of all Holder R-continuous functions f on [0, 13 such that f(0)= 0. The norm is defined by

23*

356

Part 5 . Applications

25.6.2. If 1

p

< m, 0 < 3, < 1, and DC

= lip

t J., then

's

R,: f(t) + g(s) = - (s - ty--l f(t)dt

440

defines an operator R, E 2(Lp[0,13, C1[O, 11) which is called the Riemunn-Limwille operator of fractional intqrution; cf. 22.7.5 and [ZYG, vol. 11, p. 1381. I n particular, for a = 1, we get the integration q e r a t w 8

R:f ( t , + S(S)

=

J f ( t ) dt. 0

25.6.3. According to 22.7.4 and 22.73 the following result is R special case of 22.4.13. Proposition. If 2

F

< p < 00

uruE 0

< I < l,'? - llp, then

€2 E Vp(Ld0,11, Cn[O, 11). Proof. Put (Y := lip f 3, and B := 1 - l/p - 3.. Because of R,,, = R,R, and > 112 we get the diagram €2

,Lp[O, 11

ClO, 11 JP

The assertion now follows froin 17.3.5.

26.6.4. The Brownian ?notion can be described by the so-called cylindricul Wiener defined on Co[O, 11 := { f C[O, 11: f ( 0 ) = 0). probabilit?yg := R(yLZIO,ll)

28.6.5. Immediately from 25.6.3, 25.4.8 and 25.5.4 we have the Theorem. The cylindrical Wiener probability e can be uniquely extended to the Wiener probability @ which .is regular on the Bore1 a-algebra B(C,[O, 11).

26.6.6. Finally we show that the Wiener probability is concentrated on certain linear subsets of C,,[O, I]. Theorem. If 0

< i, < 1/2, then P(CJ0, 11) = 1.

Proof. Choose some exponent p such that 112 - A > 1/21 > 0. Then we have R E ?&,(,(LBIO, 11, C1[O, 11) by 25.6.3. Hence el := R l ( y L g [ O is , l la) cylindrical Radon pro-

bability on C#I, 11. Moreover, @ is the image of el with respect to t.he embedding map from C2[0,11 into Co[O,11. This proves the assertion. Remark. The above result means that almost every sainple path is Holder A-continuous €or 0 < 1 < 112. Remark. Let us mention that C1[O,13 with 1/2 < i( < 1 is a @null set; cf. [KUO, p. 451 and [YEH, p. 4361.

25. Radonifying Operators

36.7.

357

Notes

Cylindrical probabilities on special linear topological Hausdorff spaces were first studied by -1.N. KOLWOQOROV in his fundamental monograph [KOL]. Further results were obtained by several Soviet mathematicians; cf. [GEL]. Finally, L. SCHWAETZ [l], [2], [3] introduced the concept of a radonifying operator. This famous theory is presented in [SERI,, exp. 1--6, 11-17, 24-26]. See also [BAD, exp. 121. The reader should also consult the monograph [SWA]. It wns S. K W A P I E[3], ~ [4], 151 who discovered the coincidence of absolutely p-summing and p-radonifying operators for 1 < p < cy?. There is an extensive literature dealing with Gaussian probabilities on Banach spaces; cf. [BAH], [KUO], [YEH]. Recommendutions for further reading:

[RAU], [BOU,, chap. 1x1, [HAL], [SHI], [SKO], [XIA]. 1’. ASSOUAD 113, G. BENNETTCS], R.. M. DUDLEY [l], D. J. H. GARLIXG[4J, [7], W. LINDE [ i j . [4]$W. LIXDE/A.PIETSCH [l], B. YAUREY [2], N. WIENER [l].

26.

RierJz Theory

The background of this chapter is the classical theory of compact operators created by I?. RIESZ.Looking for significant properties of I - 23,where 8 is compact, A. F. RUSTON axiomatically introduced the class of so-called Riesz operators in 1954. Our main purpose is to investigate operator ideals U such that all components

U(E,E)consist of Riesz operators only. The largest operator ideal of this kind was defined by D. KLEINECKE. Further examples are the ideals of compact, strictly singular, and strictly cosingular operators. It is essential t,o emphasize the great influence of the famous Soviet School on the development of this theory. First we recall the Riesz decomposition of operators possessing finite ascent and finite descent. Secondly, some elementary results concerning spectral theory in Banach algebras are summarized. Then we introduce the basic concept of a @-isomorphism due to F.V.ATKINSON(1951). The next step is the theory of quasicompact operators developed by K. YOSIDA(1939). Using these preliminaries we define Riesz operators and establish their main properties. The last section is devoted to ideals of Riesz operators. Throughout this chapter all Banach spaces under consideration are complex.

26.1.

Riesz Decomposition

26.1.1. I n this section some purely algebraic results are collected; cf. [TAY, p. 2711. We always consider a linear map T acting in a linear space E. 26.1.2. The map T has finite ascent if there exists an integer k 2 0 such that N ( T k )= N(Tk+l),The smallest such integer is denoted by n N ( T ) .Then we have N(Th)= N(Tk)for h > 12 2 nN(T). 26.1.3. The map T has finite descent if there exists an integer k 2 0 such that M(Tk)= L!kf(l'k+l). The smallest such integer is denoted by nnr(T).Then we have N ( ! P )= M(Tk) for h > k 2 niM(T). 26.1.4. Proposition. Let T have finite ascent and finite rle-scent. Then na(T) ad naf(T)are equal. Remark. The common value of nLV(T) and ,z,(T) is denoted by a(T). 26.1.5. We now describe the so-called Riax decomposition. Theorem. Let T have finite ascent and finite descent. il'hen E is the direct sum of N ( P ) and M ( T k )for k 2 n(T).

26. Riosz Theorv

26.2.

369

Spectral Theory in Banaoh Algebras

26.2.1. In this section let 9f be a complex Banach algebra with identity I ; of. [HILJ. 86.2.2. For S E 9f we put

~ ( 8:= ) (1E V : I The element #(A)

- 23

:= S(I -

is invertible).

is called the Fredholm resolvent.

26.2.3. An easy computation proves the Lemma. For all il E e(S) we have ( I - 1~9)-~ =I

+ AS(l).

86.2.4. Theorem. T?he function S(.) i s analytic on the open set e(8). 26.2.6. Theorem. The condii?i~ns lim IlS"l[l/"= 0 und ~ ( 8 =)V are equivalent. n

26.3.

@-Isomorphisms

26.3.1. An operator T E B(E,F ) is called a @-isomorphismif it has a finite climensional null space and a finite codimensional closed range. R e mark. Usually @-isomorphismsare called Freclholm operators or Noether operators; cf. [SEC, p. 1061 and [PRO, p. 161. Remark. Let us mention that H ( T )is automatically closed if it has finite codimension; cf. [CAR, p. 371.

26.3.2. We now prove a fundamental characterization. Theorem. An operator T E B(E,F ) i s a @-isomorphism i f and only i f there exist operators U , B E B(F, E ) , X E g ( E ,E ) , und Y E $ ( F , P)such that

UT

=

IE - X

and

TV

= I, -

Y.

Proof. Let N := N ( T )and 2M := I M ( T ) . By A.3.4 everyd5-isoniorphismT E B(E,F) admits a factorization T =; JToQ, where To is an isomorphism. Moreover, it follows from B.4.4 and B.4.5 that there are X, E B(E/N, E ) and YoE B(F, M ) with QX, =IE,N and Y,J = IM. If

X := Ie - XoQ, Y := IF - J Y , , and 5' = V := .XoT;'Yo, then

UT

==

XoT,'(YoJ) T0Q = XoQ = I,

-

S

-

Y.

a d

T B = JTo(QXo)T i l Y o = J Y ,

=I,

iV and M(Y)= M , the operators X and Y are finite. Conversely, let us suppose that UT = I E - X a.i;d T V = IF - Y , i%-hcre X and Y are finite. TheE N ( T ) 5 N(IE - X ) . Hence Y ( T )E Dim ( E ) .The range of T is closed, since X(T ) = X ( I F - Y)+ Mo with a Gnite dimensional subspace Me. T h i s proves Since X(X)

7

that M ( T ) E Cod (3').

360

Part 5. Applications

26.3.3. The next statement is the main result of the classical Riesz theory. T h e o r e m . For 8 E R ( E , E ) the operator I - S is a @-isomorphism. R e m a r k . If we consider operators acting in a fixed Banach space E , then the identity n a p I E is simply denoted by I.

26.3.4. Let Q, bc the quotient map from B(E, E ) onto the so-called Calkin aEgebra

O(E, E ) / R ( E ,E ) . Then we have the T h e o r e m . An operator T E B(E, E ) i s a @-.isomorphi.m if a d only if @(T)is iiwertible in B(E, E ) / R ( E ,E ) . P r o o f . The formulas UT = I - X and T P = I - P stated in 26.3.2 yield that @ ( T )admits a left-hand inverse and a right-hand inverse in B(E,E)/JI(E,E). Conversely, if @(T)is invertiblc in B(E,E ) / R ( E ,E ) , then UT = I - X and TV = I - Y, where T:, V E P(E, E ) and X, Y E R ( E ,E). By 26.3.3 we can find Uo,Vo E B(E, E ) and X,,Yo E %(E,E ) such that Uo(I - X ) = I - X , and ( I - Y ) Vo = I - Yo. Hence U0UT = I - Xo and T V V , = I - Yo. This completes the proof.

36.4.

Quasi-Compact Operators

26.4.1. An operator S O(E,E) is called quasi-compact if there exist a n operator K E R ( E , E ) and a natural number m such that llSm - Kjj < 1. 26.4.2. Lemma. For every qwtsi-compact operator S f B(E, E ) and E > 0 there exist an operator L E R(E,E ) and a natural number 9% such that \ISn- LII < E .

Proof. Choose K E R(E, E ) and m such that llSm- KII < 1. Let us define the operators Kh := Shm- (Sm- K ) h for h = 1, 2, , Then K h E R(E, E ) . Moreover, llShm- R h l l 5 \ISm- Kllh < E for h large enough.

..

26.4.3. P r o p o s i t i o n . Let S E O(E,E ) be quasi-compact. Then I - S is a @-isomorhism.

Proof. We choose K E R ( E , E ) and m with IISm- KI] < 1. If A := Sm- K , then I - A is invertible. Consequently,

(I - A)-l(I + s+ * * *

+ s-1) (I-

As)

= I - (I - A ) - l K

and

(I - S)(I f S

+ - + 8-1) ( I - A)-1 = I - K ( I - A)-1. **

Finally, the assertion follows from 26.3.4.

26.4.4. P r o p o s i t i o n . Let S O(E, E ) be quasi-compact. Then I ascent.

-

S has finite

Proof. Suppose that the sequence of subspaces N ( ( I - 8)k) is strictly increasing. Then, by E.l.l, there are xk E X ( ( I - S)k)such that l l q l l = 1 and llzk - 211 2 1/2 for all 2 E N ( ( 1 - A’)k-l). We now choose L E R(E, E ) and n with /IS” < 1,’s.

26. Riesz Theory

361

If xi := xk - S n q , then

x i = (I

8

+ + sn-l)(1 - 8 )xk E N ( ( I - s)k-l)

and

Lxk

xk - x! - (fin- L ) 3 .

Consequently, for h

> k we have

[ILxh - Lxkl] 2 11xk - (x:

+ xk

-

xi)ll - 2 118' - LII

> 1/6.

Therefore (Lxk) can contain no convergent subsequence, and this contradicts the fact that L is compact.

26.4.6. Proposition. Lef S

< B(E, E )

be quasi-compact. Then I - 8 has finit

dcscent.

Proof. Suppose that the sequence of subspaces M ( ( 1 - 8)k) is strictly decreasing. Then, by E.1.1, there are %k E M ( ( I - S ) k )such that llxkll = 1 and llzk - 211 2 1/2 for all x E H ( ( I - S)k+l).We now choose L E R(E,E ) and n with 11s" - L11 < ll6. If x i := xk - Saxk, then

xi

=(I

+ s+ + * *

( I - S ) Xk E M ( ( 1--- S)k+l)

s 1 )

and

Lxk = xk - x i - (8" - L) 9 . Consequently, for h j/L%k

< k we have

- Lxkll

2 1121 - (x! - xk

f

%!!I1 - 2 118' -

> 1/6-

Therefore (Lxk) can contain no Convergent subsequence, and this contradicts t h e fact that L is compact.

Riesz Operators

26.6.

26.5.1. An operator S E B(E, E ) is called a Riesz operator if I - ji,Sis a @-isomorphism for all complex numbers A. 26.5.2. First we give the basic examples; cf. 26.3.3. Theorem. Every S E R(E, E ) is a Riesz operator.

26.5.3. Lemma. Let S E O(E,E ) be a Riesz operator. Then A S is quasi-conzpmA for rcll complex numbers A. Proof. According to 26.3.4 we have Q(@(S)) = V. Hence, by 26.2.5, it follows thct lim ~ ~ ~ @ ( S= ) n0, ~where / ~ l ' n111.111 is the norm of thecalkin algebra B(E,E)/&(E,E). U

Since, given 2. E %?, there is it natural number m such that //l@(itS)mllil'm c 1, we can choose K E R(E, E ) with Il(A8)" - K(I < 1.

26.5.4. A coinpiex number & is called a characteristic value of the Riesz operatm S - AoS)-f (01. We put n(Ao)I== n(1 - A ,&').

if N ( I

I

Remark. Obviously A, is a characteristic value if and only if 12,'

is an eigenvalue

362

Part 5. Applications

26.5.5. Theorem. For every Riesx operator S E E(E,E ) the set of characteristic values has no finite point of accumulation. Proof. Suppose that 8 has a convergent sequence ( 1 , ) of distinct characteristic o with x k = 1fiXk:lr. Since x l , .., xk are linearly indevalues. Then there exists x k pendent, the sequence of subspaces Nk spanned by { x l , ...,xk} is strictly increasing. Consequently, by E . l . l , there are yk E Nk such that l\1/kll = 1 and llyk - yll 2 1/2 for y E Nk-1. We now choose L E B(E, E ) and n with Il(nS)n- L(I < 116, where A := lim Then there exists ko such that Il(&s)" - L11 < 1/6 for k 2 ko. Since

.

+

k

(I - (&&)") xh = (1 - (&/&)m) x h for h = 1, ...,n , the element y: := 1/k - ( j l k S ) n 1/k belongs to Nk-1. Moreover, Lyk = Y k - ?-/: - (&@" - L,1/kFor R > k 2 ko it follows that

- Lykl! 2 lk/h - (d - 1/k f

lb%h

d)ll - ll(hs)"- LII - l (&s)m

- LII >

Thus (@k) can contain no convergent subsequence, and this contradicts the fact that L is compact. R e m a r k . The set of characteristic values is a t most countable.

26.6.6. T h e o r e m . If I i s not a characteristic value of the R i m operator 15' E 2(E,E ) , then 1 E e(S). Proof. Since N ( I - 1.S) = lo), we have n N ( I - AS) = 0. Consequently, nirr(I- I S ) = 0 by 26.1.4. This means that M ( I - AS) = E . Therefore ( I exists. 26.5.7. As a summary of the results given in 26.1 and 26.4 me formulate the Theorern. If A. is a characteristic value of the Riesz operator S E f?(E,E ) , then, setting p := a(,lo),the following properties hold: (1) The subspaces N ( ( I - ?.oS)k) are finite dimensional and (0)

... c N ( ( I

-

1.~8)p-i)

c N((I-

a,,s)p) = N ( ( I - a o s ) p + l ) = ... .

(2) The subspaces M ( ( I - l o S ) k )are finite codimensional and

M ( ( I - 1oS)P-l) 2 M ( ( I - Il.oS)P) = N ( ( I - A$)P+') Furthermore, E is the direct sicm of W ( ( I - IoS)P)and M ( ( I - AoS)P).

E

3 *-. 3

=

.

36.5.8. Theorem. If lo ,is a characteristic value of the Riesz operator S E B(E,E), then there exists a decomposition 8 = 8, -k Snfwith the follounhg properties: (1) il, i s the only character&tic aalue of Sa E g(E, E ) , (2) I,o is not a characteristic value of Sll f g ( E , X ) , (3) S-VSJf = S,S, = 0. Proof. We represent E as the direct sum of the subspaces N := N ( ( I - AoS)p) and M := X ( ( I - 4,S)p). Let PN and PMbe the projections from E onto N along M and from E onto M along N , respectively. Put S, := PNSPNand SM:= P M ~ P , . Since N and M are invariant under S, we have Sx = Savxfor x E N and Sx = SMx for x E M .

26. Riesz Theory

363

(1) Suppose that I f I , is a Characteristic value of 8,. Then there exists x f o with (I - AS,) x = 0. Hence x E 37 and Sx = S,X. Since ( I - ?&')p x = o and x = B x , it follows that (1 - Ao/?,)p x = 0. This implies x = 0, which is a contradiction. (2) Suppose that, & is a characteristic value of S., Then there exists xo $. o with ( I - A&,) zo= 0. Hence xo E M and Sxo = SMxW Now it follows from ( I - A,#) xo = o that xo E N . Since N n M = ( o } , we obtain xo = 0, which is impossible. (3) The assertion follows from PNPM= P B ~ P= N 0.

26.6.

@-Injectionsand @-Surjections

26.6.1. An operator T E e ( E , F ) is called a @-injection if it has a finite dimensional null space and a closed range. 26.6.2. Obviously we have the following criterion. Proposition. An operator T E B(E, F ) is a @-injection if and only if there d s t s a finite codirnewionalsubqwm. M of E amh that T J S is an injection.

26.6.3. We recall that 6 denotes the ideal of Kato operators. Theorem. Let S E G(E, P ) .If T E 2(E, F ) 13a @-injection,then so is S $- T . Proof. By 26.6.2 we can find M E Cod ( E ) such that TJ& is an injection. Suppose that S T is not a @-injection.Then, by 1.9.1, there is an infinite dimensional subspaoe M , of M with ]l(S T )J&]I < j(TJ:) 5 j(TJ&). Now it follows from

+

+

i(sJ&o) 2 j(TJ$J - Il(S + T)J%,11 > 0 that SJ& is an injection. Therefore &lo must be finite dimensional, which is a contradiction.

26.6.4. Lemma. Let T E B(E,F ) be an injection. If A E 2 ( E , F ) and lIAl1 < j ( T ) , then T A .is an injection. Moreover, N ( T )E Cod ( F ) and M ( T A ) E Cod ( F ) are equivalent.

+

+

Proof. By 13.3.11 we have

+ LA)2 j ( T )- iiAll whenever 0 5 I In particular, T + A is an injection. j(T

4 1.

Suppose that one and only one of the subspaces M ( T )and X(1' codimension. Then, given E > 0, we can find lo,ill E [0, 13 such that

:Ifo := X ( T and (Io -

< F.

+ I.&)

E Cod (Fj. M I := M ( T

+ AlA)

By E.l.l there are yI,y2, ... E UFwith

+ A ) has finite

Cod ( F ) ,

364

Part 5. Applications

On the other hand, sime F / M o is finite dimensional, we have flyi(Mo)- yk(Mo)li< E for some i =+ k.

c'hoose x E E such that ( j V )-

lIzji - ?Jk -

IIAIl) l!.~!l5 j(T , 2 ljzji

(T + &A) xII < E . Then it follows froin

+ AoA)1/415 iU' -+ AoA)4 -

ykll

-+

E

52 +E

that

which becoiiies false for

E

small enough. This coinpletes the proof.

26.6.5. Theorem. Every S E G(E,E ) is a Riesz opercrtor. Proof. We see froin 26.6.3 that I - ils is a @-injection for all complex numbers A. Given & E V , by 26.6.2 we can find M E Cod ( E ) such that ( I - A,#) J$ is a n injection. Let 11. - ilol 1 1 8 1 1< j ( ( I - A0S)J:l). Then, by the preceding lemma, the subspaces N((1- &S) J:) and M ( ( 1 - ils) are finite (infinite) dimensionaJ simultaneously. This statement remains true for M(1 - &S) and M ( I - AS). Therefore, the complex plane is the union of the disjoint open subsets

JC)

Go := ( 7 6 E %: M ( I - 2 8 ) E Cod ( F ) } and GI

:= (3. E %':

M(I - 2 8 )

6 Cod (3')).

Since 0 E Go, i t follows t,hnt GI= 0. Consequently I - ih' is a @-isomorphismfor all complex numbers A.

96.6.6. An opemtor T E 2(E,F ) is called a @-surjectiow,if it has a finite codimensiond closed range. 96.6.7. Obviously we have the following criterion. (I

P r o p o s i t i o n . An operator T E B(E, F ) i s a @-surjection if and only i f there &ts finite dimensional subspnce N of E such that Q:.T is a surjectioit.

96.6.8. We recall that 2 denotes the ideal of Pekzyiiski operators. Using the method of 26.6.3 we get the T h e o r e m . Let 8 E Z ( E , F ) . If T E B(E, F ) i s a @-swjection, the72 so is S

+ T.

26.6.9. The ncxt result follows from 26.6.4 by dualization; cf. B.3.8. Lemma. Let T 2 O(E,P) be a surjection. If A f e ( E ,F ) and IlAII < q(T), the)% T A i s a suvjection. Noreover, N(T)E Din1 ( E ) and N(T A ) E Din1 ( E ) are equivalent.

+

+

20. Riesz Theory

866

26.6.10. Analogously to 26.6.5 we have the Theorem. Every S E z ( E , F ) is a Biesz operator. €3) be the set of all operators S E E(E, E ) such that the pertur26.6.11. Let bation S + T of any @-injection T B(E, E ) is also a @-injection. Analogously Z,(E, E’) is defined to be the perturbation set of @-surjections.

26.6.12. A proof of the following result is given by [CAR, p. 971. Theorem. G,(E, E ) und Z,(E’, E ) are c l a d ideals in the nlqebru B(E,E ) . Remark. It is unknown whether the components Ga(E, E ) and &(E, E ) satisfy the compatibility condition stated in 1.1.3. The answer to this problem is affirmative if and only if Ga(E, E ) = G(E,$2) and Z@(E,E ) = Z(E, E ) for all Banach spaces E .

26.7.

Ideals of Riesz Operators

26.7.1. First of all let us inmtion that, in general, the set of Riesz operators is not an ideal in the algebra B(E, B). To give a Counterexample we consider the Cartesian square E x E of any infinite dimensional Banach space. Put Sl(zl)z2):= (0,x,) and S&, x2) := (z2,0). Obviously S, and S2are nilpotent. Hence they must be Riesz operators. On the other hand, S, S2 and XIS2cannot be Rieszian, since the characteristic value 12, = 1 has infinite algebraic multiplicity.

+

denotes the ideal of Gohberg operators; uf. 4.9.5. By 26.7.2. Let us recall that 4.3.8 and 26.3.2 we have the Theorem. R .is the largest wpercrttw d e a l 8wh that all co?nponen,ztsR(E,E) COPZs&t of Riesz operators only.

26.7.3. In the following diagram thc arrows point from the smaller operator ideals at the larger ones:

R

All inclusions are strict.

366

Part 5. Applications

26.8.

Notes

As everybody knows the famous spectral theory of compact operators was created by F. RIESZ [l]. Further contributions are due to many authors. I n particular, @-isomorphisms were investigated by F. V. ATKINSON [l] and I. C. GOHBERO/N.G. KREJN[I]. The concept of a quasi-compact operator was found by K. Yosma 111, and A. F. RUSTON[4] introduced the so-called Riesz operators. The largest ideal consisting of Riesz operators first appeared in a paper of D. KLEINECKE[I]. See also I. C. GOHBERG/A. S. XARKUS/I. A. FELDMAN [l] and B. YOOD [I]. Remmmendations for further reading: [CAR], [DUN], [GOL], [HEU], [KAT], [PEO], [RIE], [ROL], [SEC], [TAY], [ZAN].

s. R. CARADUS[I], D. HIL3ERT [l], T. KATO[I], S. N. GAEHOWSKIJ/A. 8. DIKAESKIJ [I], A. PIETSCE [I], A. F. RFSTON[5], &I. SCHECHTER [l], E. SCHMIDT [l], [Z].

27.

Fredholm Theory

The famous Fredholm theory of integral operators with continuous kernel was the most important starting-point of functional analysis. All determinant-free results have been generalized to the Riesz theory developed in the preceding chapter. On the other hand, for a long time no one succeeded in constructing determinants for operators in arbitrary Bansch spaces. This problem was independently solved by A. F. RUSTON, A. GBOTHENDIECK,and T. L E ~ A ~ ~inS the K I early fifties. We begin with a fundamental characterization of Riesz operators in terms of the meromorphic behaviour of their Fredholm resolvent. By Weierstrass's theorem, there exists an entire complex function the zeros of which are the characteristic values of the given operator. Then the Fredholm resolvent is the quotient of an entire operator-valued function and the so-called Fredholin divisor just described. The main purpose of this chapter is a direct construction of Fredholm divisors. Models are the characteristic polynomials of matrices, H. v. KOCH'Sdefinition of infinite determinants (1900)' and the above mentioned theory of integral operators. As a first step we form Fredholrn divisors for nuclear operators in ZI. Using the ooncept of related operators the results can be carried over to some classes of operators acting in arbitrary Banach spaces. I n particular, we obtain the Ruston-Grothendieck determinant theory of nuclear operators. It was proved by I. S c m (1909) that for every Hilbert-Schmidt operator the sequence of reciprocal characteristic values is square summable. A general theorem of this kind, concerning Gp-operators with 0 < p c 00, is due to H. WEYL(1949). Again using the concept of related operators we study the distribution of characteristic values for both (r,1, 2)-nuclear and absolutely 2-summing operators. Finally, a trace formula is established which goes back to A.CROTHENDIECK (1955) and V. B. LIDSKIJ(1959). Throughout this chapter all Banach spaces under consideration are complex.

27.1.

Fredholm Divisors

27.1.1. The main result of this section is the following characterization of Riesz operators in terms of the meroniorphic behaviour of t,heir Fredholm resolvent. Theorem. An operator IS E B(E,E ) is a Riesz operator if and o d y if for every complex number lothere exists e > 0 such that

for 0 < /A - AOl n = 0, 1,

...

< e,

where IS-p(lbo), ...,IS+@,) E g ( E , E ) and Sn(Ao)E B(E,23) for

Proof. Let IS be a Ricsz operator. If Jo E e(S),then by 26.2.4 the function S(.)hae a Taylor expansion at A,. We now consider a characteristic value A,. Then thers

368

Part 5 . Applications

+

exists a decomposition S = BN SM according to 26.5.8. The finite operator SAv has an g ( E, ,!?)-valuedFredholm resolvent

+

for i, A0; cf. [ZAN, p. 3461. On the other hand, SM(.)is analytic in a certain neighbourhood of &. It follows from SNSM= SMSN= 0 that S(A) = s&) s,().). This gives the desired Laurent expansion.

+

Conversely, let us suppose that the above condition is satisfied. Then the complement of e(S) has no finite point of accumulation. Since the coefficients of all principal parts are finite operators, the function S(.) is analytic modulo R(E, E ) in the whole coniplex plane. Hence, S is a Riesz operator by 26.3.4. Remark. The operator SAV can be obtained from the Laurent expansioii of S(A) at &, since S,(A) coincides with the principal part and 8, = S,(O). '37.1.2. We now state an improvement of the previous theorem.

Proposition. Let S E 2(E,E ) be a Riesz operator. Then tfhe Fredholrn re.sdvent hns CI pok of order n(1,) for every characteristic value lo. Proof. Let

for 0

< /;I- jlol < e

such that S-,(L,)

+ 0. Then lim (1 - 1,). s(2)5 = o for all

x 5 E m d n > p . Wenowsuppose that 2 E N ( ( I (I -

z = ((a

- a),

ri-*lo

It follows from

s +- (I - ns)p x

by niultiplication with ( I - AS)-1that

+ (I - A s ) n - l z

= 0.

i E 1. tends to lo we obtain ( I - l,,S)n-l z = 0. This proves that we have ~ ( (-1$S).-1) = N ( ( I - i.,S)n) for n > p . Consequently, ~(1,)5 p .

Conversely, ( I - H ) S ( i ) = 8 iinplies ( I - 1,S) S(1) - (A - 2,) SS(1) -::S. Comparing coefficients we see that

(I - /'.,A') S-,(i,,)

= SK,(L0),

( I - 1,s)As-p+l(ao) = xs-,(a,). ( I - RoS) S&))

= 0.

27. Fredliolm Theory

The last equation means that &,(Ao) (I - M)*1

= (I

369

= A,,AS’LS-~(~.~). On the other hand, we have

- jl,,AS’)P-Z SS-.z(&)= - - -

= SP-lS-,(&).

Consequently, ( I - &!3)P--1

K1(&) = A;p+lS&,)

Choose 2, E E with &-,(lo) so

+

0.

S-l(2.0)xo E “((I - @)P)

and ( I - 2&)P fLl(&) = 0.

Then \ N ( ( I - l.os)p-l).

Hence p I n(Ao).

27.1.3. For every characteristic value A,, of a Riesz operator S E E(E, E ) the nlgebraic multiplicity is defined by := dim N ( ( 1 - @)p),

where p := ...(lo).

5 a(&). Proof. Choose 2 E N ( ( I - A&)”) \ N ( ( I - 2J3)P-l) and put := ( I - &8)k-1 z for k = 1, ...,p . Then sl,..., zpare linearly independent and belong to N ( ( I - &9)p). 27.1.4. Lemma. n(&)

27.1.5. An entire complex function d is called Fredholm divisor of the Riesz operator S if the zeros of d coincide with the characteristic values of S. The order of every zero being the same as the algebraic multiplicity of the corresponding characteristic value. 27.1.6. By Weierstrsss’s theorem [TIT, p. 2461 and 26.5.5 we have the P r o p o s i t io n. Every Riesz operator possesses cc Predholm divisor. 27.1.7. Finally, a very important result is shown.

Theorem. Let S E f!(E, E ) be a Rieez operator with a Fredhdm divisor 00

d(l) =2 6,P

for all 1 E 27.

0

Then there czists an entire E(E, E)-valued funci?ion m

D(i.1 = 2 D,A”

for all ilE %?

0

such that

Moreover, Do = 6,s and D, = 6,S f D,-,S for

n =

1,2,

...

Proof. Obviously D(A) := d ( l ) B(L) is an analytic function on e(S).By 27.1.2 and 27.1.4 all characteristic values of S are removable singularities of D. Consequently, a

24 Pietsch. Operator

3'70

Pnrt 5. Applicat,iorls

there exists an analytic extension to the whole complex plane. Comparing coefficient.;. from D(3.) (I- 1.8) = d ( i ) S we obtain the desired recurrence forrriuh. Remark. Let [2t, A] b e

EL

qiimsi-normed operator ideal. If the Riesz opzrator

S E 2 ( E , E ) bt-lo~gsto 3 , then so c!oc.s D, for n = 0, 1, . . . and 17eIVei <- !t.

m

2 DJqLis A-mIJ0

~

27.2.

FredhoIni Deterniinants

272.1. Let iis recall the rcprcscntation theorem of nuclear operators in 2,; T h e o r e m . An operator S matrix (aie)szich that

/s(tk)

==

(E

GLkte)

k=l

cf. 6.2.G.

g(ll,El) is nuclear if and only if there is an iGjiLjiiit'fC CQ

alzd

2 SUP jG& < t = l

CO.

k

Iw this caae, ~ ( 8=)2:sup lailtj and trace (8)= i=1

k

qj.

j=1

R e m a r k . In the s ~ q u cwe l identify operators and their representing matriceii.

27.2.2. We now state Hadamrd's imxpality, without proof; cf. [DUN, p. 10181. Lemma. Let (c,,) be an (n, n)-matrix. Then

47.2.3. As a n immediate consequence for every infinitenuclear matrix (aik) we have the 1, e ni m a. Let

27. Fredholm Theory

P r o o f . We have

nnp2

M

;SUP l@j,jl j,,...,

I-

.**

fi!

].==I

SUP

j

I@j,jl

j

27.2.5. The following estimate can be obtained in the same way. Lemma. Let

27.2.6. As an immediate consequence of 27.2.1 and 27.2.5 we have the i s nuclear in &,and

Lemma. The operator 0,= (n

N(D,,) 5

+

1)(0+1)'2

n!

N(AS)~+~ for n = 0, 1, ...

27.2.7. We now define the Fredholm determinant. Proposition. Let 00

d(1) := C 6,??

for all

1 E %'.

0

Then d i s a n entire complex function. Proof. From 27.2.4 we conclude that 00

2'

00

nnla

18"q 5 2'0

0

n!

h'(S)n 111".

The convergence of the right-hand series can be checked by the ratio test.

27.2.5. The definition of the first Frcdholm minor is made analogously. Proposition. Let 00

D(1) := 2 DJ"

for all 1 E V.

0

Then D i s a n entire %(11, 1,)-valued function. 24"

371

372

Part 5. Applications

27.2.9. We now show the same recurrence formula as in 27.1.7. Lemma. 0,= 6,s

+ Dn-lS for

1%=

1, 2,

...

Proof. The assertion can be established by the expansion of Afk in terms of the elements of the first row.

8

= Bncik

+i=l 1uij A;;'.

27.2.10. The next lemma follows immediately from the definitions of d, and Dn-l. Lemma. S,, =

1 -trace n

for n = 1, 2,

...

27.2.11. Vc'c now are in a position to prove the main rosults of this section. Theorem. If d(1)

+ 0,then R

e(S).Moreover,

Proof. From 27.2.9 we see t,hat

D(A) = d(A) s

+ AD(i.) s = d ( l ) s + ilsD(A).

Consequently,

This proves that il

p(S) and the first. formula. Furthermore, m

27.2.12. The next statement is a counterpart of the preceding result. Theorem. If d(A0) = 0, then 3.0 i s a characteristic value of S. Proof. Since lois a pole of d'ld, the same is true for the Fredholm resolvent. This proves that Lo 6 e(8).

27. Fredholm Theory

1

27.2.13. L e m m a . 8,

=

n-1

-- ,r

6k

373

trace (LP-~)for n = 1,2, ...

12 k=o

Proof. Using D, = 8,s

+ D,-,S

we obtain by induction

n-1

2 8kSnWk,

Dn-1 =

k=O

and the conclusion follows from 27.2.10. 87.2.14. The preceding leiiinia shows that the Fredholni determinant of S is uniquely determined by o k := trace (Sk)for k = 1, 2 , . .. This can be seen more explicitly in Plemelj’s formula. P r o p o s i t ion. 1

01

.. 0

0

0

.. 0 0 n! en-,

en-2

en-3

an

en-,

en-2

.. a1 n .. U2 GI

Proof. By 27.2.13 we have I

h-

These equations can be written as

-

6, 1 6, * e,

+ d2

6,

+ d2 + a2

*

-

en-2

*

2

*

en-3

+ + an-, + + a,-1

(n - 1)

+ 8,. n = -a,. Applying Cramer’s rule to this linear system in 6,, ..., 6, we obtain the desired dl en-,

*

en-2

**.

*

el

formula. 27.2.15. The next result implies that the Fredholm determinant d of every finite operator is a polynoniial. The degree of d is denoted by deg (d). m

P r o p o s i t i o n . If S = C

(I?

9xi, then

1

d(A) = det

(&ik - ?.(xk,

ni

Proof. P u t A := 2 cci representing matrix tik

=

(xk,

aJ) a d

3 e j and X

ai) for i, k = 1,

otherwise.

deg ( d ) 5 rank (8). nz

:= 2 ek

8 xk. Then

1

..., nz

#

T := A X possesses the

374

P a r t 5. AaDlications

Since S = X A , by B.1.4, me obtain trace (8”)= trace (XASn-l)= trace (AS”-lX)= trace (7’”). According to 27.3.14 the operators S and 7’ have the sainr Fredholm determinant. Using the matrix ( z i k )we see that

In particular, 6, = 0 for n. > ?n. This proves that the Fredholni detcririinant of S coincides with the characteristic polynomial of the ( n z , nz)-matrix ((q, aJ).

27.2.16. In the sequel we have to distinguish between different operators. For this purpose, if S E %(Z1, ZI), the Fredholm determinant of S and thcir coefficients are denoted by d(A, S) and an(&’). respectively. SfiIc %(11, ll). Furthermm, suppose that S P r o p o s i t i o n . Let S,SN, and = SfiISN= 0. T h e n d(A, S ) = a(?&, 8,) d(A, Sip,).

= flN

+ S,v1

Proof. The following formula will be shown by induction:

Obviously 6,(S) = 6,(S,) B,(S,). Kow suppose that (*) is correct for n trace = trace (SF-’) trace (Sg--“) we obtain 1 rn-1 & ( S ) = -&(S)trace ( t P k ) nz k=o

+

z

I

m-1

k

1

m-I

k

< m. Since

27. Fredholm Theory

375

27.2.17. Finally, we check the fundamental Theorem. For e w r y operator S E !&(Z1, diviwr.

I,) the Fredholm determinant .iS a Fredliolwi

Proof. The characteristic values of S and the zeros of d(3,, S)coincide by 27.6.11 and 27.2.12. Let a'(&, S) = 0 and use the decomposition S = SA,+- 8, described in 86.5.8. m

.

Consider some representation SN = 2 ai 0 xi such that (q,.., x,) is n basis of 1

X := ?$((I - nos)"), where p := ~ ( 2 ~Suppose ) . that det ((xk, ai)) = 0. Then there m

m

exists a linear combination x :=

?@k 1

+ 0 with 2

?.k(xk.

ai) = 0 for i = 1. ...,m.

k=l

Consequently, Sx = Ssz = 0. Therefore, it follows from ( I - & 5 ) p x = o that s = o, which is a contradiction. This proves that det ((xk, a i ) ) =# 0. By 27.2.15 we know that the Fredholm determinant of S , is a polynoinial of degree nz = a(&). Since Izo is the only characteristic value of XAv, we have

d(a, s,)

==

(1

-~i;i~)m.

On the otber hand, 26.5.8 (2) and 27.2.12 imply d(Jbo.S,w)=/= 0. Finally, the factorization d(2,S) = d ( l , 5,) d(il, S,) shows that lois a zero of d ( l , S) wvit,h order nz.

37.3.

Related Operators

2i.3.1. Operators:& E B(E, E)and T E B ( F , F ) are related if there cxist:d m d B r B(F, E ) such that S = R-4 and T = A B :

E

B(E, 3')

5.3.2. Related operators have many properties in cominoii. P r o p o s i t i o n . Let S E

B(E,E ) and T E B(F,F ) be related. Then p(8)

P r o o f . An easT; ccjniputation shows that S(2) = S and T(3.)= T US'(%) B for all ii E ~ ( 8 ) .

+

+- IBT(Z.)A

=~(1).

for all 7. E ?(T)

25.3.3. P r o p o s i t i o n . Everyoperutor SE B(E, 14;) relatad toa Riesz operntor T E 2 ( E , F ) dw 7 Biesz operator. Moreover, S and T hnec the some characteristic values with the s m i e crlrjebrnic niultiplicities. @ Proof. Suppose that S = BA and T = BB, where ,4 E B(E, F ) and €3 E 2(F,E ) . Sinoa 8 is a Riesz operator, given 3. E V, wc can find U 5 2 ( E , E:) and X E g ( E , E ) 1'9

376

Part 5. Applications

with U(IE - ils)= IE - X . Put Uo := I F + AT

+ PAUBT, then

UO(IF- RT) = IF - 12T2 + A2AUBT(I, - AT)

+ A2AU(IE As)BT = I F - i2T2 + A’A(IE X ) BT

= I F - li2Ta

-

-

=;-

I F - A’AXBT.

Hence Uo(IF - LT)= IF - X , wit,hX o := IZAXBT. Analogously we may construct Vo E e ( F , F) and Yo E g(P,P ) such that (IF- AT) JT, = IF - Yo. So IF - I T is a @-isomorphism.Obviously A induces a one-to-one map from N((I E- Uj”)onto

N((IF- AT)n) for

18

=

1,2, ... with inverse jJ3

(-AT)”l. In particular,

k=l

it follows that) dim ( N ( ( I ,- AS).)) = dim ( N((I F- AT);)). 27.3.4. As an immediate conscquerice of 27.3.3 we obtain the

T

Proposition. If the Riesz operator S E B(E, E ) is related to the Riesz operaior O(P,F ) , then every Predholm divisor of T is also a Fredholm divisor of S.

27.3.6. Now some important examples of related operators are considered. Proposition. Every operator S E %(E, E ) G re&d to an operator T E %(Zl, Zl). eo

Proof. Let S = 1ciaj $j)xi be any nuclear representation such that llajil and llxill = 1. Put

=

1

ca

-4%:= ((z,

ai)), sO(‘$j) :=

(ojtj), x(fk):=

fkxk. 1

Then S admits the factorization R

x

A 4

as described in 6.3.3. Observe that S = XSoA and T := SOAXare related. Moreover, T is a nuclear operator in 1, with the representing matrix (ci(rk, ai)).

27.3.6. By 27.3.4 and 27.3.5 we can carry over the results of Sect,ioii 97.2 to nuclear operators in arbitrary Banach spaces. ca

Theorem. Let S E R ( E , E ) . Por any nudear represet~ationS = 2 opj @ xi put 1 So := 1 and

27. Fredholm Theory

377

Then W

d(1) :=

2' 6 , P

for all 1 E V

0

is a Fredholm divisor of S .

Remark. The Fredholm divisor obtained in this way depends on the specid choice of the nuclear representation. This fact is a coiisequence of the negative answer to the approximation problem.

27.3.7. We may ask whether the determinant theory of nuclear operators can be T E %(C[O, 11, derived from the classical Fredholm theory. P r o b l e m . Is every operator S E x ( E , E) related to some integral operator T f a(C[O, 13, C[O, 11) admitting a representation by a continuous kernel?

37.3.8. Proposition. Every operator S f !4,,1,2,(E,E ) , where 0 < r 2 2, is related to an operator T f G&, h). Moreover, given E > 0, the operator T may be chosen such that 5 (1 E ) N ( r , l , z ) ( 4 -

+

W

Proof. Let S -=

cia9@ xi be 1

uj

2 0, Mu9) 5 (1

+

8

(r, 1, S)-nuclear representation with

8) N(r,2,1)(8), w2(ai) =

1,

ww(xk) =

1.

If W

~4% := ((2,a*)), o(li):= (C;-"'[j),

x(ik)

:=

lkc'$2q, 1

then we obtain the factorization S

E

+E

Observe that 8 = X D A and T := DAX are related. Since A X is represented by the matrix (C;!'(Z~, ai)),it follows that GL

Furthermore,

I(Xk, C&i)l5 zy

Ci}.

373 -

Part 5. Applications

Romark. We refer to [DUX, p. 11061, [GOH, pp. 1%- 1711, a n d I.RIN, pp. 109 to 1411 for a determinant theory of G;,-operatorsin I,. The Bretllzolin divisors obtaint7il thcw can be carried over to ( r , 1, 2)-nuclear operators by 27.3.4.

27.3.9. Proposition. E u e y operator S E ?&(E', E ) i.s related to a Hilbert-Schmidt operator T E G2(L2(K,y ) , L,(K, ,a)), where ,u is a probability on a suitable conipnct Hqusdorff space K. Moreover. the operator T may be chosen such that S2(T)5 P2(S). Proof. We consider a factorization

E

S

YE

1.

A c C(K)

'LdI', P )

J,

such that I!Xll llAll 5 P2(S) and y ( K )= 1. Then T := J2AX is a Hilbert-Schmidt operator snd

SAT) 5 P Z ( J 2 ) ll-4Xll

5 P,(S) *

Remark. Using Cadernan's theory of Hilbert-Schmidt operators, cf. [SMI] and [ZAM], we can construct Fredholm divisors for absolutely 2-summing operators.

Distribution of Characteristic Values

27.4.

27.41. We know from 26.5.5 (remark) that every Riesz operator S has a t most a cowtablo set of distinct characteristic values A, with j E J . Put

M := ((i,k): j E J , k

=

1, ...,a(&)] and

:= I$.

Then (Am) with m E M is called the family of characteristic values, counted according t o their algebraic multiplicities. In the following this notation is used without further explanation.

27.4.2. Lemma. Let X E Q(€I,Ii). Then thew is an orthorzormnl furriily (x,~,)i ~ d h nE M such tibut ?bm(S~m, x,) = 1. Proof. Consider the operators A j induced by I - i.,S in N , := N ( ( I - ~,X)"('J)) for j E J . Let {Y(],~): k = 1, ..., .(A,)] be a, basis of 3, such that the representing inatrix (ai,) of A, has superdiagoilal form. Take any order in J , and then use the lexicographical order in M . Construct an orthonormal family (zm)by ap2lying Sclmidt's procedure t o (ym). L c t L,"denote the subspace spanned by (rl,...,xm} or (yl,...,y,]. Since A, is nilpotent. me have = 0 not only for A > I:, but also for 11, k. Hence 1=

k- 1

27. Fredholm Theory

379

This means that (I- 2,s) ym E Lm+.Moreover, we always have (I- 2,s) y, E L,,. Consequently ( I - 2,s) xmE L,-l. Therefore xmis orthogonal to (I-- 2,s) xm. This = 1. proves that Ibm(Sx,,q,&)

27.4.3. Theorein. Let 0 < p

< cw mid 8 E bp(H,H ) . Then

Proof. If 1 s p < ca, then the assertion follows from 15.5.7 snd 27.4.2. For proofs in the remaining case 0 < p < 1 we refer to [GQH, p, 411 and [ZAN, p. $401. Remark. As recently proved by W. B. JORNSON/H. ROXIG/B.MAUREYIJ. R. RETHERFORD [I] the above rzsult can be extended to 9lP-operatorsin Banach spaces. A somewiiat weaker statement, is due to A. S. XARKUS/V.J . NACEAV El]. 27.4.4. As a consequence of 27.3.8 a i d 27.4.3 we obtain the Theorem. Let 0 < r 5 2 and S E %(r,l,2)(E, E ) . Then

37.4.6. Theorem. Let 0

< r 5 1 rrnd S 6 !12(r,l,ll(E, E). Then

5 X{,,,,,,(S), Proof. By 18.1.5 every opwator YP,1,2@)

where

1/p::

I/? -

112.

X E !12(7,1,1)(&,E ) is ( p , 1, ;?)-nuclearand

5 %*I,l)(W*

Remark. Let S,, be the operator in 1: gensrated by the nnitary (n,%)-matrix ,

Then S(r,l,l)(Sn) 5 n*/r-1’2.On the other hand, 8, has n charac-

teristk values with /I,/ = 1. This shows that p is the best possible exponect in the above result.

27.4.6. The next statcirierit follows froiii 27.3.9 am1 27.4.3. Theorem. Let S E ?&(E, E ) . I’hc-n

25.4.7. We now state, without proof, a striking result which has been recently Kijmc/B. MACREY/J.It. RETHERFORD [l]. obtained by W. B. JOHSSOR/€€. Theoreni. Let 2

< r < 00

and S

f

(Pr(E;E). Then

Remark. Since $,(H, H ) =- G2(H,H ) , for 0 cannot be less than 2.

< r 5 2 the exponmt of

con\rergencc

37.4.8. A t this point some results ahout entire functions are collectcd [LEW, p. 211.

Let (A,) with

rtL

M be any family of complex numbers suth that

[Aml-l M

< 00.

380

Part 5. Applications

Then p(A) := R(1 - f./A,)

for all f. E 0

M

is called the canonical prod,&. I n the trivial case 211 = 0 we put p(1) = 1.

Next we observe that the entire function p just defined is of order 1 at most and of mininiuni type. Lemma. Let

> 0. Then there exists e > 0 such that

E

\p(A)l 2 p exp ( E 121) for aEZ iE %?.

Proof. First choose a finitc subset m of M and then g

Since 1

> 0 such that

+ 111. 5 exp ( / A / ) ,we obtain

Theorem. Let d be an entire function with the followingproperties: (1) Given E > 0, there exists e > 0 such that

Id(A)l 2 Q exp ( E 121) for all f. E V. (2) I f (A,) with m E dl denotes the family of zeros, counted according to their order, then 2 lAm1--1 < do. ‘M

(3) d(0) = 1. Under these conditims 42)=

n (1 - ]./Am)

for all 1 E %?.

M

Proof. We know from Hadamard’s factorization theorem [LEW, p. 331 that d ( l ) = ( ~ ( 1 ,exp ) ( a &I where ), p is the canonical product formed with the zeros of d. Since d(0) = p(0) exp ( a ) ,we have oc = 0. The preceding lemma tells us that I, is of order 1 at most and of minimum type. Suppose that #I 0. Then exp (PA) has order 1 and nornial type I@/.By [LEW, p. 211 the product d must be of order 1 and normal type 1/31, as well. But this contradicts (1). Therefore, we have ,8 = 0.

+

+

27.4.9. We next state an important result which improves 27.3.6.

Theorem. Let 8 E !R(,,l,2)(E, E ) . For an arbitrary (1, 1, 2)-nuclear repre+sentation 00

AS’ = 1 u p , ;,

2,

put So := 1 and

1

6, :=

(--l)a 2 ail. . . ui, det ((xfs,aj,>). j,. ....jn=l O0

Then, 00

&A*

d(3.) := 0

for

all Iz E %

27. Fredholm Theory

381

is a Fredblm div-kor of 8.Moreover,

d(1) =

n (1

- A/&,)

for all l E V.

All

Proof. The first part of the assertion follows from 27.3.6. Therefore it remains to show the product representation of d. Without loss of generality we may suppose thah uj 2 0, wz(ai) = 1, and w,(xk) = 1. Then Hadamard's inequality implies

Using 27.4.8 (lemma) we obtain

m

5

n(1-k 0~14)5 e exp

(E

Ill),

j-1

for some p > 0, where E > 0 is given. By 27.4.4, moreover, the conclusion follows from 27.4.8 (t.heorem).

llml-l < 00. Pinally, M

27.4.10. Lemma. (trace (S)l S X(l,l,2)(S)for all S E $(E, E ) . m

Proof. Let S =

uiai @ xi be a finite representation such that 1

According to 18.1.16 we obtain Itrace (#)I

5 N:l.l,dS)

= %,w(S).

By the preceding lemma, the trace admits a unique continuous extension to the E). whole space %(l,l.2)(E, Proposition. Every operator S E %(l,l,2)(E, E ) has a well-defined truce which can be computed by 00

trace (8)= 2' c j ( x j ,aj) 1 W

for

any (1, 1, 2)-nuclear representation S =

dini @

xj.

1

27.4.11. Pinally, we prove a generalization of the Grothendieck-Lidskijtrace formula

Theorem. If S E %(l,l,2)(E, E),then trace (8)= 2 l/lm. M

Proof. We know from 27.4.9 that

382

Pert 5. Applications

and

for any (1, 1, 2)-nuclear represtntat’ion of S. Therefore, the assertion follows iron1 27.4. I0 (proposition). R e m a r k . Since %(l,l,2)(U, H ) = G,(H, U ) ,we obtain RS a special case Lidskij’s tracs formula for nuclear operators acting in a Hilbert space. On the other hand, by %(2,3,1,1) E !Rn,,,,,,,, the result is also true for (2/3, 1, 1)-nuclear operators. In general, the trace formula does not hold for (r, 1, 1)-nuclear operators with 3/3 < r 5 1 ; cf. 10.4.5 (remark).

27.5.

Notes

Determinants of integral operators have been defined by I. FREDHOLX [i]. See also T. CABLE[I]. The abstract determinant theory for nuclear operators in Banach spaces was developed [6], A. F. RUSTON[i], [2], and T. L E ~ A ~ ~Ll]. S KFurther I contributions by A. GROTHENDIECK are due to R. SIKORSKI [i],[2]. In this chapter we have presented a new approach discovered [i].There is an extensive literature dealing with the distributions of charaoby A. PIETSCH teristic values. The first result of this kind was proved by I. SCHUR[I]. Further references are 8. H. CHANG111, K. FAX[l], [2], A. HORN[i],A. PIETSCH [17], and, above all, H. WEYL[I], [2].The so-called trace formula is due to A. GROTRENDIECK [6] and V. B. LIDSKIJ [l]. See also J. A. ERDOS[i].

MAN

Recommendations for further reading:

CGOHI, CRIPU’I, [ S W , CYOSI, [ZANI. C. TV. HA [2], E. HELLINQER/~. TOEPLITZ [i], H. vox KOCH[I], H. KONIG[4], A. S. MARKUS/V.J. MACAEV[l], N. J. XIELSEN111, A. F.RUSTOX[4], [Ti], B. SIMON [i].

25.

Structure Theory of Bnnach Spaces

There is a close relationship between the theory of operator ideals and the geomctry of Banach spaces. In this chapter we only deal with a few aspects.

TTsing sonic results about operator ideals we get deep structure theorem. So the ftmious principle of local reflexivity follows from the fact that B(E,P)' and %(F, E ) are metricdly isomorphic for all finite dimensional Banach spaces E. Moreover, w estimate projection constants by the help of the absolutely 2-summing norm. This yields a short prooI of the fundamental Kadec-Snobar lemma. Opcrator ideals can also be used to characterize classical Banach spaces. Here the quotient of two operators ideals plays a significant role.

Since the coiriponents U ( E ,a) of any normed operator ideal are likewise Banach spaces, we get a, lot of new examples with interesting properties. In particular, we mention the Banach spaces G,(H, K ) ,where 1 p < 00. So, for example, it has b e s observed by Y. GORDONand D. R. LEWISthat %(H, K ) and B(H, 9) do not havelocal unconditional structure.

28.1.

Principle of Local Reflexivity

58.1.1. We start this section with Helly's Lemma. Given M E Dim (El) and E > 0, for every xi E E" there exists ,z, f. E suck thut 1 ) ~ ! 1 2 (1 + E ) IIxJl and (zo,a) = (xi,a ) whenever a E M . Proof. Put

N

:= (z E

E : (z, a) = 0 for a E N )

and

NO0 := (x" E": (x", a) = 0 for a E N}. By B.3.10 we have the diagram

where K is a, metric isomorphism from E / N onto E"INoo. Choose xo E E such4hat K&E,x,,= and [[zo// S (1 e) l[x:II. Then Q$+i - KEzo)= o implies 26 - KEzoE NOO. Hence (ro,a) = (z:, a) for all a E M .

Qs0z{

+

3a

Part 6. Applications

228.1.2. We are nov ready to check a fundamental Lemma. Suppose that E is finite dinzensionaZ. Let S E B(E,F')and M E Dim (P'). Given E > 0, there i s R E E(E,8)such that lIR[l I (1 E ) llS\l and (Rx,b) = (8qb) fordZxEEandbEM. Proof. Observe that

+

M @ E := 1

bi @ x i : xl,..., x, E E,bl, ..., b, E M , n = 1, 2,

...

i q a finite dimensional subspace of % ( F ,E). By 9.2.3 we may identify B(E,P)' and %(F, E ) a8 well as %(F,E)' and B(E, F").Now the assertion follows from Helly's

lemma.

28.1.3. As an improvement of the preceding result we get the principle of local refZezitdy. Lemma. Suppose that E is finite dimensional. Let S E B(E, F")and M E Dim (F'). Given E > 0, then the!reexists R E B(E,F ) such that IlRll 5 (1 E ) 11511,(Rx,b)=(Sx,b) for d x E E and b E M , as well us KFRx = Sx whenever SX E M(KF). Proof. Put Eo := (x E E:SX E M ( K F ) )and denote the injection from Eo info E by J . Choose any surjection Q E B(E, Eo) for which QJ is the identity map of E,. Obviously, x -+ K;'Sx defines an operator So E f?(Eo,F) with llSo(li IlSiI and KpSo = SJ. Let ( M i ) with i E I bc a directed family of subspaces M iE Dim (F')such that F' = U Mi and M i 2 M . By 28.1.2 we can find Ri E B(8,F ) with llRill 5 (1 E ) llSl1

+

+

I

and (Rix, b) = (Sx, b) for all x E E and b E Mi.This yields (RiJx,b) = ( S g ,b) for all x E Eo and b M i . So, according to 9.2.3, the directed faniily (RiJ) tends to So in the weak E(Eo.F)'-topology. Let C be the convex set of all operators RJ, where R E E(E, F ) , such that /]Rll5 (1 E ) ljSjl and (Rx, b} = (Sx,b) whenever x E E and b E M . Clearly RiJ E C. Therefore So belongs to the weak closure of C. Let us recall that we have the same closed convex subsets for the weak topology and the norm topology of B(Eo,P);cf. [DUN, p. 4221. Hence there exists R, E B(E, F ) with RJ E C and ljBoJ - Soil I]&//d E /IS//. Finally, put R := (So- RoJ)Q Roe Then JJRII5 (1 2 ~11 )611. If x E E and b E a,using KFSO = SJ we get

+

+

(AX, bj = (SOQX,bj - (ROJQX,bj

+

+ (Rex, b) = (SX,b}.

Furthermore, Sx E M ( K F )implies z = JQx and therefore

KFRx = KFSOQX+ K,R,(x - JQx)= SX. 28.2.

Geometric Parameters Related to Operator Ideals

28.2.1. Let A be a quasi-norm defined in some operator ideal 31. Then for every finite dimensional Banaoh space E we may consider the value A(IE) which is inva,riant under metric isomorphisms. For certain quasi-norms A it turns out that A(IB) is only a function of dim ( E ) . In other cases A(I,) also depends on the geometric structure of E .

28. Structure Theory of Banach Spaces

385

98.2.2. Wo begin with a trivial example.

Lemma. llI,ll = 1.

28.2.3. As a counterpart of the preceding result we have the Lemma. If dim ( E )= n, then X(IE)= n. Proof. By B.4.8 we can find a finite representation n

ui

IB =

0xi

1

such that IIcci// = 1 and /\sill= 1. Therefore K ( I E )5 n. Conversely, by 6.8.3 and 6.8.5, n = trace (I,) 5 SO(IB) = N(IE).

28.2.4. We now state the most interesting result of t,his section. The following proof was given by S . KWAPIEA. Lemma. If dim ( E ) = 12, then P2(IE)= dZ. Proof. L e t x ~ , . . x,,, . , E E and define the operator X E E(lF, E ) by m

X(&):=

c tixi. 1

Then llXll = wZ(zi). Consider the factorization

I

+ J

where N := N ( X ) and M := M ( X ) ; cf. A.3.4. Clearly H := Zr/N is also a Hilbert %lie. Hence space and dim ( H ) 9 n. By 17.5.3 we have P2(IH)= S 2 ( 1 H )

PAX) 5 IIJx,ll PZ(4d 11&11 5 nl’z IIXII. I t follows from zi = Xei and w2(ei) = 1 that

lZ(xi)= 12(Xei)5 PJX) w2(ei) 5 n%v2(xi). so P2(I,) 5 d ’ 2 . Conversely, using 24.6.5 and 28.2.3 we get n = N(IE)5 P2(IE)2.

* 28.2.5. According to the following characterization &,,(IE) is called the projecEion constant of E. 25 Pic!tsoh. Operator

386

Part 5. Applications

Proposition. Let E be a finite dimensiomlBanuch space. Then L,(Ig) equals the infimum of all positive numbers cr for which the following condition is satisfied: If E, is any Banach space containing E as a subspace, then there existsaprojectimz P E O(Eo, E,) such that E = M ( P ) and III’li 5 u. Proof. For zation

E

> 0 we pnt u := (1 + E ) Lm(IE).By 19.3.9 we can find a factori-

(1x11

IIAJl5 CT. Suppose that E is a subspace of E,. Since L,(O,p) has the with metric extecsioii property, A E O(E,L,(O, p)) extends to A , E O(Eo,L,(Q, p)) such that IiAoii = IIAjl. Then P := J$XAo is a projection on E, for which E = M ( P ) and iJP// u. Hence c satisfies the above condition. Recall that E is a suhspace of Einf:= l,(UEc). If P denotes any projection in Einj with E = M ( P ) ,then we have Lm(IE)2 IlPil. This fact completes the proof.

28.2.6. The following estimate, together with 28.2.5, yields immediately a famous result due to &I. J. K4DEC/&I. G. SNOBAR [I]; cf. B.4.9. Lemma. If dim (E’) = n, then Lm(IE) =( P2(IE)= n1I2. Remark. Let I,, be the identity map of I;. Then, by 9.1.8 and 19.3.10, we have Lm(Ifl)Pl(Ifl)= ‘n. Consequently, it follows from 22.1.1 and 22.1.3 t,hat Jim n-l’zLm(In) = lini dkza,, =~ ~ c ; : . n

So in the above leniina the exponent of n cannot be improved. 28.2.7. The value of the projection constant strongly depends on the geonletry of the Banach space in question. Let us mention the following result; cf. D. J. H. GARLING[IS].

Proposition. Let E be an n-dirtiensitmab Bmach space. Thcn L,(IE) = 1 i/ and only i f E .is metl-ically isomorphic to.:1

28.3.

Isomorphic Charact,crizationsof Classical Banach Spaces

28.3.1. We start this section with a basic result which follows from 19.6.2. Theorem 1. A Banach space 3’ belongs to L, i f and only if @ F ( E ,F ) for all Banach spaces E. By duality we get

&,(E, P)

28. Structure Theory of Banach Spaces

387

Theorem 2 . A Banach space E belmgs to Lp if and only if !&.(E, F ) & S Y ( E , F ) for all Bunach spuces F. 28.3.2. The next criterion is a direct consequencc of 8.2.5 and the previous statement; of. also 17.3.5. Theorem. A Banach s p c e E belongs to L, if and only if F(E, F ) = S ( E ,F ) for all Baimch spaces F. 28.3.3. We now formulate, without proof, a similar result which was obtained by

D. R. LEWIS/C.P. STEGALL[l]. For this purpose let us recall that F, is the ideal of all Banach spaces having the lifting property; cf. C.3.8. if and o d y if F ( E , F ) = R ( E , F ) for

Theorem. A Banuch spuce E belongs to F;*' all B a d spaces P .

28.3.4. In the following H denotes the ideal of all Banach spaces isomorphic to some Hilbert space. Remark. Obviously, by 19.3.8, we have H = L,. 28.3.5. Next we state a special case of 28.3.1. Theorem. 1. A B a m h space F belongs to for all Banach spaces E .

H if and only if !#P1(E, P ) V2(E,F )

The dual version looks as follows.

F) Theorem 2. A Banach space E belongs to H if and only if !&(E, F ) E Vgua1(E, for aU Banach spaces F . 28.3.6. The following open problem is closely related to some results of Section 17.6. Conjecture 1. A Banach space F belongs to H if and only if @ p l ( E , P )E a 2 ( E , P ) for all Banach spaces E. Conjecture 2. A Banach spacl: E belongs to H if and only if V P , ( EF) ,

s S z ( E ,F )

for all Bansch spaces F.

28.3.7. The next question arises from 11.5.2 and 11.6.2. Conjecture 1. ABanach spaceFbelongs to H ifandonlyif S2(E,F ) = U","(E, F ) for all Banach spaces E . Conjecture 2. A Banach space E belongs to H if andonlyifU,(E, P)= U;"j(E,F ) for all Banach spaces F .

28.3.8. We now prove another basic result. Theorem. f i t I 5 p

2 5q5

00.

.F

Then H = L F n L y .

Proof. Since the normed operator ideal $ is maximal, it follows from 22.1.5 that H & L F . By duality we get H Ly.This proves that H LinJ n L r . 25*

Port 5 . Applications

388

Lp Ly.

Conversely, suppose that E’ n Then JE E &, and QE E f&. Hence 22.5.1 yields JzISl&BE 4. Finally, using the injectivity and the surjectivity of $jwe obtain I E E .Sj and therefore E E H.

28.3.9. As an immediate consequence of 21.4.6 we get a mpplement of the preceding result. Theorem. Let 0 < p

< 00. Then H == C(2,p)n T(2,ph.

28.3.10. Theorem. The Banach space F belongs to H if ultd onlg i f 2(Z17 F ) = g(ll,F ) . Proof. By 22.4.4 we have 2(Z1, F ) = ?J3(Zl, F ) whenever F E H. Conversely, let us suppose that all operators S E .(?(I,, F ) are absolutely sumnung. Then 2(Zl,F ) = p2(Z1, F ) . It turns out that we also have .(?(Zl(I), F ) = ?J32(Z1(I)7 F) for every index set I . Since Fur :=Z1(UF),we get Q p E clg,(FsUr,P ) B(Fs”‘, F ) . Now the surjectivity of 8 implies IT E .Sj and therefore F E H. Remark. As recently shown by S. V. KISLJAKOV [1] and G. PTSIER [4]it does not follow from B(E,E,) = ?J3(E,12) that E belongs to L,.

28.4.

The Structure of Certain Ideal Components

28.4.1. Let [U, A] be a normed operator ideal. Then U(E, F ) is also a Banach space, and we may ask “HOWcertain properties of E and F are transmitted to U(E, F)?” In particular, it is interesting to know for which Banach spaces E and F the corresponding coniponent U ( E ,P ) belongs to a given space ideal B. The following necessary condition can he easily checked. Proposition. #uppose that E + (0) and F =/= ( o f . Them %(E,F ) E B implies E E Bdual and 3’E B. Proof. Take yo E P and 6, E F‘ such that yo + o and (yo, 6,) = 1. By setting Ja := a tgyo for a E E‘ and &[S]:= S’b, for S E U(E, F ) we obtain operators J O(E’, U(E,F ) ) and Q E O(%Z(E,P),E ) with IE, = &J. Therefore E’ U(E,F ) by 13.4.4.So U(E,P ) E B implies E‘ E B. The conclusion F E B can be proved with the same method.

<

28.4.2. To give an example we now mention a converse statement due to Y. GORDON/

D. R. LEWTS/J. R. RICTHERFORD [2]. Theorem. Let E and F be reflexive Banach spaces. Then Q ( E ,F ) is. reflexive, as well. Remark. For further results of this kind the reader is referred to S . HEIYRICH P I 7

PI.

28.4.3. Soine of the following results, where H and h-are Hilbert spaces, have been 123. According to T.FIGIEL/G. PISIER[ 11 obtained by N. TOMCZAK~AEGERMANN the assertions can also be proved by the help of certain convexity and smoothness properties.

28. Structure Theory of Banach Spaces

Proposition 1. If 1 Qq(H7

W

E

389

-< q 5 2 and 0 < p < s < q, then

T(8.p)-

< 00 and 0 < p < 00, then Gq(H, K ) T ( 2 . p ) . Proposition 3. I f 1 5 q 5 2 and 0 < p < CQ, then GqW, K ) E C ( 2 , p )* Remark. The above statements are related to 22.3.2, 22.3.3, and 22.3.4. Proposition 2. If 2 5 q

28.4.4. As observed by Y. GORDON/D. R. LEWIS[2], [3] the Banach spaces %(H, K ) and B(H, K ) do not have iocal unconditional structure; cf. 23.3.7. A much more general result is due to D. R. LEWIS [3]. Proposition. Let [a, -41 be a normed operutor ideal on the &s.s of N-ilbert spnces. Thcn a(&, Z2) E I, implies SI = G2.

Miscellaneous Results

28.5.

28.5.1. A sequence ( e j ) is called an unconditional basis of the Banach space E if 8

every x E E admits a unique representation x = C Ejei, where Cj E X , such that 1

([iei)

is summable. If lleill = 1, then the basis is said to be normalized.

28.5.2. We shall need the following lemma; cf. [SIN, pp. 499-5001. Lemma. If (ei) i s a n unconditional basis, then there exists a constant ,o 2 3 that

whenever

5

jv7il

and n

=

SZLC~~

1,2, ...

28.5.3. Suppose that (ei) and ( f i ) are unconditional bases of the Banach spaces E 8

and F , respectively. These bases are said to be cquivulent of the series 2' liei. and 1

;a

1.1-ifi 1

converge for the same set of sequences (Aj). Then we can define an isomorphism

1 € B(E,P)with l e i = f i for i = 1,2,

...

28.5.4, We are now able to formulate a fundamental Theorem. The Banach spaces I,, l,, and c, have a normalized unconditimtal basis

which is unique u p to equivaleme. Proof. First observe that the unit sequences el, e,, ... form a normalized unconditional basis of I,, 12, and c,. 00 (1)Let (ti)be any normalized unconditional basis of I,. Clearly is convergent 1 whenever (A,) E I,. a, Conversely, we can show that (Ai) E lI for Wery x = &fi. Since (&fi) is summable 1 the Orlicz property of I , implies

390

Part 5. Applications

So, by setting Sx := (Ai) we obtain a n operator 6' from I , into 1,. Hence S E ?$(Z1, I,) by 22.4.4. This proves that W

W

zl l i l z I l ~ i W r l l z< =

1

00 *

1

(2) Let ( f i ) be any normalized unconditiocal basis of 12. Then, by 28.5.2, there exists a constant e 2 1 such that

for all A1,

...,1, E X

and el = f l ,

...,

E,

= &l.Consequently

1 2: l * &fi

S erz(Aiji).

Observe that 1, is of Rademacher (2,2)-type. More precisely, a; shownin the proof of 22.3.2, we have r,(&fi) 5 lz(lzifi).Hence

W

This proves that 2 &fi is convergent for all (Ai) E .2, 1

M

Conversely, since ( A i f i ) is summable for every x = implies W

&fi, the Orlice property of l2 1

00

L1'I&Iz = Z1 IIWiIE <

00-

(3) The Banach space co can be treated by duality; [SIN, p. 5351. 28.5.6. Denote by T := (5 E %: ICl = 1) the unit circle in the complex plane equipped with ths normalized Lebesgue measure. Let A ( T ) and Hl( T)be the subspaces of C ( T )and L,(T),respectively, generated 97.

by the polynomials p(5) :=

2 L X ~ C where ~,

(xo,

cil,

. .., a, E %.

0

28.6.6. P r o p o s i t i o n . A ( T ) @ Lz. P r o o f . Suppose that there exists a surjection Q from soine Banacli space L , onto A ( T ) .If J , denotes the embedding map from A ( T ) into Hl(T), then we have J1Q E &(Aw, HI(!/!')). Observe that H l ( T ) has the Radon-Nikodym property, since [7]. Now it followa froiii it is a separable dual Banach space; cf. A.PZLCZY~SKI 24.4.3 2nd 24.6.1 (theorem) that JIQis l-nuclear and therefore compact. Using the injectivity of R we get J, E B(A(T),H l ( T ) ) ,which is a contradiction.

28.5.7. Next we prove 'chat A ( T ) does not have local unconditional stracture. The space ideal I, is defined in 23.3.7. P r o p o s i t i o n . A ( T ) 6 I,.

Proof. Suppose that A ( T ) (5 I,. Then 23.4.1 yield$ J, E f&(L4(!/!'), H,(T)). Using a result of A. PELCZP~SICI [7] we get J1 E R ( A ( T ) HI(")). , This contradiction colnpletes the proof.

28. Structnre Theory of Banach Spaces

39 1

28.5.8. I n order to prove the following theorem we need a fundamental statement which is due to H. P. ROSEWTHAL [5] and E. E. DOR[l] in the real and complex case, respectively. See also [LIT, vol. I, p. 991.

Lemma. Let (y,) be a bounded sequence in a Banach space F. Then there exGts a subsequence (y,,) satGfying one of the following two mutually exclusive alternatives: (1) (y,J .is a weak Cawhy sequence. ( 2 ) W e can find some injection J E 2(Z,, B ) such thut (yfiJ = (Jei), where (ei)is the canonical basis in 1'.

We now formulate the

vF1

Theorem. %-I o R = o R =3 ; ' o R for 1 < r < 00. Mowover, an operator S E 2(E,F ) belolzgs to these quotients i f and only if every sequence (Sx,)with x,,E UE

ndmits a weak Cauchy subsequence. Proof. First of all, it follows from ? F O R

sr

vr

2?that

sv;'oJt&3;1oR.

Let X E 2(E,F ) satisfy the above property. Then every sequence (Sx,)with x, f UE admits a weak Cauchy subsequence (8xn,).Hence, for all B E B(F, Fo), we see that (BXX,,)is norm convergent by 1.6.3. This proves that BS E B(E, Fo) and therefore S E %-1 o a.

Conversely, let S E 2(E, I?) belong t o 3 ; ' o R, where 1 < r < 00. Puthermore, suppose that S does not fulfil the above condition. Then, by the prececlinglemma, there exists a bounded sequence (xi) in E and a n injection J E 2(ll, F ) such that (Sxi) = (Jej).Let Qo be any surjection from I , onto 12. By 22.4.3 and 22.4.4 we have Qo E Zr(l1,Z2). Using a factorization as described in 19.2.6 (2) we can find some extension Q E 3 , ( F , Z2) such that Qo = QJ. Therefore (QSx,) = (Qoei). Finally, QS f B(E, 12) implies Q,, E Jt(ll, &), which is a contradiction.

28.6.

Notes

The famous principle of local refiexivity mas found by J. LIXDENSTRAUSS/H. P. ROSENTHAL 111 and strengthened by W. B. JOHNSON/H. P. ROSENTHAL/M. ZIPPIS [i]. The simple proof given in this chapter is due to D. W. DEAN[l]. Geometric parameters of finite dimensional Bantlch spaces associated with quasi-norms of operator i d d s wcre computed by several *iuthors;cf. D. J. H. G 4 R L I N G / Y . GOBDON [l],Y. GOEEOX[l], [ 2 ] ,[3], Y. GonnoN/D. R. LEWIS 131, and M. G. SNOBAR [i]. The most striking result, namely the exart value of the P,-norm. was obtained by S. KWAPIEB(unpublished). As an immediate consequemo he got a sharp estimate of the projection constants for finite dimensional Banach spaces; cf. M. J. KADEC/ ?$I.G. SNOBAR[i]. The first isomorphic characterization of Hilbert spaces by the help of operator ideals has been given by J. S. COHEN[i].The extensions of this result to Lp-spacesare meiniy due t o S. KWAPIE~~ [GI. The same author [7] proved that every Banach space with (2,p)-typeand (2,p)-cotypeisisomorphicto a Hilbert space. We also refer the render t o D. R. LEWIS/C.P. STEGALL[I]. Several results cowerning the structure of certain ideal components [l], Y. GORDON/D.R. LEWIS[2], [3j, Y. GORDOK/ appeared in J. ARAZT/S.LINDENSTRAUSS D. R. LEWIS/J.R. RETHERFORD [2], S. HEINRICH [i], [2], J. R. HOLUB [7],[S], C. 9. XCCARTHH [l], and N. TOMCZAK-JAEGERMAXN [2]. In particular, the Banach space O(E,F ) has been investigated by M. FEDER/P. 8 LPEAR 111, J. R. HOLUB[GI, N.J. KALTOX [i], W.RUCXLE[I].

392

Part 5. Applications

R. SCHATTEN [l]. and A. E. TONG[S]. Miscellaneous results are taken from L. E. DOR[l], J. LINDENSTRAUSS/A. PELCZY~SKI [l], A. PEECZYI~KI El], [7]. H. P. ROSNNTHAL [5], and I,. WEIS [l]. Recommendations for further roading :

[DIE],[LIN], [LOTl. B. BEAUZAMY [l],D.I. CARTWRIGHT/H. P. LOTZ[l], T. FIGIEL/G.PISIER [l], Y. FRIEDMAS D. J. H. GARLIXG[6], Y. GORDON/D.R. LEWIS/J.R. RETHERFORD [l], A. GROTHESDIECK [3], B. GRUNBAUM [l], V. J. GURLRIJ/&~. 3. KADEC/V. J. NACAEV [l], E.HELLY[lj, -1.R. HOLUB [5], M. J . KADEC/B.S. MITJAGIN [I], D. R. LEWIS 111, [3], W. LINDE 141, J . LINDENSTRAUSS/L. TZAFRIRI [l], J. LINDE~STRAUSS/M. ZIPPIN 111, M. 8. MACPHAIL [I], B. MAITREPIG.PISIER[2], J. S. MORRELL/J. R. RETRERFORD [l], G. PISIER[4], [51, D. J. RANDTKE [a], J. R. RETEERBORD/C. P. STEGALL [l], B. ROSENBBRQER 131, l€. P. ROSEKTHAL [7], D. RUTOVITZ [2]. [I],

29.

Operator Ideals on Locally Convex Spaces

This final chapter is devoted to the theoiy of operator ideals on locally convex spaces. Many concepts can be taken over from the Banach space theory directly. On the other hand, there are some new features such as invariancc under the formation of Cartesian products. The most important problem is the extension of operator ideals from the class of Banach spaces to the class of locally convex spaces. Our main purpose, howcvcr, is to deduce the theory of nuclear locally convex spaces from the theory of opesatar ideals.

29.1.

Locally Convex Spaces

49.1.1. A locally cmuex space is a real or complex linear space U together with a system '$ of semi-norms possessing the following properties : (PI) Let pl. p2 E 8. Then there are a semi-norm p E $ and a constant p > 0 such that rnax ( p l ( x ) p, z ( x ) ) ep(x) whenever x E 77. (P2) If p ( x ) = 0 for all p E 3, then x = 0.

29.1.2. In the following U , T7, and W always denote real or complex locally comex spaces with corresponding semi-norm systems '$, 0,and W. The class of all locally convex spaces is denoted by L. Warning. Let us recall that, in the preceding chapters, L was used to denote the class of all Banach spaces.

29.1.3. By the semi-norm system 3 we may generate a topology on the locally convex space U ; cf. [KOT, p. 2061. In this way U becomes a linear topological Hausdorff space.

29.1.4. Let M be a closed linear subset of the locally convex space V. Then VQ:= M becomes a locally convex space if we take the system 0, of semi-norms qo(y0) := q.(JYo) for Yo E

vo,

where q E 0 and J denotes the canonical map from Vo into V. Every locally convex space Voobtained in this way is called a subspnce of V.

29.1.5. Let N be a closed linear subset of a locally convex space U.Then U, := U/W becomes a locally convex space if we take the family Po of semi-norms po(ro):= inf

(~(2 : &Z )

=50)

for

~0

E UQ,

where p E and Q denotes the canonical map fmm U onto 77,. Every locally convex space 0, obtained in this way is called a quotient space of U.

394

Part 5. Applications

29.1.6. A linear map S from U into V is continuous if for every semi-norm q E D there are a semi-norm p E '$3 and a constant CT 2 0 such that q ( 8 x ) 5 ap(x) for all x E hi.

A continuous linear map is simply called an operator. 29.1.7. I n the following 2 denotes the class of all operators between locally convex spaces, and 2 ( U , V ) is the set of t>hoseoperators acting from U into V. Warning. Let us recall that, in the preceding chapters, 2 was used to denote the class of all operators between Banach spaces.

29.1.8. A one-to-one linear map T from U onto V is called an isomorphism if both T and T-l are continuous. In this case the locally convex spaces U and V are said to be .isomorphic. The identity map of a locally convex space U is denoted by Iu. 39.1.9. Let ns mention that U is isomorphic to a compleniented subspace of V if and only if there are operators J E O(U, 7) and Q E 2 ( V , U ) such that Iu = QJ. In V. this case, we write U

<

29.6.10. Let ( U i )be a family of locally convex spaces Ui with corresponding seminorm systems pi, where i € I . Then the set U of all families x = (xi) such that xi E Ui is a linear space. Furthermore, by putting

p ( 5 ) := 2 Pi(Xj) for x i

= (Xi),

where pi E Pi and i g ( I ) , me define a semi-norm system '$3. The locally convex spacc )((Ui, I ) obtained in this way is called the Cartesian product of ( U i ) . Remark. The Cartesian product of two locally convex spaces U, and U , is denoted by U , x U2.

29.1.11. Let ( V , )and (Vi) be families of locally convex spaces. Suppose that (ai) is a family of operators S i E i?(U,,rz). Then (xi)--f (Xiz,) defines an operator )((X,, I) from )((Ui,I)into )(( V i , I) which is called the Cartesian product of (Si). 29.1.12. Let p be a senii-norin on the linear space U. Then N ( p ) := {xo E U : p ( x @= ) 0) Is a linear subset. Form the quotient space U ( p ) := U,"(p) ~ ( p:= ) .i: N ( p ) . By setting

+

with the elements

lix(p)ll := p ( z ) for z E C we define a norm on UCpj. Finally. let o ( p ) be the Banach space obtainell by conlpleting the normec! linear space U ( p ) , and denote the quotient map from U into 'DY I&).

w

29.1.13. Lemma. Let S E g ( U , F). If p E q(A!x) 5 G p ( Z )

f01.all Z

f

U,

4 and q E C are semi-nonm with

29. Operator Ideals on Locally Convex Spaces

395

then there ezists a unique operator S ( p , q) E ~ ( l ( p 7(q)) ) , such that

U

S

+V

Proof. Since X(N(p)) 2 N ( q ) , the operator S(p, q) x ( p ) := (Xx)(q) for x E U is well-defined, and ((S(p,q)ll 5 u. So we obtain &p, q ) by continuous extension. Remark. When we shall use &(p,q), it will always be assumed that (*) holds.

29.2.

Operator Ideals and Space Ideals

39.2.1. An operator ideal 'u is a subclass of 2 such that the components

U(U,V ) := 4c n 2 ( U , V ) satisfy the following conditions: (01,) 1% E U, where Zdenotes the l-dimensional locally convex space. (01,) It follows from S1, S, E %(U,V )that 8, d2 E %(U,V ) . (01,) If T E O(U,, U ) ,S E %(U, V ) , and R E 2 ( V , V,), then RXT E %(U,, Vo).

+

Remark. Most of the definit,ionscan be carried over from Banach space theory. If no confusion seems to be possible we shall do so without further explanations. '29.2.2. A space ideal A is a subclass of L with the following properties: (SI,) The 1-dimensionallocally convex space .X belongs to A. (SI,)It follows froin U1, U , E A that U,x 0, E A. (SI,) Let U,, U. Then U E A implies U, E A.

<

39.2.3. Let 'u be a n operator ideal. Then Space (Ju) is the class of all locdly convex spaces U such that Iu E 'u. 29.2.4. Analogously to 2.1.3 we have the

Theorem. Space ('67) is a sptce ideal. 29.2.6. Let A be a space ideal. An operator S E j2( U,V ) is called A-factorable if there C-iistB a factorization S = Y A such that A E O ( 0 , M ) , Y E(M. V ) , m d .M E A. Tho class of all A-factorable operators is denoted by Op (A). :6

29.2.6. i n the saine way as in 2.2.2

37e

get the

Thcor em. Op (A) is a7~operator idecr2. 39.2.7. Xow the preceding result is iniproved; cf. 2.2.7

Theor em. Let A be a space ideal. Then Op (A) 12the smallest operator d e a l '4 with A = Space (a).

396

Part 5. Applications

29.3.

Injective and Surjective Ideals

29.3.1. An operator ideal rU is called injective if for every subspace V , of a locally convex space V and for every operator So 6 2 ( U , V,) with J S , E % ( U , TI), where J denotes the canonical map from V , into V , it follons that So E % ( U , V,). 29.3.2. A space ideal A is callnd injective if every subspace of a locally convex spacr V f A also belongs to A. 29.3.3. Proposition. For every injective operator ideal 2l the space ideal Spaco (a) i s injective, as cvell. 29.3.4. Proposition. Por every injective space ideal A the operaior ideal Op (A) i s injective, us well. 29.3.6. An operator ided t!? is called surjective if for every quotient space Uo of R locally convex space U and for every opcrator SQE g(U,, V ) with SQQE n ( U , V ) , where Q denotes the canonical imp from U onto U,, it follows that X,,E 21(Uo, V ) . 29.3.6. A space ideal A is callec! surjective if every quotient space of a locally convex space U E 2l also belongs to A. 29.3.7. Proposition. For every surjective operator ideal 8 the space ideal Space (U) is surjective, as well. 29.3.8. Proposition. For every surjective space iderll A the operator ideal Op (A) is surjective, as well. 29.3.9. We now restate some well-known results from the theory of locally convex spaces, [KoT, pp. 301,371, and 4371 and [SAF,, pp. 61 and 1461. P r o p o s i t i o n 1. The ideal of semi-reflexive spaces i s injective, but not surjective. P r o p o s i t i o n 2. Il’he ideal of quasi-barrelled spaces i s surjectice, but not injective.

39.3.10. Since tile intersection of injective operator ideals is also injective, for every given operator ideal we can define an injective hull. Analogously, there exists a surjective hull. In contrast to the Banach space theory, no direct construction of these hulls is known at present.

29.4.

Cartesian Ideals

29.4.1. An operator ideal 3c is called Carfesicln if X(S4, I ) f %(X(ui, I),X (Vj, I ) ) for every family ( f i t ) of operators X i E U(Ui, Vi). 29.4.2. A space ideal A is called Cartesian if X ( U 1 I, ) E A for every fainiIy (Ui) of locally convex spaces UTiE A. 29.4.3. Proposition. Por every Cartesian. operator ideal 2l the space ideal Space (a) & Cartesian, as well. 29.4.4. Proposition. For every Cartesian space deal A the operator ideal Op (A) i s Cartesian, as well.

29. Operator Ideals on Locally Convex Spaces

397

29.4.5. For proofs of the following statements we refer to [KOT, pp. 302 and 3711 and [SAF,, p. 1461.

P r o p o s i t i o n 1. The ideal of semi-reflexive spaces is Cartesian. Proposition 2. The ideal of quasi-barrelled spaces is Cartesian. 29.4.6. By analogy with Cartesian invariance, we may consider ideals which are closed under the formation of locally convex direct sumfi.

29.5.

Extensions of Operator Ideals

29.5.1. I n this section Sodenotes some operator ideal on the class of Banach spaaes. An operator ideal U defined on the class of locally convex spaces is called an extension if N ( E , F ) = Uo(E,F ) for all Bana,ch spaces E and F . 29.5.2. Every operator ideal a. adinits six natural extensions

T the definition of which is analogous to that given in Section 15.6.

+

29.5.3. Proposition. ?@' %Fp.

Proof. It can easily be seen that

!&rpis Cartesian, while n';t

is not.

29.6.4. From our point of view the right-superior extension is the most important one. Let us therefore recall that %;;Up consists of all operators S e ( U , V ) such that, given B 2(V, F),there exists a factorization BS = S,A, where A f S(U, E ) and So E U&E,P). 29.5.5. Proposition. Let S E 2 ( U , V ) . Then S E Urp(CJ, V ) if and only if forevery semi-norm q E E l there exist8 CL semi-norm p E $'3 such thnt S ( p ,q) E %,,(o(p), p(q)).

Proof. According to 29.1.13 we have the diagrams

398

Part 6. Applications

and

which prove the sufficiency and the necessity, respectively. 29.5.6. P r o p o s i t i o n . Let !H0 and Bobe operator i d e a l on the elms of B u m h spaces. Thm o %;;Up 5 (a, o ?Bo)rup.

9$rp

Proof. We consider operators T E %:”(U, V ) and S E S;;Up(V, W ) . For every operator C E f!(W, G ) there are operators So E ?Ho(P,G ) and B E f!(V, P) such that CS = SOB.Next we choose operators To b o ( E ,F ) and A E B(U, E ) with BT = ToA Consequently, we have the diagram :

uvw -

E-F-G

T

S

TO

SO

Finally, it follows from SOToE 8, o So(E,G) that ST E (U, o !30)rup (U, W). 29.5.7. Lemma. Let V , be a subspace of the locall!/ convex space V . Then for ewery semi-.norm qo E 0, there exists a semi-norm q E Q such that J(qo,q ) i s an injection.

Proof. Accordiiig t o 29.1.4 every semi-norm qo E SZ, isobtainedby qo(yo):= q(Jyo) for yo E V,, where p 6 Q. Then

Consequently, the continuous extension J(qo,q ) is a n injection. P r o p o s i t i o n . For every injective operator ideal a, the right-superior extension’8Y” i s injective, as well.

Proof. Using the notation of 29.1.4 we consider a n operator So E B(U, V,) with S := J S , E U;;Up(U, V ) . Take a semi-norm qo E Qo. By the preceding lemma there exists a semi-norm q E El such that j ( q o , q ) is a n injection. Applying 29.5.5 we can choose a semi-norm p E $‘3 with g ( p , q ) E N o ( 6 ( p ) F , ( q ) ) . This yields the diagram which is printed a t the top of the following page. CoiLsequently,for every qo E &, there is a p E ’$ such that s0(p,qo) E !Ho(O(p),vo(q,,)). This proves So E W,”;l”(U,Vo).

29. Operator Ideals on Locally Convex Spaces

399

29.5.8. Lemma. Let Uo be a quotient 8pme of the locally convex S(pace U. Then for every semi-norm p E !$ there exists a semi-norm pa E Posuch that &I, po)is a, surjectim. P r o o f . According to 29.1.5 we obtain a semi-norm po c @ ' , by setting p o ( ~ o:= ) inf ( p ( x ) Qx : = xo} for zoE Uo. If Zo(po)E ~ o ( p o then ) , there are zonE 77,

with

W

W

1

1

~ O ~ P=OZ) X O ~ ~ P O ) and Z ll~on(p0)Il5 (1 + E ) llStP~)llChoose x,

c

U such that zo,,= Qz, and p(x,,) 5 (1 + E ) po(xo,J.Then we have m

ZO(P0)

Moreover,

= @P, Po) Z(P) with

Z(P):=2 z,,(p)E n P )* 1

W

l l ~ ( ~ ,I ) l lZ lIxn(P)ll 5 (1 1

+

m

E)

.Z IIxon(~o)Il5 (1 + 8)' l l ~ o @ ~ ) l l * 1

Therefore @ ( p ,po) is a surjection. P r o p o sit i o n. For every SurjeCCive operutor ideal gothe right-superior extensiota is surjective, as well. Proof. Using the notation of 29.1.5 we consider a n operator So c E(Uo, V ) with S := SoQ E Ut;""(U,V ) . Fix a semi-norm q 6 0. Applying 29.5.5 we can choose a v(q)), and by the preceding lemma there semi-norm p E '$ with g ( p , q) E ?&(c(p), exists a semi-norm po E Posuch that Q(p, pa)is a surjection. This leads to the diagram :

%rp

400

Part 5. Applications

Consequently, for every q t Q there is po E This proves that So E Uyp(U,,V ) .

qosuch that 8,(p0, q ) :ao(i7b0), r(q)).

29.5.9. Proposit.ion. The right-8uperior extembn Carte&n.

a?”of

every operator ideal Uoi s

P r o o f . Let S, E Urp(UJ,V,) for i 5 I . By 29.1.10 every semi-norin y of has the form

d ~:=):.’

qi(yi) for

Y

x ( Jrt, I )

(~i),

1

where qi E Ci and i E %(I). Let us choose semi-norms p i E %) E ? l o ( m P J >%)). Then

pi such that

we have

A%%,

p ( 5 ) :=

pi(5i)

for x

= (xi)

1

defines a semi-norm p on li := X (Ui, I ) . Observe that o ( p )and v ( q )are isomorphic to the finite Cartesian products x ( o J p t ) ,i) and )( ( v i ( q , ) i), , respectively. Let S := x (Si,I).,Then it turns out that S(p, p) and )( ( 8 i ( p i ,qi), i ) coincide. Therefore 8 ( p , q ) € N o ( o ( p ) q(y)). , This completes the proof.

29.6.

Grothendieck Space Ideals

29.6.1. Of special interest are those ideals A of locally convex spaces which can be generated by the right-superior extension at;”” of some operator ideal 910 defined on the class of Banach spaces. Then A := Space (9frp) is called a Grothendieck q c e ideal and we write A = Groth (ao). 49.6.2. First a consequence of 29.5.5 is st,ated. Theorem. A locally convex space U belongs to Groth (No)if a d only if for every semi-norm po E ‘$3 there exists a semi-norm p E 9 such that l ~ ( ppo) , E Uo.

29.6.5. We now consider important examples of locally convex spaces. Let (znK) be an infinite matrix such that

0

< nI1;5 z2k5 -.-

Then all sequences 5

= &)

for II. = 1,2,

...

with

.-.

form a linear space U,and p , 5 p 2 5 is a systeni of norms. The coniplete nietrisable locally convex space obtained in this way is denoted by Z2(znf). Furthermore, the operator I,(&) := (n&) is an isomorphism between

and the Hilbert space 12.

29. Operator Ideals on Locally Convex Spaces

Finally, let n 5 rn and I m n ( ( k )

:= (;cnkn;&).Then

401

we have the diagram :

T

12

I,,

)12

29.6.4. Let us recall that I, consists of all sequences (a,) E I such that the operator S E 2(Z2,Z2) defined by S&) := (antn)is contained in a,. Theorem. The locally convex spwe 12(7dnk) belongs to Groth ( N o ) i f and only i f for Every natural nwnber n there exists u natural number n1 such that n 5 m and (~nln;;,

nn2n&

* *

.)

E IN,-

Proof. The assertion follows immediately from the preceding results. 39.6.6. Proposition. For every injective operator ideal Uothe Qrothendieck space ideal Groth (ao) i s injective, as well. Proof, Use 29.3.3 and 29.5.7. 38.6.6. Proposition. For every surjective operaior ideal Uothe Grothendieck space &deal Groth (Uo)is surjective, as well. Proof. Use 29.3.7 and 29.5.8. 29.6.7. Proposition. Every Grothendieck space ideal i s Carteskn. Proof. Use 29.4.3 and 29.5.9.

29.6.8. Let 8, and bobe operator ideals defined on the class of Banach spaces. Then we have the following lemmas which are elementary, but very important. Lemma 1. Groth (a,) = Groth (8;)for TZ = 1 , 2 , ... Lemma 2. If 9l0 2 So,then Groth (ao) Groth (so). Lemma 3. Groth (ao) n Groth (b,) = Groth (ao n bo) -- Groth (U, o B,). 39.6.9. We now consider a trivial example. Proposition. Groth (8)is the ideal of all locally convex spaces equipped with the weak topology. 29.6.10. For the definition of Schwartz spaces we refer to [GAR, p. 1191. Proposition. Groth (R)i s the ideal of Schwartz spaces. 29.6.11. Next a counterexainple is given. Proposition. The ideal of normable locally coneen spaces is not Grothendieckiun. Proof. This space ideal fails to be Cartesian. 26 Pietach, Operator

402

Part 5. Applications

Remark. The ideal of Monte1 spaces is Cartesian, but not Grothendieckian (letter from A. P E E C Z Y ~ K I ) . 29.6.12. Let A be a Grothendieck space ideal. Then, by definition, there exists an However, no method seems to be operator ideal a0such that A = Groth (ao). known how to construct Nofrom A. More generally, we have the open

Problem. Charact,erize Grothendieck space ideals by means of certain “inner” properties !

29.7.

Nuclear Locally Convex Spaces

29.7.1. A locally convex space belonging to N :=Groth (!It)is called nuclear. 29.7.2. A Iarge part of the theory of nuclear locally convex spaces is concentrated in the next theorem which summarizes many results about products of operator ideals on the class of Banach spaces. Theorem. The Grothend,ieckspace ideal N can be generated b y the foE1om.q operator .iaeals:

ap,aipnj,qlyf w o < p < O 0 , forO
@, %t.p.n) ‘PI

%, 3,

+ l/r > l/p i- l/q,

Proof. The theoretical tools are the lemmas formulated in 29.6.8. (1) The relation N = Groth

(ap), where 0 < p <

00,

follows from

(18.6.4) !It2 E a,, (14.2.3)

(a,)”E aplnfor 0 < p < 00,

(18.6.3) U1

3.

The inclusions (trivial) 3, (14.2.3)

Wij

(nl,”j)*En$@

for O < p < m , for 0

< p < 00,

(14.2.13) a;”] E U p

imply N = Groth (@:j) N = Groth (87).

whenever 0 < p

( 2 ) The formula N = Groth

E ts, (14.3.3) ( g p pE a, (14-3.11)

< 00.

(ap)with 0 < p <

9II

(14.3.10) tlIzwp.

In the same way we can check that

for 0 < p

< 00,

00

is induced by

29. Operator Ideals on Locally Convex Spaces

403

(3)Because of

we have N

= Groth

+

+

(18-6.1) a ( r , p . q ) E are (18.6-2) a ( 7 , p . g ) 2 ar (18.6-3) 91, E S ( r . 1 ~ ) (18-1-5) 9 2 ~ r o , ~C=, ~%(r.p.q) ~

for 1 l j r > lip ljq 2 1, for 1lP llq 5 1 , for O < r s l , for 1/ro = l / r I/P’ 1/d,

+

+

+

(a,,,,,,)) whenever 0 < r <

00

and

1

+ l/r > 11p + l/q.

(4) Furthermore, it follows from

(22.1.8) 93(2,2,2) G br for 0 < r < 00, (20.2.4) (?&)” G V2 for 0 < r 5 2n, (17.6.4) (‘p2)2 G 912 that N = Groth (Q,) whenever 0 < r (5) The relation N

= Groth

< 00.

(ar) = Groth ( Z r )with 1 5 r <

00

can be checked by

(18.2.4) 2 a, for 1 5 r < 0 0 , (19.21) S, & 3, for 1 5 r < 00, (20.2.4) (3,)” !& for 1 5 r I :2n, R e m a r k . The list of operator ideals given in the above theorem is of course incomplete.

29.7.3. P r o b l e m . Which procedures new possess the property that N = Groth (a) implies N = Groth

(anew)?

R e m a r k . The answer is affirmative for the procedure min.

29.7.4. Till now it seems unknown for which exponents we have N = Groth (!$ir,p,q,). I n particular, the following problems arise. C o n j e c t u r e 1. N = Grot,h (!&r,2)) for 2

< r < 00.

R e m a r k . It was proved by B. MAUREYIA.PELCZY~~SKI [l] that Groth (!&r,P)) is contained in the ideal of Schwartz spaces. C o n j e c t u r e 2. N = Groth (!&r,2,2))

for 1 < r

< 2.

R e m a r k . We see from ! l l E ?&I,2,2) 5 p2that N = Groth (!J3(l,2,2)).On the other hand, there are infinite dimensional Banach spaces belonging to Groth (Ipc2,2,2)).

29.7.6. T h e o r e m . The Grothendieclc space ideal N is injective and surjective. P r o o f . According to 29.6.5 and29.6.6 theassertionfollows from N = G r ~ t h ( ! J l ( ~ ,;~ , ~ ) ) cf. 18.1.8.

29.7.6. Finally, we state a famous Theorem. If 1 s p

< 00

and 1 < q

5 00, then

N = Groth (CP)n Groth (acp)and 2G*

N = Groth

(a1)n Groth (I&).

404

Part 5. Applications

Proof. Since 5 2, for 1 5 r 5 00, we have N & Groth (ar).On the other hand, it follows from 22.4.2 and 29.6.8 that Groth (2J n Groth

(a,) = Groth

o2 , )

s Groth (3,)

for s > niax (2, p ) . Considering the dual operator ideals we obtain Groth (g1)n Gr0t.h (2,) = Groth (2, o 2,)

for

8

29.8.

Groth (3;"")

> max ( 2 , q'), Strictly Nuclear Locally Convex Spaces

29.8.1. A locally convex space belonging to No:= Groth (I,is )called strictly w l e a r 29.8.2. I n contrast to 29.7.2, the Grothendieck space ideal Nocan only be generated by the operator ideal so. Theorem. Let Uo be a n operator ideal

012.

No= Groth (a,) impllies !Ito= 2l0.

the class of Banach spaces. Then

Proof. In the following we use locally convex spaces as constructed in 29.6.3. Furthermore, let a, := I,, be the sequence ideal corresponding to lo. (1) Consider (uk)E ,l with 0 < a, 5 1, and generate the locally convex space U := Z,(oi"). Since (uy-l!m

o;/?&-l/?n 3

,...) E

we have U E N, = Groth

for n

<m,

(a,,).Hence, there exists a nat.ura1 number m such that

It follows from 1 >= cri-l/m2 uk > 0 that (bk) i a,,. This proves that 1, E a,. (2) Consider (ck)E a. with 0 < uk 5 1, and generate the locally convex space V := &(a;"). Since (by-",

uy-",

...) E a.

for m

<m,

we have V E Groth (go)= No.Hence, there exists a natural number m such that (up, u p ,

...) E lo -

So (uk) E b. This proves that a, &. (3) Finally, we see from a, = lo that a. and socoincide on the class of Hilbert spaces. So we get go= go,since is the unique extension of its Hilbert space part ;

xo

of. 18.7.9. 29.8.3. Theorem. The Grothendieck spuce ideal Noi s injective and surjective. Proof. According to 29.6.5 and 29.6.6 the assertion follows from 18.7.7.

29. Operator Ideals on Locally Convex Spaces

29.9.

44%

Notes

The theory of nuclear locally convex spaccs is mainly due to A. GROTHENDIECK; of. [GRO]. A simplified treatement was given in [PIE]. Other classes of locally convex spaces with special properties have been investigated by several authors; cf. [KoT]. So-called varieties of locally convex spaces were studied by J. DIESTEL/&A. MORRIS/S.A. SAXON[l]. The relationship between ideals of operators on the class of Banach spaces and certain ideals of locally convex [15]. Tho extension theory of operator ideals is spaces has been pointed out by A. PIETSCH treated in this monograph for the first time. Recommendations for further reading: [BOUtI, [GAR& [JAR], [SAFJ.

H. APIOLA[l], [2], C. BESSAGA/A. PELCZYBSKI [2], B. S. BBUDOVSKIJ[l], M. DE WILDE[l], E. DUBINSKY/M. S. RAMANUJAN [ 2 ] , A. GROTHENDIECK [I], H. J. JUNEK [2], [3], G. KOTHE [3], A. MARTINEAU[I], H. MILLINGTON[l], B. S. MITJAGIN [I], A. PELCZY~SKI [2], A. PIETSCH [3], M.8. RAMANUJAN [3], I). RANDTKE [l], B. ROSENBEBGEB [2], E. SCHOUK [I], T. TERZIOGLV

[ll.

Epilogue

Finally, we list some subjects which could not be covered in this monograph. First of all let us mention the theory of nornied tensor products. As stated before concepts like those of a cross-norm are the historical background of the theory of operator ideals. Since and no modern presentation is available, we refer to the original work of A. GROTHENDIECE R. SCHATTEN. The most striking paper is “RBsumit de la thborie mbtrique des produits tenSHIGA. soriels topologiques”. A short, but excellent survey was given by I. AMEMIYA/K. Certainly the attentive reader has observed that we made consistent use of ideas taken from the theory of categories. However, we were not able to deal with functors on the category of Banach spaces. This concept is due to B. S. MITJAQI~-/A. S. SHVARTS. A full presentation may be found in lecture notes of J. CIGLER and P. W. MICHOR.

As observed by I. C. GOHBERG/M. G. KREJNinterpolation methods are of great significance in the theory of normed operator ideals on the class of Hilbert spaces. Moreover, i t is also possible to construct new normed operator ideals by means of interpolation; cf. A. PIETSCHl H. TRIEBEL and J. PEETRE/G.SPARR. There is an extensive literature dealing with operators in Banach lattices. We only refer to the concepts of a cone absolutely summing and a majorizing operator as defined in H. H. SCEAEFER’S second monograph. As shown before special operator ideals play an important role in spectral theory. I n particular, we have presented the famous Riesz theory. But nothing could be said about the completeness of principle elements and on the invariant subspace problem; ef. A. S. MARKUS, J. R. RINGROSE, and the encyclopaedic monograph of Pi. DIXFORD/J.T. SCHWARTZ. Furthermore, let us mention that the theory of decomposing operators yields a new approach to obtain general eigenelement expansions; cf. K. MAURIN. Analogously to the last chapter it is possible to develop a theory of operator ideals on the class of bornological spaces. I n this context the concept of nuclearity was investigated in the lecture notes Of H. HOOBE-XLEND. We now give a small selection of references. f&asi-normed tensor products:

[GRO], [SAT,], [SEM,, exp. 9, 101. I. AMEMIYA/~<. SHIGA[l], S. CHEVET [l], A. GROTHEKDIFCK [4], J. R. HOLUB[l], V.-K. KHOAN [I], P. SAPHAR [2], [3], (41. Categories of Banach spaces: [CIG], [MIH].

J. CIGLER[I],

v. L. LEVIN[I], P. W. XICHOR[I], B. s. &hJAGIN/A.S. SHVARTS[I].

Interpolation theory: [BER], [BUT], [TRI].

s.

n. F R E I T ~[I],G CH. GAPAILLARD/PHAM THE L A 1 [I], pu’. v. JIIROSEIX [I], B. &fITJAQIX [31, C. MERUCCI/PEABXTHE LAI [l], R. OLOFF [2], PHAM THE LAI [l], J. PEETBE 111, [2],

Epilogue

407

J. PEETRE/G.SPARR[l]. A. PERSSOX [l], A. F’IETSCH[S], [12], A. PIETSCH/H. TRIEBEL[I], Br. RIESZ111.

Operators on Banach lattices: iSAF,].

D. I. CARTWRIQHT/H. P. LOTZ[l], D.J. H. GARLING[7], V. L. LEVIN[l], [21, [l], U. SCHLOTTERBECK 111, H. U. SCHWARZ [3].

K.J. NIELSEN

Spectral theory:

[DUN], [GOH], [MAU,], [RIN]. 9. S. MAXKUS[2], H. WITTMER [l].

Homological spaces: [HOG].

H. BUCHWALTER [l], 5. P. LIGAUD[l].

While this monograph was in print several new results were obtained which could not be included. Here we list some of them.

1. Clearly we have I Io b E I In B for arbitrary operator ideals. S. HEIXRICII: [4] found necessary conditions such that equality holds. Theorem. Let I Iand b be closed operator ideals. If ‘Zl i s surjective and b i s injective, then SIob=IInb.

Eemark. For example, we have D ! o ?B= b n 8. 0. The preceding result is an immediate consequence of the following interpolation theorems also due to S. HEINBICH [4].

Theorem 1. Let 91 be am injective closed operator ideal. Suppose that E a’s a n intermediute P ) , then S E %(E,, P ) q a c e of {Eo,El) possessing I<-type 3 with 0 < 3 < 1. If S E *(Ex, implies S E U(E,F ) . Theorem 2. Let I Ibe a surjective closed operator ideal. Suppose that F i s an intermediate space of {Fo,FJ possessing J-type 0 with 0 < 3 < 1. If S E %(E,F A ) .then S E %(E, F,) implies S E %(E,F). Remark. The above statements are generalizations of an interpolation theorem known for the ideal of compact operators; cf. [BER, p. 561 and A. PERSSON [I].

3. The famous Davis-Figiel-Johnson-Pebzyriskifactorization theorem was extended to other operator ideals by B.BEAUZANY [2] and S. HEIXRICH [4]. I n order to state HEINRICH’S version an operator ideal I Iis called 1,-stable if every operator S E E(E,(E,J, l P ( F n ) )such that Q,Sb, E %(Em,F,) for m,n = 1,2, ... belongs t o ZI. Theorem. Let I Ibe a n operator ideal which is injectire, surjective, and p-stable for sonap p 5 p 5 cs. Then SI = Op[Space(%)].

7dh1

Remark. It turns out that % is p-stable for 1 < p and B-1 o a are costable.

< m.

Moreover, the operator ideals

408

Epilogue

4. Extending a result of H. KONIG[a] the author [26] was able to prove the following miiltiplication formula which is an improvement of 14.2.3and 15.5.9.

Theorem.

If 0 < p , p -<

cy)

and l!r = 1/27

+ l/q, then %, o u, = a,.

5. An analogous result is true for the entropy ideals; cf. 14.3.3. Theorem. If 0 < p , q

/*

cv and

I j r == l i p

4-119, thric t, o t, = t,.

6. Using ultraproduct techniques the conjecture 19.1.7 was verified by S. HEINBICH/A. PIETSCH

PI. 7. The following characterization is due to A. PIETSCH [27]. Theorem. An operator S E B(E,F ) is inessential if and ordy if all opeintors I , L E (!l F , E ) haxe a finite dimensional null space.

8. Let 0 < p

< co. An inessential operator S E B(E,F) is said to be of

2 jl,(L~S’)/-p<

cy)

€or all L

E

- 1,s with

Weyl tgpe l,, if

2(F,E ) .

dl

The class of these operator IS denoted by ?Bp. Clearly ?lB, satisfies (01,)and (O12).However. as observed by A. PIETSCB [23],the property (01,)fails. Therefore 8,is not an operator ideal.

9. The following statement x a s also proved by A. PIETSCH [%I. Theorem. Let U be an operator ideal with % 5 2Bp. Then WUZn & % for 11 2 p. Remark. Using recent results of H. KONO we obtam from the above t
Bibliography

The bibliography is divided into the following parts: 1. textbooks, monographs, and lecture notes, 2. classical papers, 3. papers, 4. references added in proof.

W e refer to textbooks, monographs, and lecture notcs by symbols built from three characteristic letters of the author’s name. If there are editions in different languages, then we have used that which is marked by *. Papers listed in the other parts of the bibliography are referrcd by the nanie(s) of their author(s) and a finite ordinal number indicating some chronological order.

1. Textbooks, Monographs, and Lecture Notes AEIEZER,N. I./GLAZMAX, I. 31. [AHI] Theory of linear operators in Hilbert spaces, Moscow 1950, Berlin 1964*, New York 1961163. BADRIHIAN, A. [BAD] S6minaire sup les fonctions albatoires lineaires et les mesures cylindriques, Lecture Notes in Xath. 139, Berlin-Heidelberg-Kew York 1970. BADBXKIAX, A./CHEVET, s. [BAH] Mesures cylindriques, Espaces de Wiener et fonctions albatoires Gaussieimes, Lecture Notes in Nath. 379, Berlin-Heidelberg-New York 1974. BAXACH, S. [BAN] Th6orie des o p h t i o n s lin&aires,Warsaw 1932. BAUER,H. [BAW Wahrbcheinlichkeitstheorieund Grundziige der NaBtheorie, Berlin-New York 1974. BERGH,J./LOFSTROY,J. [BER] Interpolation spaces, Berlin-Heidelberg-Xe’ew York 1976. BOURBAEI,N. [BOU,] Topologie g&nCrale,chap. IX, Paris 1948*, Moscow 1976. [BOUJ Espaces vectoriels topologiques, chap. 1-11, Paris 1953*, Moscow 1959; chap. I11 to V, Paris 1955*, Xoscom 1959. [BOU,] Integration, chap. I-IV, Paris 1952*, Moscow 1967; chap. V, Paris 1956*, Moscow 1967; chap. VI, Paris 1959*, Moscow 1970; chap. IX, Paris 1969*, Moscow 1977. BUTZEB,P. L./BEREXS, H. [BUT] Semi-groups of operators and approximation, Berlin-Heidelberg-Kew York 1Oti’i. CARADUS,S. R./PFAFFENBEROER, W.E./YooD, B. [CAR] Calkin algebras and algebras of operators in Banach spaces, New York 1974. CIGLER, J. [CIG] Funktoren auf Kategorien von Banachraumen, Lecture Notes, Univ. Vienna 19’74. DAY,M. M. [DAY] Normed linear spaces (third editim), Berlin-Heidelberg-New York 1973. DIESTEL,J. [DJE] Geometry of Banach spaces, Lecture Notes in Math. 485, Berlin-Heidelberg-New York 1975.

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DIESTEL,J./UIIL, J. J. (Junior) [DIU] The theory of vector measures, Providencc 1977. DINOULEANU, N. [ D I N Vector Measures, Berlin 1966. DUNFORD, N./SCEWARTZ, J. T. [DUN] Linear operators, vol. I, New York-London 1958*, Moscow 1962; vol. 11, New York-London 1963*, Moscow 1966.

GAR-, H.G./DE WILDE,M./SCEMETS, J. [GAR] Analyse fonctionelle I, Basel-Stuttgart 1968. GELFAND,I. MJVILENKIN,N. J. [GEL] Generalized functions IV, 3foscow 1961, Berlin 1964*, New York 1964. GOHBERG,I. C./KREJN,M. G. [GOH] Introduction to the theory of linear nonselfadjoint operators in Rilbert space, Moscow 1965, Providence 1969*, Paris 1971. GOLDBERG, s. [GOL] Unbounded linear operators, New Pork 1966. GROTHENDIECK, A. [GIRO] Produits tensoriels topologiques et, espaces nncl&aires,Mem. Amer. Math. SOC.16, Providence 1955.

HALMOS, P. [HAL] Measure theory, Princeton-New Pork-Toronto -London 1960. HARDY, G. H./LITTLEWOOD. J. E./WLYA,G. [HAR] Inequalities, Cambridge 1952. HEUSER,H. [HEU] Funktionalanalysis, Stuttgart 1975. HELE, E./PHILLIPS,R. S. [HIL] Functional analysis and semi-groups, Providence 1957, Moscow 1962*. HOQBE-NLEND, H. [HOG] Thhorie des bornologies et applications, Lecture Notes in Math. 213, Berlin-Heidelberg-New York 1971.

IONESCU TULCE.4, A./IONESCU TULCEA,c. [ION] Topics in the theory of lifting, Berlin-Heidelberg-New

York 1969.

Jmcaow, H. [JAR] Ruclear locally convex spaces, Lecture Notes, Univ. Maryland 1976.

BACZMARZ, S./STEIKHAUS, H. [KAC] Theorie der Orthogonalreihen, Warszawa -Lwou- 1935*, Moscow 1958. l i a ~ r ~J.s P. ~, [KAH] Some random series of fonctions, Lexington 1968*, Moscow 1973. KATO,T. [KAT] Perturbation theory for linear operators, Berlin-Heidelberg-New York 1966*, Moscow 1972. KAWATA, T. [KAW] Fourier analysiv in probability theory, New York-San Francisco-London 1972. ROLXO~OROV, A. N. [KOL] Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin 1933*, Moscow 1974. KOTHE,G. [KC)T] Topological vector spaces I , Berlin-Heidelberg-New York 1960* (German), Berlin-Heidelberg-Xew York 1969. KRASNOSELSEIJ, M. A./RUTICKI, I. B. [KRA] Convex functions and Orlica spaces, Moscow 1958*, Groningen 1961.

1. Textbooks, Monographs, and Lecture Notes

411

ICuo, HUI-HSIUKG [KUO] Gaussian measures in Banach spaces, Lecture Notes in Math. 463, Berlin-Heidelberg-New York 1975.

LACEY,H. E. [LAC] The isometric theory of classical Banach spaces, Berlin- Heidelberg-New York 1974. LEWIN,B. J. [LEW] Nullstellenverteilung ganzer Funktionen, Moscow 1966, Berlin 1962*. LINDENSTRAUSS, J./TZAFRIRI, L. [LIN] Classical Banach spaces, Lecture Notes in Math. 338, Berlin-Heidelberg-New York 1973. [ L I q Classical Banach spaces, vol. I : Sequence spaces, Berlin-Heidelberg-Xew York 1977, vol. 11: Function spaces, Berlin-Heidelberg-New York 1979. LOBENTZ, G. G. [LOR] Approximation of functions, Xew York 1966. LOTZ,H. P. [LOTI Grothendieck ideals of operators in Banach spaces, Lecture Notes, Univ. Illinois, Urbana 1973. NAURIN,K. [MAU,] Methods of Hilbert spaces, Warsaw 1959, Moscow 1965*. [MAU,] General eigenfuiiction expansions and unitary representation of topological groups, Warsaw 1968. XICHOR,P. W. [MIH] Functors and categories of Banach spaces, Lecture Notes in Math. 651, BerlinHeidelberg-New York 1978. PIETSCH, A. [PIE] Nuclear locally convex spaces, Berlin 1965, Moscow 1967, Bcrlin-Heidelberg-New York 1972*. PR~SSDORF, S. [PR&J Einige Klassen singuliirer Gleicliungen, Berlin 1974.

RIESZ,F./SZ.-NAGY, B. [RIE] Lepons d’analyse fonctionelle, Budapest 1953, Berlin 1968*. J. R. RINQROSE, ERIK] Compact non-self-adjoint operators, London 1971. PBZEWORSKA-ROLEWICZ, D./ROLEWICZ, S. [ROL] Equations in linear spaces, Warsaw 1968. SCHAEFER, H. H. [SAF,] Topological vector spaces, New York-London 1966*, Moscow 1971. [SBF,] Banach lattices and positive operators, Berlin-Heidelberg-New York 1974. SCXSTTEN,R. [SAT,] A theory of cross-spaces, Princeton 1950. [SAT,] Norm ideals of completely continuous operators, Berlin-Heidelberg-New York 1960. SCHECHTER, M. [SEC] Principles of functional analysis, New York-London 1971. SEXIdDENI, z. [SED] Banach spaces of continuous functions I, Warsaw 1971. S6minaire LAURENT SCHWARTZ [SEM,] Applications radonifiantes, Paris 1969/70.

412

Bibliogritphy

SBminaire GOVLAOUIC-SCEWARTZ [SEMd Equations aux dBrivBes partielles et analyses fonctionelle, Paris 1970/71. [SEM,] Equations aux dbrivhs partielles et analyses fonctionelle, Paris 1971/72. SBminaire MAUREY-SCHWARTZ [SEMJ Espaces LP et applications radonifiantes, Paris 1972/73. [SEM,] Espaces LP, applications radonifiantes et ghmbtrie des espaces de Banach, Paris 1973174. [SEK] Espaces Lp,applications radonifiantes et ghm6trie des espaces de Banach, Paris 1974175. [SEM,] Espaces Lp, applications radonifiantes et g6omBtrie des espaces de Banach, Paris 1975/76. SBminaire [SEM,] La ghm6trie des espaces de Banach, Paris 1977178. S m o v , G. E./TIN, FAN-DYK [SHI] lctegral, measure, and derivative on linear spaces, Moscow 1967. SIRGER,I. [SIN] Bases in Banach spaces I, Berlin-Heidelberg-New York 1970. SROROHOU, A. V. [SKO] Integration in Hilbert spaces, Berlin-Heidelberg-New York 1974". Moscow 1975. SMITHIES, F. [SMI] Integral equations, Cambridge (New Pork) 1958. SCHRARTZ, L. [SWA] Radon measures on arbitrary topological spaces and cylindrical measures, Oxford (Bombay) 1973.

T LEILOONSKI, G. I. [TAR] Seminar on functional operators and equations, Lecture Xotes in Xath. 33, BerlinHeidelberg-New York 1967. TAYLOR, A. E. [TAY] Introduction to functional analysis, ?Jew York 1958. TITCHXARSH, E. c. [TIT] The theory of functions, Oxford 1950. TRIEBEL, H. [TBI] Interpolation theory, function spaces, differential operators, Berlin 1978. VAHANIJA,

N.x.

F A R ] Probabilistics distributions in linear spaces (Russian), Tbilissi 1971.

X u , DAO-XIXQ [XIA] Measure and integration theory on infinite-dimensional spaces, Sew York 1972. YEH, J. [YEH] Stochastic processes and the Wiener integral, Ken; York 1973. YOSIDA,K. [YOS] Functional analysis, Berlin -Gottingen-Heidelberg 1965*, Moscow 1967.

ZAANEN,A. c. [ZAN] Linear Analysis (third reprint), Groningen 1960.

ZYQMUND, A. [ZYG] Trigonoiiictric series, Cambridge 1959*, Moscow 1965.

2. Classical Papers

413

2. Classical Papers This part of the bibliography is dedicated to those mathematicians who laid the foundations for the development of the theory of operator ideals.

BOCHNER, S. [i] Integration von Funktionen, deren Werte die Elemente eines Vektorraumos sind, Fund. Math. 20 (1933), 262-276. BOENENBLUST, F. [l] On axiomatic characterization of Lp spaces, Duke Math. J. 6 (1940), 627-640. CALKIN,J. W. [i] Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math. 42 (1941), 839-873. CABLEMIIN,T. [l] Zur Theorie der linearen Integralgleichungen, Nath. Z. 9 (1921), 196-217. CEANC,S. H. [i] On the distribution of the characteristic values and singular values of linear integral equations, Trans. Amer. Math. SOC.67 (1949), 351 -367.

DUHBORD, N./PETTIS, B. J. [I] Linear operations on summable functions, Trans. Amer. Math. SOC. 47 (1940),323-392. DVOBIETZKY, A./ROGERS,C. A. [i] Absolute and unconditional convergence in normed linear spaces, Pror. Nat. Acad. Sci. USA 86 (1930), 192-197. FAN,Ky [i] One theorem of Weyl concerning eigenvalues of linear transformations, Proc. Xat. Aoad. Sci. USA 35 (1949), 652-655; 36 (1950), 31-36. FREDHOLY,1. [l] Sur une classe d’bquations fonctionelles, Scta Math. 27 (1903), 365-390. GAXTMACHER, V. R. [i] Uber schwach totalstetige Operatoren, Mat. Sb. 7 (1940), 301-308. GELFAND, I. M. [l] Abstrakte Funktionen und lineare Operatoren, Mat. Sb. 4 (1938), 235-286. GOURSAT, E. [i] Sur un cas Blbmentaire de 1’Bquntion de Fredhoim, Bull. SOC.Math. France 36 (1907). 163-173.

J. E. HARDY,G. H./LITTLEWOOD, [I] Bilinear forms bounded in spaces ( p , q), Quart. J. Math. (Oxford) 6 (1934), 241-254. HELLINQEB, E./TOEPLITZ, 0. [I] Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten, Encyklopiidie der Math. Wiss. 11, 3, 2 (1927), 1335-1601. HELLY,E. [I] nber Systeme linearer Gleichungen n i t unendlich vielen Unbekannten, Monatsh. fur Math. u. Phys. 31 (1921), 60-91.

HILBERT, D. [I] Grundziige einer allgemeinen Theorie der linearen Integralgleichungen IV, Xachr. Akad. Wiss. Gottingen, Math.-Phys. K1. (1905), 157-227.

HORW, A. [l] On the singular values of a product of completely continuous operators, Proc. Nat. Acacl. Sci. USA 36 (1960), 374-375.

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PHILLIPS, R. S. [l] On linear transformations, Trans. Amer. Math. SOC.48 (1940), 516-541. PITT,H. R. [l] A note on bilinear forms, J. London Nath. SOC.11 (1936), 174-180. PONTRJACIN, L. S./SCHNIRELMANN, L. G. [l] Sur une propri6th m6trique de la dimension, Ann. of Math. 33 (1932), 156-162. RIESZ. F. [l] uber lineare Funktionalgleichungen, Acta Math. 41 (1918), 71 -98. RIESZ,M. [l] Sur le maximum des formes bilineaires et sur les fonctionelles linbaires, Acta Math. 49 (1926), 455-497. SCHATTEN, R. [l] The cross-space of linear transformations, Ann. of Math. 47 (1946), 73-84.

3.Papers

415

SCHAUDER, J. [l] tfber lineare vollstetige Funktionaloperationen, Studia Math. 2 (1930), 183-196. SCHMIDT, E. 111 Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann. 63 (1907), 433-476; 64 (1907), 161-174. [2] uber die Auflosung linearer Gleichungen mit unendlich vielen Unbekannten, Rend. del Circolo Mat. Palermo 25 (1908), 53-77. SCHUR, I. [l] vber die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen, Math. Bnn. 66 (1909), 488-510. [2] Uber lineare Transformationen in der Theorie der unendlichen Reihen, J. Reine Angew. Math. 151 (1921), 79-111. WEYL,H. [l] Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann. 71 (1912), 441-479. [2] Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Net. Acad. Sci. USA 35 (1949), 408-411. WIENER,N. [l] Generalized harmonic analysis, Acta Math. 55 (1930), 117-258. YOSIDA,K. [l] Mean ergodic theorems in Banach spaces, Proc. Imp. Acad. Tokyo 14 (1938), 292-294.

3. Papers The new era in the theory of operator ideals starts, by definition, with the fundamental work of A. GROTHENDIEOH. Therefore, this part of the bibliography contains those papers which appeared after 1950. AMEMIYA, I./SHIGA, I(. [l] On tensor products of Banach spaces, Kodai Math. Sem. Rep. 9 (1957), 161-178; MR 21 # 3766. APIOLA, H. [l] Duality between spaces of p-summable sequences, (p, 9)-summing operators, and characterization of nuclearity, Math. Ann. 219 (1976), 53-64; MR 52 # 11530. [2] Some permanence properties of locally convex spaces defined by norm space ideals of operators, Studia Math. 55 (1976), 265-278; MR 54 # 3345. ARAZY,J./~NDPNSTRAUSS, J. [l] Some linear topological properties of the space C pof operators on Hilbert space, Compositio Math. 30 (1975), 81-111; MR 51 # 8876. P. ASSOUAD, [l] Applications sommantes et radonifiantes, Ann. Inst. Fourier (Grenoble) 22 (1972), 81 -93; MR 49 # 5906. ATKINSOB, F. V. [l] The normal solvability of linear equations in normed spaces (Russian), Mat. Sb. 28 (1951), 3-14; MR 13, p. 46. BADRIKIAN, A. [I] Proi6gomi?mes au calcui de probabilit6s dans les Banach, Lecture Notes in Math. 539, pp. 1-166, Berlin-Heidelberg-New York 1976. BAUHARDT, W. [I] Hilbert-Zahlen von Operatoren in Banachraumen, Math. Nachr. 79 (1977), 181 187.

-

416

Bibliography

BAUMBACH, G./LINDE,W. [I] Asymptotic behavior of p-absolutely summing norms of identity operators, Math. Nachr. 78 (1977), 193-196.

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Index of Axioms

(01,)

operator ideals

(XI,) sequence ideals (FI,) (SI,) (QOI,) (QNI,) (QFI,)

family ideals space ideals quasi-norms of operators quasi-norms of sequences quasi-norms of families (OS,) s-numbers of operators (NS,) s-numbers of sequences

1.1.1, 29.2.1 13.1.1 16.1.1 2.1.1, 29.2.2 6.1.1 13.2.1 16.1.3 11.1.1 13.7.2

Index of Operator Ideals

Operator ideals and their quasi-norms are denoted by capital bold Gothic and capital bold Roman letters, respectively. The corresponding space ideal is marked by capital bold sanserif letters. Example : operator ideal a, quasi-norm A, space ideal A.

Symbol

Ideal of

...

Gp)-operators operators of ( 8 , p)-cotype dualisable operators p-dominated operators ( p , q)-dominated operators Gp-operators finite operators * discretely p-factorable operators 29*

Point 14.2.4 21.4.1 4.4.10 17.4.7 17.4.1 14.3.1

B.hl 19.3.11

440

Symbol

Index of Operator Ideals

Ideal of

...

approximable operators @(*)-operators Hilbert operators N,-Hilbert operators Gr)-operators integral operators r-integral operators ( r ,p, p)-intogral operators o-integral operators compact operators p-compact operators ( p , q)-compact operators arbitrary operators p-factorable operators ( p ,p)-factorable operators ( 8 , p)-mixing operators nuclear operators r-nuclear operators ( r , p , q)-nuclear operators o-nuclear operators strictly nuclear operators absolutely summing operators absolutely r-srxniming operators absolutely ( r , p)-suu”irningoperators absolutely ( r ,p , p)-summing operators absolutely z-summing operat>ors decomposing operators p-decomposing operators inessential operators p-radonifying operators strictly singular operators Gp-operatorson Hilbert spaces GF)-operators Gf)-operators strictly cosingnlar operator operators of (8, p)-type unconditionally suinming operators operators of ultra (8,p)-type conipletely continuous operators weakly compact operators separable operators Radon-Nikodym operators

Point

1.3.1 14.1.11 6.6.1 5.4.1 14.2.15 6.4.1 19.2.1 19.1.1 23.3.1 1.4.1 18.3.1 18.4.1 A.3.2 19.3.1 19.4.1 20.1.1 8.3.1 18.2.1 18.1.1 23.2.1 18.7.1 6.5.1 17.3.1 17.2.1 17.1.1 23.1.1 24.42 24.5.3 4.3.5

25.4.2 1.9.2 15.5.1 14.2.1 14.1.1 1.10.2 21.2.1 1.7.1 21.3.1 1.6.1 1.5.1 1.8.1 24.2.6

Index of Family Ideals

Family ideals and bheir qtiasi-norms are denoted by small bold Gothic and small bold Roman letters, respectively. Example : family ideal a, quasi-norm a.

...

Symbol

Ideal of

I

bounded families absolutely p-summable families mixed (s,p)-summable families weakly p-summable families

1, m(8jP)

5

Point A.4.1 16.2.1 16.4.1 16.3.1

Index of Scalar Sequence Ideals

Scalar sequence ideals and t,heir quasi-norms are denoted by small Gothic and small Roman letters, respectively. Example : scalar sequence ideal a, quasi-norm a.

Symbol CO

i 1 1, I,,.,, I@ Gai

Ideal of ...

Point

zero sequences finite sequences bounded sequences absolutely p-summable sequences ( p ,q)-Lorentz sequences Q-Orlioz seque'nces @-Sargent sequences

13.1.6 13.1.4 A.4.1 13.2.10 13.9.1 13.11.1 13.10.1

Index of Procedures

~~

Symbol

Procedure

Point

clos rad dual reg inj sur max min

closure radical dual ideal regular hull injective hull surjective hull maximal hull minimal kernel adj oint ideal

4.2.1 4.3.1 4.4.1, 8.2.1 4.5.1, 8.3.1 4.6.1, 8.4.1 4.7.1, 8.5.1 4.9.2, 8.7.1, 13.5.1 4.8.2, 8.6.1, 13.4.1 9.1.1, 13.6.1

*

Index of Extensions

Symbol

Extension

Point

SUP inf mP rin lUP lin

superior inferior right-superior right -inferior left-superior left-inferior

15.6.2 15.6.4 15.6.6, 29.5.4 15.6.8 15.6.10 15.6.12

Index of Symbols

Fixed ndatim coniplex field n-dimensional complex linear spaoe Banach function space; C.1.5 Banach function space; 25.5.1 Banach sequence space; C.1.7 set of all e = (el, .,E,) with ~i = f1 ; E.5.1 single unit family (1) i-th unit family characteristic family of M Kronecker's symbol normalized n-dimensional Gauss probability chracteristic function of B injection defined by n ;A.4.3 real or complex field n-dimensional real or complex linear space Banach function space; C.1.2 Banach sequence space; (2.1.3 Banach vector space; C.1.3 stable law; 21.1.1 %-dimensionalstable law; 21.1.3 set of natural numbers [ 1 , 2 , . .) n-dimensional canonical projection surjection defined by n ; A.4.3 real field set of non-negative real numbers n-dimensional real linear space normalized rotation invariant measure on the n-dimensional sphere closed unit ball of lp closed unit ball of 1; Banach function space; 22.7.1

..

.

Let E be any Banach space LT.3

R' E" p n j

Fur 1,

closed unit ball; A.1.3 dual Banach space ; A.2.3 bidual Banach space; 4.2.4 Banach space I,( UEp) ; (3.3.3 Banach space ll(UE);C.3.7 identity map; A.1.4

444

Index of Symbols

KE JE QE

dim ( E ) Dim ( E ) Cod ( E ) Car (3)

W)

evaluation map ; A.2.4 canonical injection froin ik: into EinJ; C.3.3 canonicnl surjection from E'"' onto E ; C.3.7 dimension of E collection of all finite diiaensiond subspaces collection of all finite codiinensional subspaces collestion of Cartesian Banach spaces; 5.3.7 algebra of cylindrical subsets

Let LV aid N be subspccs of E dim (N) cod

(w)

J% 0%

8N(E)

dimension of M oodimension of N embedding map froin ill into E ; A.1.5 canonical map froin E onto EIN; A.1.6 o-algebra of cylindrical subsets

Let E and F be Baitack spaces

2(E,F )

EXF E

N-

F

Banach space of operat.ors froin E into F Cartesian product of E and F ;B.4.11 E is isomorphic to some complemented subspace of F equivalence defined by E P and F E

<

<

Let S be any operator from E into F /IS// S'

8'' S" M(S)

N(S) rank ( S ) trace (8) S(4

norm; A.3.2 dual operator; A.3.5 bidual operntor ; 14.3.6 strongly bidual operator; 1.5.5 range; A.3.4 null space; ,4.3.4 rank; B.l.l trace; B.1.4 Fredlioliii resolvent S(I - IZs)-l; 36.2.2

Let (Q, ,u) be any memure spuce LJQ, ,u) IP I(P4

!Mp

Banach function space; C.l.l embedding map froin La@, p ) into LP(f2$ p) embedding map froin Lq@, p ) into Lp(O,/A) norm on L@, p ) ; C.f.1

Index of Symbols

Let K be any co?npact Hausdorff space

WK) C(K) WK) JP

W(K) 11.1101

Bore1 o-algebra; C.1.8 Banach function space; C.1.5 collection of open subsets canonical map from C ( K )into L,(K, p ) set of all regular Bore1 probabilities; (2.1.8 norm of C ( K ) ;C.1.5

Let I be any index set

S(I)

%(I) ZP(4

collection of all finite subsets Banach fanlily space; (2.1.7 Banach fainily space; C.1.3

Let (Ei) with i E I be any family of Ba.nach spaces l,(Ei, I ) Jk Qk

(Eih

Cartesian I,-product; C.4.1 canonical injection froin Ek into lp(Idi,I);(3.4.1 canonical surjection from Zp(Ei,I ) onto Zk; C.4.1 ultraproduct; 8.8.1

Notations in the theory of locally convex spaces

WP) O(P)

gP)

0% a) X(Oi, I ) U V

<

space; 29.1.12 related Banach space; 29.1.12 equivalence class ; 29.1.12 induced operator; 29.1.13 Cartesian product; 29.1.10 U is isomorphic t.0 some coinpleniented subspa,ceof V l i d

Let p be any non-tbegntive ?umber [PI

'P P',P*

greatest integer not exceeding p see 21.1.2 dual exponent, l/p lip' = 1

+

f(n)x g(n) f ( n ) 5 cg(n) and g(n) 5 cf(n) for n = 1, 2, ... It4 norin (*, .) scalar vroduct .> natural pairing; A.2.1 1

(a,

445

446

Index of Symbols

Universal constants anp

CG CK CP CSP

p) 89

quotient c!$/cg); 22.1.1 Grothendieck’s constant,; 22.4.5 converse Khintchine’s constant ; E.5.3 Khintchine’s constant; E.5.2 absolute p-th moment of pus;21.1.2 absolute p-th moment of p ; ; 22.1.1

Index of Subjects

absolutely - p-summable family 16.2.1 summing operator 6.5.1 - r-summing operator 17.3.1 - (r,p)-slimming operator 17.2.1 - (7, p, q)-summing operator 17.1.1 - t-summing operator 23.1.1 abstract - Lp-space C.2.1 Loo-space C.2.3 additive s-function 11.8.1 adjoint - operator D.2.2 - operator ideal 9.1.1 - sequence ideal 13.6.1 a,lgebraicmultiplicity 27.1.3 approximable operator 1.3.1 approximation number - of an operator 11.2.1 - of a sequence 13.7.3 approximation property 10.1.1 ascent 26.1.2

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Banach - space A.1.2 - space idea,] 2.1.1 basis 23.2.3, 28.5.1 bidual - Banach space A.2.4 - operator A.3.6 Bochner integral 24.1.9 Bore1 - probability C.1.8, 25.1.2 - a-algebra (2.1.8, 25.1.1 Brownian mot,ion 25.6.4 Calkin algebra 26.3.4 Cartesian - Banachspace 2.1.4 - operator ideal 29.4.1 - space ideal 29.4.2 Cartesian product - of Banach spaces B.4.11 - of operators 29.1.11 ’ - of locally convex spaces 29.1.10

Ctlrtesian $,-product C.4.1 characteristic - family A.4.4 - value 26.5.4 closed operator ideal 4.2.4 closure of an operator ideal 4.2 1 compact operator 1.4.1 p-compact operator 18.3.1 (p, 9)-compact operator 18.4.1 complemented subspace B.4.3 completely continuous operator 1.6.1 completely symmetric - operator ideal 4.4.5 quasi-normed operator ideal 5.2.4 - 8-function 11.7.1 COtYpe Banach space of (s,p)-cotype 21.4.11 operator of (sr,p)-cotype 21.3.1 cylindrical - probability 25.3.1 - Gauss probability 26.5.3 - Radon probability 25.3.5 - set 25.2.2

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decomposable operator 24.2.1, 24.4.1, 245.1 decomposing operator 24.4.2 p-decomposing operator 24.5.3 desuent 26.1.3 dimension of a Hilbert space D.1.3 direct sum B.4.2 discretely p-factorable operator 19.3.11 dominated operator p-dominated operator 17.4.7 (p, q)-dominated operator 17.4.1 domination property - of a norrned operator ideal 15.4.1 - of a normed sequence ideal 13.8.1 dual - Banach space A.2.2 - exponent C.1.4 - operator A.3.5 - operator ideal 4.4.1 - quasi-normed operator ideal 8.2.1 - space ideal 4.44 dualisable operator 4.4.10

448

Index of Subjects

Dunford-Pettis property 3.2.8 -- operator 3.2.5

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elementary operator B.1.5 entropy number -, inner 12.1.6 -, outer 12.1.2 equivalent quasi-norms 6.1.7 evaluation map A.2.4 extension property C.3.1 extension of an operator ideal from Banach spaces to locally convex spaces 29.5.1 - from Hilbert spaces to Banach spares 15.6.1 factorable operator A-factorable operator 2.2.1, 29.2.5 p-factorable operator 19.3.1 (p, q)-factorable operator 19.4.1 family A.4.1 family ideal 16.1.1 finite ascent 26.1.2 - descent 26.1.3 operator B.l.l - sequence 13.1.4 finite dimensional Banach space 2.3.1 finite nuclear norm 6.8.1 Fredholm coefficient 27.2.4 determinant 27.2.7 -- divisor 27.1.5 minor 27.2.8 - operator 26.3.1 - resolvent 26.2.2 fractional integration 22.7.5 function s-function of operators 11.1.1 s-function of sequences 13.7.2 functional A.2.1

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Gauss probability -, cylindrical 25.5.3 -, normalized cvlindrica! 25.5.2 Gelfand number 11.5.1 Gohberg operator 4.3.5 Grothendieck - constant 22.4.5 - operator 3.2.6 space ideal 29.6.1

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Hadamard inequality 27.2.2 Hilbert -- number 11.4.1 - operator 6.6.1 - space D.l.l N,-Hilbert operator 5.4.1 Hilbert-Schmidt operator 15.5.5 HBlder - space 22.7.1 - A-continuous function 26.6.1 hull procedure - on the class of operator ideals 4.1.2 - on the class of quasi-iiormed operator ideals 8.1.2 - on the set of quasi-nornied sequenre ideals 13.3.2 hyperorthogonal basis 23.2.3 idem potent - procedure 4.1.2 - operator ideal 3.1.5 -- quasi-normed operator ideal 7.1.5 identity map A.1.4 inemential operator 4.3.5 inferior extension 15.6.4 injection B.3.2 @-injection 26.6.1 injection modulus B.3.1 injective -- operator ideal 4.6.8, 29.3.1 - quasi-normed operator ideal 8.4.8 - space ideal 4.6.8,29.3.2 - s-function 11.5.4 injective hull - of an operator ideal 4.6.1 - of a quasi-normed operator ided 8.4.1 inner entropy number 12.1.6 integral operator 6.4.1 r-integral operator 19.2.1 (P, p, q)-integral operator 19.1.1 a-integral operator 23.3.1 intermediate space C.5.3 isomorphic - Banach spaces B.2.2 - locally convex spaces 29.1.8 isomorphism - between Banach spaces B.2.1 - between locally convex spaces 29.1.8 @-isomorphism 26.3.1 Eat0 operator 1.9.2 kernel 24.2.1, 24.4.1, 24.5.1

Index of Subjects ~

449

~~

kernel procedure - on the class of operJtor ideals 4.1.3 - on the class of quasi-normed operator ideals 8.1.2 - on the set of quasi-normed sequence ideals 13.3.2 Khintchine - constant E.5.2, E.5.3 - inequality E.5.2 Kolmogorov number 11.6.1 left-inferior extension 15.6.12 left-superior extension 15.6.10 lift.ing property C.3.5 limit order 14.4.1 locally convex space 29.1.1 locally convex space ided 29.2.2 local unconditional structure 23.3.7 Lorentz sequence ideal 13.9.1 maximal - operator ideal 4.9.6 - quasi-normed operator ideal 8.7.7 - quasi-normed sequence ideal 13.5.5 - 8-function 11.10.1 maximal hull - of an operator ideal 4.0.2 - of a quasi-normed operator ideal 8.7.1 - of e quasi-normed sequence ideal 13.3.1 metric - approximation property 10.2.1 - extension property C.3.1 - injection B.3.2 - isomorphism B.2.1 - lifting property (3.3.5 - surjection B.3.6 metrically isomorphic Banach spnce €3.2.2 minimal - operator ideal 4.8.6 - quasi-normed operator ideal 8.6.5 - quasi-normed sequence ideal 13.4.5 minimal kernel - of an operator ideal 4.8.2 - of a quasi-normed operator ideal 8.6.1 -- of a quasi-normed sequence idonl 13.4.1 mixed (8, p)-sumniabk fdnnlj 16.4.1 mixing operator (8, p)-mixing operator 20.1.1 moment -, absolute p-th 25.3.10 -, week p-th 23.3.8

monotone procedure 4.1.2 multiplicative 8-function 11.9.1 Noether operator 26.3.1 non-increasing rearrangement 13.7.6 norm - of family ideal 16.1.7 - of an operator ideal 6.2.1 - of a sequence ideal 13.2.6 normed -- family ideel 1 6 . 1 3 - operator ideal 6.2.2 - sequence ideal 13.2.7 nuclear locally convex space 29.7.1 nuclear operator 6.3.1 r-nuclear operator 18.2.1 (r,p, p)-nuclear operator 18.1.1 u-nuclear operator 25.2.1 null space 8.3.4 niim ber 8-number of operators 11.1.1 8-number of sequences 13.7.2 operator A.3.1, 29.1.6 Ep-operator 14.2.1 &p-operator 14.3.1 gp-operntor 14.2.15 Gp-operator 15.5.1 G;)-operator 142.1 @)-operator 14.1.1 operator ideal - on Banach spaces 1.1. L - on locally convex spew 29.2.1 Orlicz property 17.2.8 Orlicz sequence ideal 13.11.1 orthogonal projection B.4.6 outer entropy number 12.1.2 partially isometric operator D.2.4 Pelczyriski - property 2.0.1 - operator 1.10.2 perfect - normed operator ideal 9.3.3 - normed sequence ideal 13.6.6 positive operator D.2.3 procedure - on the class of operator ideals 4.1.1 - on the class of quasi-normed operator iC!eals 8.1.1 *. - on the set of q:iasi-normcd sequence ideals 13.3.1

450

Index of Subjects

product - of operator ideals 3.1.1 - of quasi-norms 7.1.1 projcction B.4.1 projection constant 28.2.5 proper - operator ideal 2.3.3 - sequence ideal 13.1.7 pseudo-s-function 12.1.1

right-inferior extension 15.6.8 right-superior extension 15.6.6 Rosenthal operator 3.2.4

Sargent sequence ideal 13.10.1 Schmidt factorizatiori D.3.3 - representation D.3.2 Schur property 2.5.1 quasi-compact operator 26.4.1 Schwartz space 29.6.10 quasi-dual Banach space 4.4.9 self-adjoint quasi-norm - norrned operator ideal 9.4.1 - of a family ideal 16.1.3 - normed sequence ideal 13.6.8 - of an operatm ideal 6.1.1 separable -- of sequence ideal 13.2.1 - Banach space 2.7.1 quasi-normed - operator 1.S.l - family ideal 16.1.5 sequence ideal 13.1.1 - operator ideel 6.1.3 simple Banach space 5.2.1 - sequence ideal 13.2.3 small operator ideal 14.5.1 quotient space A.1.6, 29.1.5 space quotient L -s ace 22.2.1 p. - of operator ideals 3.2.1 space ideal 2.1.1, 29.2.2 - of quasi-norms 7.2.1 Bobolev Rademacher type limit order 22.7.3 Banach spaces of Rademacher (8, p)-twypvpe - space 22.7.1 21.2.12 Sobolev-Slobodetzkij space 22.7.1 operators of Rademacher (8, a)-type stable law 21.1.1 21.2.6 strictly cosingular operator 1.10.2 radical 4.3.1 strictly nuclear Radon probability 28.1.2 - locally convex space 29.8.1 radonifying operator - operator 18.7.1 p-radonifying operator 25.4.2 strictly singular operator 1.9.2 Radon-Nikodym strongly - derivative 24.3.4 - bidual operator 1.5.5 - operator 24.2.6 - p-factorable operator 19.3.11 -- property 24.2.9 - r-integral operator 19.2.15 range A.3.4 subspace A.1.5, 29.1.4 rank B.l.l summable family A.4.6 reflexive Banach space 2.4.1 superior extension 15.6.2 regular surjection B.3.6 - Bore1 probability C.1.8, 25.1.2 @-surjection 26.6.6 - operator ideal 4.5.5 surjection modulus B.3.4 - quasi-normed operator ideal 8.3.4 surjective - s-function 11.7.1 - operator ideal 4.7.8, 29.3.5 regular hull - quasi-normed opeartor ideal 8.5.6 - of an operator ideal 4.5.1 surjective - of a quasi-normed operator ideal 8.3.1 - space ideal 4.7.8, 29.3.6 related operators 27.3.1 - s-function 11.6.4 Riesz surjective hull - decomposition 26.1.5 - of an operator ideal 4.7.1 - operator 26.5.1 - of a quasi-normed operator ideal 8.5.1

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Index of Subjects symmetric operator ideal 4.4.5 - s-function 11.7.1

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trace B.1.4 trace formula 27.4.11 type Banach space of (s,p)-type 21.2.12 operator of (8,p)-type 21.2.1 operator of ultra (s,p)-type 21.3.1 type of an intermediate space J-type C.5.6 K-type C.5.4 ultraproduct - of Banach spaces 8.8.1 - of operators 8.8.3 ultrastable - quasi-normed operator ideal 8.8.5 - 8-function 11.10.5 unconditionally summing operator 1.7.1 unit family A.4.4

451

viiriation 24.3.2 vector function 24.1.1 -, p-integrable 24.1.9 -, p-measurable 24.1.3 -, ,u-simple 24.1.2 vector measure 24.3.1 -, p-continuous 24.3.4 -, p-differentiable 24.3.4 weak operator topology A.3.3 weak topology A.2.3, A.2.5 weakly compact operator 1.5.1 weakly lower semi-continuous quasi-norm of an operator ideal 8.7.16 quasi-norm of a sequence ideal 13.5.7 - s-function 11.10.7 weakly singular integral operator 22.7.5 weakly p-sunimable family 16.3.1 Wiener probability, cylindrical 25.6.4

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