Kantorovitz Topics in Operator Semigroups

2010
ISBN: 978-0-8176-4932-6
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 281, 266 Seiten

Reihe: Progress in Mathematics

This monograph is concerned with the interplay between the theory of operator semigroups and spectral theory. The basics on operator semigroups are concisely covered in this self-contained text. Part I deals with the Hille--Yosida and Lumer--Phillips characterizations of semigroup generators, the Trotter--Kato approximation theorem, Kato's unified treatment of the exponential formula and the Trotter product formula, the Hille--Phillips perturbation theorem, and Stone's representation of unitary semigroups. Part II explores generalizations of spectral theory's connection to operator semigroups.

Autoren/Hrsg.

Kantorovitz, Shmuel

Inhaltsverzeichnis

1;Contents;7
2;Preface;10
3;General Theory;13
3.1;A Basic Theory;14
3.1.1;A.1 Overview;14
3.1.2;A.2 The Generator;16
3.1.3;A.3 Type and Spectrum;20
3.1.4;A.4 Uniform Continuity;21
3.1.5;A.5 Core for the Generator;22
3.1.6;A.6 The Resolvent;24
3.1.7;A.7 Pseudo-Resolvents;26
3.1.8;A.8 The Laplace Transform;28
3.1.9;A.9 Abstract Potentials;29
3.1.10;A.10 The Hille–Yosida Theorem;31
3.1.11;A.11 The Hille–Yosida Space;33
3.1.12;A.12 Dissipative Operators;36
3.1.13;A.13 The Trotter–Kato Convergence Theorem;39
3.1.14;A.14 Exponential Formulas;43
3.1.15;A.15 Perturbation of Generators;47
3.1.16;A.16 Groups of Operators;53
3.1.17;A.17 Bounded Groups of Operators;54
3.1.18;A.18 Stone’s Theorem;55
3.1.19;A.19 Bochner’s Theorem;58
3.2;B The Semi-Simplicity Space for Groups;60
3.2.1;B.1 The Bochner Norm;60
3.2.2;B.2 The Semi-Simplicity Space;64
3.2.3;B.3 Scalar-Type Spectral Operators;70
3.3;C Analyticity;73
3.3.1;C.1 Analytic Semigroups;73
3.3.2;C.2 The Generator of an Analytic Semigroup;75
3.4;D The Semigroup as a Function of its Generator;80
3.4.1;D.1 Noncommutative Taylor Formula;80
3.4.2;D.2 Analytic Families of Semigroups;88
3.5;E Large Parameter;96
3.5.1;E.1 Analytic Semigroups;96
3.5.2;E.2 Resolvent Iterates;99
3.5.3;E.3 Mean Stability;103
3.5.4;E.4 The Asymptotic Space;112
3.5.5;E.5 Semigroups of Isometries;116
3.5.6;E.6 The ABLV Stability Theorem;118
3.6;F Boundary Values;122
3.6.1;F.1 Regular Semigroups and Boundary Values;122
3.6.2;F.2 The Generator of a Regular Semigroup;127
3.6.3;F.3 Examples of Regular Semigroups;130
3.7;G Pre-Semigroups;140
3.7.1;G.1 The Abstract Cauchy Problem;141
3.7.2;G.2 The Exponentially Tamed Case;145
4;Integral Representations;148
4.1;A The Semi-Simplicity Space;149
4.1.1;A.1 The Real Spectrum Case;149
4.1.2;A.2 The Case R+ .(-A);162
4.2;B The Laplace–Stieltjes Space;169
4.2.1;B.1 The Laplace–Stieltjes Space;169
4.2.2;B.2 Semigroups of Closed Operators;174
4.2.3;B.3 The Integrated Laplace Space;177
4.2.4;B.4 Integrated Semigroups;181
4.3;C Families of Unbounded Symmetric Operators;184
4.3.1;C.1 Local Symmetric Semigroups;184
4.3.2;C.2 Nelson’s Analytic Vectors Theorem;188
4.3.3;C.3 Local Bounded Below Cosine Families;190
4.3.4;C.4 Local Symmetric Cosine Families;194
5;A Taste of Applications;198
5.1;Prelude;199
5.2;A Analytic Families of Evolution Systems;200
5.2.1;A.1 Coefficients Analyticity and Solutions Analyticity;200
5.2.2;A.2 Kato’s Conditions;201
5.2.3;A.3 Tanabe’s Conditions;203
5.3;B Similarity;207
5.3.1;B.1 Overview;207
5.3.2;B.2 Similarity Within the Family S + V;207
5.3.3;B.3 Similarity of Certain Perturbations;221
5.4;Miscellaneous Exercises;223
5.4.1;Abstract Landau Inequality;223
5.4.2;Variation on the Theme of Dissipativity;223
5.4.3;Resolvents of the Hille–Yosida Approximations;224
5.4.4;Adjoint Semigroup;224
5.4.5;Spectra of a Semigroup and its Generator;225
5.4.6;Compact Semigroups;226
5.4.7;Powers of the Generator;227
5.4.8;C8-semigroups;228
5.4.9;Entire Vectors;228
5.4.10;Nonhomogeneous ACP;228
5.4.11;The Graph Norm on D(A);229
5.4.12;Commutativity;229
5.4.13;Square of the Generator;229
5.4.14;Resolvents of Bounded Analytic Semigroups;230
5.4.15;A-boundedness;230
5.4.16;Unitary Vectors;231
5.4.17;Markov Semigroups;231
5.4.18;Translation Semigroup;232
5.4.19;The MacLaurin Formula for Semigroups;233
5.4.20;Restriction of Semigroup to Invariant Subspaces;233
5.4.21;Semigroups Arising from ACP;234
5.4.22;Bounded Below Semigroups;235
5.4.23;Natural Operational Calculus for Groups;235
5.4.24;Construction of Analytic Semigroups;236
5.4.25;Approximation of Co- semigroups by Uniformly Continuous Semigroups;237
5.4.26;Stability in the u.o.t;238
5.4.27;Semigroups on Hilbert Space;239
5.4.28;Stability in the u.o.t. on Hilbert Space;239
5.4.29;Hille–Yosida Space, Semi-Simplicity Space, etc.;241
5.4.30;Approximation Formula for the Integrated Semigroup;245
5.4.31;Semigroup Induced on Quotient Space;246
5.4.32;Semigroup Induced on l8(X);246
5.4.33;Semigroup Induced on a Tensor Space;247
5.4.34;Infinite Product of Semigroups;247
5.4.35;Perturbation of Generator by B B([D(A)]);248
5.4.36;Intertwining and Spectrum;249
5.4.37;Mining Lemma 2.16;250
5.4.38;The Eberlein and Schoenberg Criteria for Fourier– Stieltjes Transforms;251
5.5;Notes and References;253
5.5.1;Part I. General Theory;253
5.5.1.1;A. Basic Theory;253
5.5.1.2;B. The Semi-simplicity Space for Groups;254
5.5.1.3;C. Analyticity;254
5.5.1.4;D. The Semigroup as a Function of its Generator;254
5.5.1.5;E. Large Parameter;254
5.5.1.6;F. Boundary Values;254
5.5.1.7;G. Pre-Semigroups;254
5.5.2;Part II. Integral Representations ;255
5.5.2.1;A. The Semi-Simplicity Space;255
5.5.2.2;B. The Laplace–Stieltjes Space;255
5.5.2.3;C. Families of Unbounded Symmetric Operators;255
5.5.3;Part III. A Taste of Applications ;255
5.5.3.1;A. Dependence on Parameters;255
5.5.3.2;B. Similarity (etc.);256
6;Bibliography;257
7;Index;266