1st Edition

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

    244 Pages
    by Chapman & Hall

    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.

    Preface
    THE BANACHSTONE PROPERTY
    Introduction
    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
    Introduction
    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
    Introduction
    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
    Introduction
    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
    Introduction
    Isometries of CP
    Isometries of Symmetric Norm Ideals: Sourour’s Theorem
    Noncommutative LP Spaces
    Notes and Remarks
    Minimal and Maximal Norms
    Introduction
    An Infinite-Dimensional Space with Trivial Isometries
    Minimal Norms
    Maximal Norms and Forms of Transitivity
    Notes and Remarks
    Epilogue
    Reflexivity of the Isometry Group
    Adjoint Abelian Operators
    Almost Isometries
    Distance One Preserving Maps
    Spectral Isometries
    Isometric Equivalence
    Potpourri
    BIBLIOGRAPHY
    INDEX

    Biography

    Richard J. Fleming, James E. Jamison

    "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