Isometrics on Banach Spaces

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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.

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Table of Contents

BEGINNINGS
Introduction
Banach's Characterization of Isometries on C(Q)
The Mazur-Ulam Theorem
Orthogonality
The Wold Decomposition
Notes and Remarks
CONTINUOUS FUNCTION SPACES—THE BANACK-STONE THEOREM
Introduction
Eilenberg's Theorem
The Nonsurjective case
A Theorem of Vesentini
Notes and Remarks
THE L(p) SPACES
Introduction
Lamperti's Results
Subspaces of L(p) and the Extension Theorem
Bochner Kernels
Notes and Remarks
ISOMETRIES OF SPACES OF ANALYTIC FUNCTIONS
Introduction
Isometries of the Hardy Spaces of the disk
Bergman spaces
Bloch Spaces
S(p) Spaces
Notes and Remarks
REARRANGEMENT INVARIANT SPACES
Introduction
Lumer's Method for Orlicz Spaces
Zaidenberg's Generalization
Musielak-Orlicz Spaces
Notes and Remarks
BANACH ALGEBRAS
Introduction
Kadison's Theorem
Subdifferentiability and Kadison's Theorem
The Nonsurjective Case of Kadison's theorem
The Algebras C(1) and AC
Douglas Algebras
Notes and Remarks
BIBLIOGRAPHY
INDEX

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