Isometries on Banach spaces: vector-valued function spaces and operator spaces

Richard J. Fleming, James E. Jamison1584883863, 9781584883869, 9781420010206

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 C 0( Q , X ) to C 0( 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.

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Table of contents :
ISOMETRIES ON BANACH SPACES: VECTOR-VALUED FUNCTION SPACES, Volume 2……Page 3
Contents……Page 5
Preface……Page 7
Bibliography……Page 219
7.1. Introduction……Page 10
7.2. Strictly Convex Spaces and Jerison’s Theorem……Page 12
7.3. M Summands and Cambern’s Theorem……Page 19
7.4. Centralizers, Function Modules, and Behrends’ Theorem……Page 27
7.5. The Nonsurjective Vector-Valued Case……Page 37
7.6. The Nonsurjective Case for Nice Operators……Page 45
7.7. Notes and Remarks……Page 55
8.1. Introduction……Page 59
8.2. Lp Functions with Values in Hilbert Space……Page 61
8.3. Lp Functions with Values in Banach Space……Page 71
8.4. L2 Functions with Values in a Banach Space……Page 80
8.5. Notes and Remarks……Page 85
9.1. Introduction……Page 91
9.2. Sequence Space Decompositions……Page 92
9.3. Hermitian Elements and Orthonormal Systems……Page 105
9.4. The Case for Real Scalars: Functional Hilbertian Sums……Page 113
9.5. Decompositions with Banach Space Factors……Page 123
9.6. Notes and Remarks……Page 136
10.1. Introduction……Page 144
10.2. Morita’s Proof of Schur’s Theorem……Page 145
10.3. Isometries for ( p, k) Norms on Square Matrix Spaces……Page 147
10.4. Isometries for ( p, k) Norms on Rectangular Matrix Spaces……Page 154
10.5. Notes and Remarks……Page 162
11.1. Introduction……Page 165
11.2. Isometries of Cp……Page 166
11.3. Isometries of Symmetric Norm Ideals: Sourour’s Theorem……Page 172
11.4. Noncommutative Lp Spaces……Page 178
11.5. Notes and Remarks……Page 185
12.1. Introduction……Page 189
12.2. An Infinite-Dimensional Space with Trivial Isometries……Page 190
12.3. Minimal Norms……Page 192
12.4. Maximal Norms and Forms of Transitivity……Page 196
12.5. Notes and Remarks……Page 202
13.1. Reflexivity of the Isometry Group……Page 207
13.2. Adjoint Abelian Operators……Page 210
13.3. Almost Isometries……Page 213
13.5. Spectral Isometries……Page 216
13.6. Isometric Equivalence……Page 217
13.7. Potpourri……Page 218