1st Edition

Isometries in Banach Spaces
Vector-valued Function Spaces and Operator Spaces, Volume Two

ISBN 9781584883869
Published November 15, 2007 by Chapman and Hall/CRC
248 Pages

USD $180.00

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Book Description

A continuation of the authors’ previous book, Isometries on Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two covers much of the work that has been done on characterizing isometries on various Banach spaces.

Picking up where the first volume left off, the book begins with a chapter on the Banach–Stone property. The authors consider the case where the isometry is from C0(Q, X) to C0(K, Y) so that the property involves pairs (X, Y) of spaces. The next chapter examines spaces X for which the isometries on LP(μ, X) can be described as a generalization of the form given by Lamperti in the scalar case. The book then studies isometries on direct sums of Banach and Hilbert spaces, isometries on spaces of matrices with a variety of norms, and isometries on Schatten classes. It subsequently highlights spaces on which the group of isometries is maximal or minimal. The final chapter addresses more peripheral topics, such as adjoint abelian operators and spectral isometries.

Essentially self-contained, this reference explores a fundamental aspect of Banach space theory. Suitable for both experts and newcomers to the field, it offers many references to provide solid coverage of the literature on isometries.

Table of Contents

Strictly Convex Spaces and Jerison’s Theorem
M Summands and Cambern’s Theorem
Centralizers, Function Modules, and Behrend’s Theorem
The Nonsurjective Vector-Valued Case
The Nonsurjective Case for Nice Operators
Notes and Remarks
The Banach–Stone Property for Bochner Spaces
LP Functions with Values in Hilbert Space
LP Functions with Values in Banach Space
L2 Functions with Values in Banach Space
Notes and Remarks
Orthogonal Decompostions
Sequence Space Decompositions
Hermitian Elements and Orthonormal Systems
The Case for Real Scalars: Functional Hilbertian Sums
Decompositions with Banach Space Factors
Notes and Remarks
Matrix Spaces
Morita’s Proof of Schur’s Theorem
Isometries for (p, k) Norms on Square Matrix Spaces
Isometries for (p, k) Norms on Rectangular Matrix Spaces
Notes and Remarks
Isometries of Norm Ideals of Operators
Isometries of CP
Isometries of Symmetric Norm Ideals: Sourour’s Theorem
Noncommutative LP Spaces
Notes and Remarks
Minimal and Maximal Norms
An Infinite-Dimensional Space with Trivial Isometries
Minimal Norms
Maximal Norms and Forms of Transitivity
Notes and Remarks
Reflexivity of the Isometry Group
Adjoint Abelian Operators
Almost Isometries
Distance One Preserving Maps
Spectral Isometries
Isometric Equivalence

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"This is a well-written, highly self-contained book which presents the results and their proofs in an accessible way. The results are complemented with an interesting Notes and Remarks section at the end of each chapter which points the interested reader to paths for further investigation. An extensive bibliography is provided. The two volumes provide not only a very good introduction to the subject but also a nice reference tool for experts."
—Miguel Martin, Mathematical Reviews, Issue 2009i

Praise for Volume One:
"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