Isometries on Banach Spaces: function spaces / Edition 1

by Richard J. Fleming, James E. Jamison

Overview

Fundamental to the study of any mathematical structure is an understanding of its symmetries. In the class of Banach spaces, this leads naturally to a study of isometries-the linear transformations that preserve distances. In his foundational treatise, Banach showed that every linear isometry on the space of continuous functions on a compact metric space must transform a continuous function x into a continuous function y satisfying y(t) = h(t)x(p(t)), where p is a homeomorphism and h is identically one.

Isometries on Banach Spaces: Function Spaces is the first of two planned volumes that survey investigations of Banach-space isometries. This volume emphasizes the characterization of isometries and focuses on establishing the type of explicit, canonical form given above in a variety of settings. After an introductory discussion of isometries in general, four chapters are devoted to describing the isometries on classical function spaces. The final chapter explores isometries on Banach algebras.

This treatment provides a clear account of historically important results, exposes the principal methods of attack, and includes some results that are more recent and some that are lesser known. Unique in its focus, this book will prove useful for experts as well as beginners in the field and for those who simply want to acquaint themselves with this area of Banach space theory.

Product Details

ISBN-13:	9780367395575
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Monographs and Surveys in Pure and Applied Mathematics
Pages:	208
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Fleming, Richard J.; Jamison, James E.

Table of Contents

Preface vii

Chapter 1 Beginnings 1

1.1 Introduction 1

1.2 Banach's Characterization of Isometries on C(Q) 2

1.3 The Mazur-Ulam Theorem 6

1.4 Orthogonality 10

1.5 The Wold Decomposition 15

1.6 Notes and Remarks 19

Chapter 2 Continuous Function Spaces-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 L^p Spaces 49

3.1 Introduction 49

3.2 Lamperti's Results 50

3.3 Subspaces of L^P and the Extension Theorem 55

3.4 Bochner Kernels 67

3.5 Notes and Remarks 72

Chapter 4 Isometries of Spaces of Analytic Functions 79

4.1 Introduction 79

4.2 Isometries of the Hardy Spaces of the Disk 79

4.3 Bergman Spaces 89

4.4 Bloch Spaces 92

4.5 S^P Spaces 96

4.6 Notes and Remarks 98

Chapter 5 Rearrangement Invariant Spaces 103

5.1 Introduction 103

5.2 Lumer's Method for Orlicz Spaces 104

5.3 Zaidenberg's Generalization 118

5.4 Musielak-Orlicz Spaces 127

5.5 Notes and Remarks 142

Chapter 6 Banach Algebras 145

6.1 Introduction 145

6.2 Kadison's Theorem 146

6.3 Sub differentiability and Kadison's Theorem 151

6.4 The Nonsurjcctive Case of Kadison's Theorem 157

6.5 The Algebras C⁽¹⁾ and AC 164

6.6 Douglas Algebras 168

6.7 Notes and Remarks 171

Bibliography 181

Index 193

Editorial Reviews

"This is a very well written book. The authors have done a remarkable job in collecting this material and in exposing it in a very clear style. It will be an important reference tool for analysts, experts and nonexperts, and it will provide a clear and direct path to several topics of current research interest." - Juan J. Font, Mathematical Reviews, Issue 2004j

From the Publisher