Isometries in Banach Spaces: Vector-valued Function Spaces and Operator Spaces, Volume Two, 1st Edition (Hardback) book cover

Isometries in Banach Spaces

Vector-valued Function Spaces and Operator Spaces, Volume Two, 1st Edition

By Richard J. Fleming, James E. Jamison

Chapman and Hall/CRC

248 pages

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pub: 2007-11-15
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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.


"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

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




About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
MATHEMATICS / Functional Analysis