Table Of ContentVector Lattices and Integral Operators
Mathematics and Its Applications
Managing Editor:
M.HAZEWINKEL
Centre for Mathenwtics and Computer Science. Amsterdam. The Netherlands
Volume 358
Vector Lattices
and Integral Operators
edited by
S. s. Kutateladze
Sobolev Institute of Mathematics,
Siberian Branch of the Russian Academy of Sciences,
Novosibirsk, Siberia, Russia
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
A C.J.P. Catalogue record for this book is available from the Library of Congress
ISBN-13,978-94-01O-6571-9 e-ISBN-13: 978-94-009-0195-7
DOl: 10.1007/978-94-009-0195-7
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This is a completely revised and updated
translation of the original Russian work
Vector Lattice:r and Integral Operators,
A.V. BuldlValov, V.B. Korotkov, A.G. Kusraev,
S.S. Kutateladze and 8.M. Marn.kov,
Novosibirsk, Nauka It! 1992.
This book was typeset using AMS-TEX.
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e
1996 Kluwer Academic Publishers
Softcover reprint of the hardcover 1st edition 1996
No pan of the material protected by this copyright notice may be reproduced or
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Contents
Foreword vii
Chapter 1. Nonstandard Theory of Vector Lattices
(A. G. Kusraev and S. S. Kutateladze)
1. Vector Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Boolean-Valued Models ....................................... 16
3. Real Numbers in Boolean-Valued Models 31
4. Boolean-Valued Analysis of Vector Lattices .................. . 46
5. Fragments of Positive Operators ............................. . 59
6. Lattice-Normed Spaces ...................................... . 71
Comments .................................................. . 91
References 99
Chapter 2. Operator Classes Determined by Order
Conditions (A. V. Bukhvalov)
1. Ideal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2. The Space of Regular Operators .... .................... ...... 117
3. Spaces of Vector-Functions ............ .............. ... .... .. 132
4. Dominated Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
References 153
Chapter 3. Stably Dominated and Stably Regular Operators
(B. M. Makarov)
1. p-Absolutely Summing Operators ............................. 159
2. p-Absolutely Summing Operators in Hilbert Space . . . . . . . . . . . . 168
3. Nuclear Operators ............................................ 172
4. Stably Dominated Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
5. Coincidence of Some Classes of Operators in the Scale
of the Banach Spaces LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
VI
6. Nikishin-Maurey Factorization Theorems ..................... 215
7. Stably Regular Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8. Certain Operator Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9. Operator Spaces and Local Unconditional Structure . . . . . . . . . . 242
S. Supplement to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
References 275
Chapter 4. Integral Operators (A. V. Bukhvalov, V. B. Korotkov,
and B. M. Makarov)
1. Basic Properties of Integral Operators ....................... . 281
2. Integral Representation of Linear Operators ................. . 288
3. Applications of the Criterion for Integral Representability 301
4. Linear Operators and Vector Measures ...................... . 303
5. Integral Representation of Nonlinear Operators .............. . 305
6. Algebraic Properties of Integral Operators ................... . 308
7. Universal Integral Operators and Operators with Integral
Commutators ............................................. . 319
Comments .................................................. . 339
References .................................................. . 341
S. Supplement to Chapter 4. Integral Operators of Convolution
(V. D. Stepanov) .......................................... . 347
References 357
Chapter 5. Disjointness Preserving Operators (A. E. Gutman)
1. Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
2. Order Approximating Sets .................................... 368
3. Order Bounded Operators .................................... 378
4. The Shadow of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
5. Orthomorphisms ............................................. 398
6. Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
7. Weighted Shift Operator . .. .. . .. .. .. . .. . .. . .. .. . . .. . .. . . . . . . . 420
8. Representation of Disjointness Preserving Operators . . . . . . . . . . 430
9. Interpretation for the Properties of Operators . . . . . . . . . . . . . . . . . 437
Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
References 451
Index 455
Foreword
The theory of vector lattices, stemming from the mid-thirties, is now at the stage
where its main achievements are being summarized. The sweeping changes of the
last two decades have changed its image completely. The range of its application was
expanded and enriched so as to embrace diverse branches of the theory of functions,
geometry of Banach spaces, operator theory, convex analysis, etc. Furthermore, the
theory of vector lattices was impregnated with principally new tools and techniques
from other sections of mathematics. These circumstances gave rise to a series of mono
graphs treating separate aspects of the theory and oriented to specialists. At the same
time, the necessity of a book intended for a wider readership, reflecting the modern
diretions of research became clear. The present book is meant to be an attempt at
implementing this task. Although oriented to readers making their first acquaintance
with vector-lattice theory, it is composed so that the main topics dealt with in the
book reach the current level of research in the field, which is of interest and import
for specialists.
The monograph was conceived so as to be divisible into two parts that can be
read independently of one another. The first part is mainly Chapter 1, devoted to the
so-called Boolean-valued analysis of vector lattices. The term designates the applica
tion of the theory of Boolean-valued models by D. Scott, R. Solovay and P. Vopenka
to constructing a special realization of a (model of) vector lattice which allows one to
treat elements of the lattice under study as reals. The starting point is an E. I. Gor
don theorem claiming that the presentation of the field of reals in a Boolean-valued
model constitutes a universally complete vector lattice (an extended K-space in the
respective Russian terminology). Thus a huge part of the general theory of vector
lattices admits of a straightforward derivation by interpreting the familiar properties
vii
Vlll Foreword
of reals. This chapter also exposes Boolean-valued approaches to more advanced sec
tions of vector-lattice theory such as lattice-normed spaces and fragments of positive
operators, approaches being proposed by A. G. Kusraev and S. S. Kutateladze. The
elementary exposition of the apparatus of model theory as well as that of vector-lattice
theory makes it possible to present the material in such a form that for a reader-logician
it appears as an introduction into new fields of applications of model theory whereas
for a reader-analyst it is an introduction into applicable model theory. More advanced
topics of the abstract theory of operators in vector lattices are treated in Chapter 5
by A. E. Gutman. His research into the properties of disjointness preserving operators
is in many aspects motivated by Boolean-valued analysis, demonstrating the power of
the latter.
The second part of the monograph consists of Chapters 2-4 and its Supplement.
It deals with operator theory in spaces of measurable functions and is oriented to the
reader who is interested in functional analysis and the theory of functions. The book
treats the classes of operators that are explicitly or implicitly tied with the natural
order relation between measurable functions.
The explicit connection is discussed, for instance, in considering regular operators
which are differences of pairs of positive operators. The implicit connection relates
to integral operators whose definition is given in the conventional terms of function
theory. It turns out that the answer to the question of which operators are integral
depends upon the theory of vector lattices not only in formulation but also in proof.
Part of Chapter 2 and the first part of Chapter 4 address the answering of the just
mentioned question first raised by John von Neumann as long ago as the thirties. The
book presents an original solution to the problem that was given by A. V. Bukhvalov
(1974) and supplements it with the approaches that have appeared since then.
The theme of Chapter 3 and the second part of Chapter 4 is mainly concerned
with studying the stability of different classes of operators defined in terms of order,
where stability as regards multiplication by arbitrary continuous operators is treated.
As a rule it is relatively easy to demonstrate that such composition does not always
belong to the class considered initially. Thus the problem appears of describing sub
classes of operators stable under the indicated operation. In Chapter 3 we study various
modifications of the problem for regular and dominated operators. The main results
in this direction are due to B. M. Makarov and V. G. SamarskiI. The technique of
researching the topic turns out to be interwoven with the theory of p-absolutely sum
ming operators and operator factorization theory of E. M. Nikishin-B. Maurey. This
Foreword IX
material related to modern Banach space theory is exposed in monograph form for the
first time. The second part of Chapter 4 treats the same problem on composition but
now for the class of integral operators.
In Chapter 4 the authors try their best to provide comprehensive information on
the solution of problems raised in the book of P. Halmos and V. Sunder "Bounded
Linear Operators on L2 Spaces." The main results here are due to the contribution
of V. B. Korotkov and W. Schachermayer. A separate supplement treats a related
problem for the convolution operator which was settled by V. D. Stepanov.
The variety of addressed subjects and results determined the style of exposi
tion. Some of the more elementary material freely accessible to the reader is present
ed without proofs. All principal results are however furnished with complete proofs.
Commentaries appended to all chapters contain additional information and refer to
the literature. The contributors assume the reader to be familiar with standard courses
in the theory of functions and functional analysis.
The present collection is based on its predecessor in Russian which is enriched by
Chapter 5 written by A. E. Gutman at my request. The Russian edition was a joint
venture and a joint monograph by A. V. Bukhalov, V. B. Korotkov, A. G. Kusraev,
B. M. Makarov and S. S. Kutateladze which was published in 1992. Tumultuous events
in the former Soviet Union hinder the means of communications between the contrib
utors. As a result, I became the only one of our team who had a chance of reading the
whole manuscript in English. So, I solely bear full responsibility for all demerits of the
present edition, pretending to none of its possible merits.
s. S. K utateladze
CHAPTER 1
Nonstandard Theory of
Vector Lattices
BY
A. G. Kusraev and S. S. Kutateladze
