Isometries on Banach spaces: function spaces

196 Pages·2002·5.394 MB·English

by  Richard J. Fleming, James E. Jamison

CHAPMAN & HALUCRC
Monographs and Surveys in
Pure and Applied Mathematics 1 29
ISOMETRIES
ON BANACH SPACES:
function spaces
O 2003 by Chapman & HallICRC
CHAPMAN & HALLICRC
Monographs and Surveys in Pure and Applied Mathematics
Main Editors
H. Brezis, Universite' de Paris
R.G. Douglas, Texas A&M University
A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)
Editorial Board
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R.P. Gilbert, University of Delaware
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B. Lawson, State University of New York
B. Moodie, University of Alberta
L.E. Payne, Cornell University
D.B. Pearson, University of Hull
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I. Stakgold, University of Delaware
W.A. Strauss, Brown University
J. van der Hoek, University of Adelaide
O 2003 by Chapman & HallICRC
CHAPMAN & HALUCRC
Monographs and Surveys in
Pure and Applied Mathematics 1 29
ISOMETRIES
ON BANACH SPACES:
function spoces
RICHARD J. FLEMING
JAMES E. JAMISON
CHAPMAN & HALUCRC
A CRC Press Company
Boca Raton London NewYork Washington, D.C.
O 2003 by Chapman & HallICRC
Library of Congress Cataloging-in-PublicationD ata
Fleming, Richard J.
Isometries on Banach spaces : function spaces I by Richard J. Fleming and James E. Jamison.
p. cm. - (Chapman & HallICRC monographs and surveys in pure and applied
mathematics ; 129)
Includes bibliographical references and index.
ISBN 1-58488-040-6 (alk. paper)
1. Function spaces. 2. Banach spaces. 3. Isometrics (Mathematics) I. Jamison, James E.
11. Title. 111. Series.
QA323 .F55 2002
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Contents
Preface
Chapter 1. Beginnings
1.1. Introduction
1.2. Banach's Characterization of Isometries on C(&)
1.3. The Mazur-Ulam Theorem
1.4. Orthogonality
1.5. The Wold Decomposition
1.6. Notes and Remarks
Chapter 2. Continuous Function Spa,ces-The Banach-Stone Theorem 25
2.1. Introduction 25
2.2. Eilenberg's Theorem 26
2.3. The Nonsurjective Case 29
2.4. A Theorem of Vesentini 39
2.5. Notes and Remarks 42
Chapter 3. The LP Spaces
3.1. Introduction
3.2. Lamperti's Results
3.3. Subspaces of LP and the Extension Theorem
3.4. Bochner Kernels
3.5. Notes and Remarks
Chapter 4. Isornetries of Spaces of Analytic Functions
4.1. Introduction
4.2. Isometries of the Hardy Spaces of the Disk
4.3. Bergman Spaces
4.4. Bloch Spaces
4.5. 9 Spaces
4.6. Notes and Remarks
Chapter 5. Rearrangement Invariant Spaces
5.1. Introduction
5.2. Lumer's Method for Orlicz Spaces
5.3. Zaidenberg's Generalization
5.4. Musielak-Orlicz Spaces
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5.5. Notes and Remarks
Chapter 6. Banach Algebras
6.1. Introduction
6.2. Kadison's Theorem
6.3. Subdifferentiability and Kadison's Theorem
6.4. The Nonsurjective Case of Kadison's Theorem
6.5. The Algebras and AC
6.6. Douglas Algebras
6.7. Notes and Remarks
Bibliography
O 2003 by Chapman & HallICRC
Preface
Herman Weyl has said that in order to understand any mathematical
structure, one should investigate its group of symmetries. In the class of Ba-
nach spaces, such a goal leads naturally to a study of isometries. A principal
theme in geometry from the earliest times has been the study of transforma-
tions preserving lengths and angles. If the origins of the theory of Banach
spaces are assigned to the appearance of Banach's book in 1932, then the
study of Banach space isometries must be assigned the same starting date.
In his book, Banach included the first characterizations of all isometries on
certain classical spaces. The body of literature concerning isometries that has
grown up since that time is large, perhaps surprisingly so.
An isometry, of course, is a transformation which preserves the distance
between elements of a space. When Banach showed in 1932 that every linear
isometry on the space of continuous real valued functions on a compact metric
space must transform a continuous function x(t) into a continuous function
y(t) satisfying
-
where Ih(t)l 1 and p is a homeomorphism, he was establishing a canon-
ical form characterization which fits in an astonishing number of cases. In
this volume we are interested primarily in just such explicit descriptions of
isometries. Our approach, then, will differ from that of many authors whose
interest in isometries lies in showing which spaces are isometric to each other,
or to those whose interest is in discovering properties of topological spaces on
which the functions in the Banach spaces are defined. Such interests as these
have served as excellent motivations for the types of characterizations that
interest us. There have been several excellent surveys concerning isometries,
including those by Behrends, Loomis, Jarosz, and Jarosz and Pathak, each
of which concentrates on some subset of the whole. Our intent was to pro-
vide a survey of the entire subject, and our survey article (1993) serves as an
inspiration and guide for the current work.
Our goal has been to produce a useful resource for experts in the field as
well as beginners, and also for those who simply want to acquaint themselves
with this portion of Banach space theory. We have tried to provide some
history of the subject, some of the important results, some flavor of the wide
variety of methods used in attacking the characterization problem for various
vii
O 2003 by Chapman & HallICRC
. . .
vlll PREFACE
types of spaces, and an exhaustive bibliography. We have probably underes-
timated the enormity of such a project, and perhaps what we have produced
is more of a sampler than a full-blown survey.
We have chosen to organize the material according to the different classes
of spaces under study and this is reflected in the chapter headings. The current
volume is the first of two intended volumes, and as can be seen from the table
of contents, it is primarily concerned with isometries on function spaces. The
first chapter treats some general topics such as linearity, orthogonality, and
Wold decompositions, while Chapter 6 contains material on noncommutative
C*-algebras, but the rest of the chapters treat the classical function spaces.
The second volume will include chapters with the following titles:
Chapter 7: The Banach-Stone Property
Chapter 8: Other Vector-Valued Function Spaces
Chapter 9: Orthogonal Decompositions
Chapter 10: Matrix Spaces
Chapter 11: Norm Ideals of Operators
Chapter 12: Spaces with Trivial Isometries
Chapter 13: Epilogue
In each chapter we try to include an early result of some historical impor-
tance. Other selections are made in order to expose some one of the principal
methods that have been used, or perhaps to give an account of work that
has not received much attention. The chapters, and even the sections within
the chapters, are mostly independent of each other, and the reader can begin
at any point of interest. In making our selections we have, no doubt, left
out many others just as deserving and quite possibly more important. Hope-
fully, most of these omissions in the text are mentioned in the notes and the
bibliography.
The exposition relies mostly on the original papers, and we have tried to
report faithfully on those results, with additional clarifications when possible.
Probably we have included more detail than necessary in some instances, but
we have chosen to err on that side. There are a few places where we have
given only sketchy arguments.
In each chapter we include a section on notes and remarks which give
related results and other approaches that were not included in the main text.
We hope these sections will help to soothe those who disagree sharply with
our choice of material. For the most part, all references are given in the notes
section. The exceptions are in cases where a reference is needed to justify a
statement being made in the text.
In the bibliography we give a representative selection of the works on
isometries, and a serious investigation of all such works available should prob-
ably begin with Math Sci Net using the phrases "isometries on," "isometries
in," or "isometries for." Certainly there are many papers on vector-valued
function spaces and matrix spaces which one might be expected to be men-
tioned here, but we are saving these for the second volume. That second
O 2003 by Chapman & HallICRC
PREFACE ix
volume will include many more references which were not directly relevant in
this first one, and we also intend for Chapter 13 of Volume 2 to provide a
further guide to the literature.
We assume that our readers are familiar with the standard material in
courses in real variables, complex function theory, and functional analysis.
Terms and notation that are common in those fields we leave undefined in
the text. Page references to some of the special notation are given in the
index. Some notation, of course, serves multiple purposes which should be
understood in the context in which it appears. If some symbol or term is
encountered which is not referenced in the index, the reader should be able
to find it explained within a page or two of that location.
We have received much encouragement for this project from a number of
people over recent years, and we want to mention three people in particular
who have provided special help. Joe Diestel read portions of the work in
early stages and his kind words helped move us forward. Bill Hornor has
provided valuable advice, particularly in regard to Chapter 4 on analytic
functions. David Blecher helped immeasurably in reading much of Chapter
6, and tried to guide us in understanding the material on operator spaces and
the nonsurjective case of Kadison's theorem. However, we strongly emphasize
the fact that we alone are responsible for any existing errors.
Finally, we would both like to express our deep appreciation and love for
our wives, Diane Fleming and Jan Jamison, for their patience and devotion.
Richard J. Fleming and James E. Jamison
August 30, 2002
O 2003 by Chapman & HallICRC
CHAPTER 1
Beginnings
1.1. Introduction
Isometries are, in the most general sense, transformations which preserve
distance between elements. Such transformations are basic in the study of
geometry which is concerned with rigid motions and properties preserved by
them. The isometries of the Euclidean plane may all be described as rotations,
translations, reflections, and glide reflections, and these transformations form
a group under the operation of composition. This group is sometimes called
the Euclidean group of the plane. Of course the Euclidean group is very large
and often certain subgroups are sought which preserve some particular subset
of the plane. If S is a subset of the Euclidean plane R2, the subgroup G
which consists of all isometries which map S onto itself is called the complete
symmetry group of S. A subgroup of G is called a symmetry group of S.
The symmetry group of the unit circle given by an equation
relative to a fixed coordinate system is sometimes called the orthogonal group
in the plane. It can be seen that each transformation in this group must leave
the origin fixed and is therefore a linear transformation. This is a special case
of a more general result which we will prove shortly.
The fact that transformations in the orthogonal group are linear allows
them to be represented by matrices which are of two forms:
which represents a rotation through an angle 0; and
which represents a reflection of the plane with respect to the line given by the
equation
The rotations form a subgroup of the symmetry group called the rotation
group.
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