Isometries on Banach spaces: function spaces

Richard J. Fleming, James E. Jamison1584880406, 9781584880400

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.

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Table of contents :
ISOMETRIES ON BANACH SPACES: function spoces……Page 3
Contents……Page 5
Preface……Page 7
1.1. Introduction……Page 10
1.2. Banach’s Characterization of Isometries on C(Q)……Page 11
1.3. The Mazur-Ulam Theorem……Page 15
1.4. Orthogonality……Page 19
1.5. The Wold Decomposition……Page 24
1.6. Notes and Remarks……Page 28
2.1. Introduction……Page 33
2.2. Eilenberg’s Theorem……Page 34
2.3. The Nonsurjective Case……Page 37
2.4. A Theorem of Vesentini……Page 47
2.5. Notes and Remarks……Page 50
3.1. Introduction……Page 56
3.2. Lamperti’s Results……Page 57
3.3. Subspaces of LP and the Extension Theorem……Page 62
3.4. Bochner Kernels……Page 74
3.5. Notes and Remarks……Page 79
4.2. Isometries of the Hardy Spaces of the Disk……Page 85
4.3. Bergman Spaces……Page 95
4.4. Bloch Spaces……Page 98
4.5. SP Spaces……Page 102
4.6. Notes and Remarks……Page 104
5.1. Introduction……Page 108
5.2. Lumer’s Method for Orlicz Spaces……Page 109
5.3. Zaidenberg’s Generalization……Page 123
5.4. Musielak-Orlicz Spaces……Page 132
5.5. Notes and Remarks……Page 147
6.1. Introduction……Page 150
6.2. Kadison’s Theorem……Page 151
6.3. Subdifferentiability and Kadison’s Theorem……Page 156
6.4. The Nonsurjective Case of Kadison’s Theorem……Page 162
6.5. The Algebras C(1) and AC……Page 169
6.6. Douglas Algebras……Page 173
6.7. Notes and Remarks……Page 176
Bibliography……Page 185