Topics in Operator Semigroups (eBook)

Contents 7
Preface 10
General Theory 13
A Basic Theory 14
A.1 Overview 14
A.2 The Generator 16
A.3 Type and Spectrum 20
A.4 Uniform Continuity 21
A.5 Core for the Generator 22
A.6 The Resolvent 24
A.7 Pseudo-Resolvents 26
A.8 The Laplace Transform 28
A.9 Abstract Potentials 29
A.10 The Hille–Yosida Theorem 31
A.11 The Hille–Yosida Space 33
A.12 Dissipative Operators 36
A.13 The Trotter–Kato Convergence Theorem 39
A.14 Exponential Formulas 43
A.15 Perturbation of Generators 47
A.16 Groups of Operators 53
A.17 Bounded Groups of Operators 54
A.18 Stone’s Theorem 55
A.19 Bochner’s Theorem 58
B The Semi-Simplicity Space for Groups 60
B.1 The Bochner Norm 60
B.2 The Semi-Simplicity Space 64
B.3 Scalar-Type Spectral Operators 70
C Analyticity 73
C.1 Analytic Semigroups 73
C.2 The Generator of an Analytic Semigroup 75
D The Semigroup as a Function of its Generator 80
D.1 Noncommutative Taylor Formula 80
D.2 Analytic Families of Semigroups 88
E Large Parameter 96
E.1 Analytic Semigroups 96
E.2 Resolvent Iterates 99
E.3 Mean Stability 103
E.4 The Asymptotic Space 112
E.5 Semigroups of Isometries 116
E.6 The ABLV Stability Theorem 118
F Boundary Values 122
F.1 Regular Semigroups and Boundary Values 122
F.2 The Generator of a Regular Semigroup 127
F.3 Examples of Regular Semigroups 130
G Pre-Semigroups 140
G.1 The Abstract Cauchy Problem 141
G.2 The Exponentially Tamed Case 145
Integral Representations 148
A The Semi-Simplicity Space 149
A.1 The Real Spectrum Case 149
A.2 The Case R+ .(-A) 162
B The Laplace–Stieltjes Space 169
B.1 The Laplace–Stieltjes Space 169
B.2 Semigroups of Closed Operators 174
B.3 The Integrated Laplace Space 177
B.4 Integrated Semigroups 181
C Families of Unbounded Symmetric Operators 184
C.1 Local Symmetric Semigroups 184
C.2 Nelson’s Analytic Vectors Theorem 188
C.3 Local Bounded Below Cosine Families 190
C.4 Local Symmetric Cosine Families 194
A Taste of Applications 198
Prelude 199
A Analytic Families of Evolution Systems 200
A.1 Coefficients Analyticity and Solutions Analyticity 200
A.2 Kato’s Conditions 201
A.3 Tanabe’s Conditions 203
B Similarity 207
B.1 Overview 207
B.2 Similarity Within the Family S + V 207
B.3 Similarity of Certain Perturbations 221
Miscellaneous Exercises 223
Abstract Landau Inequality 223
Variation on the Theme of Dissipativity 223
Resolvents of the Hille–Yosida Approximations 224
Adjoint Semigroup 224
Spectra of a Semigroup and its Generator 225
Compact Semigroups 226
Powers of the Generator 227
C8-semigroups 228
Entire Vectors 228
Nonhomogeneous ACP 228
The Graph Norm on D(A) 229
Commutativity 229
Square of the Generator 229
Resolvents of Bounded Analytic Semigroups 230
A-boundedness 230
Unitary Vectors 231
Markov Semigroups 231
Translation Semigroup 232
The MacLaurin Formula for Semigroups 233
Restriction of Semigroup to Invariant Subspaces 233
Semigroups Arising from ACP 234
Bounded Below Semigroups 235
Natural Operational Calculus for Groups 235
Construction of Analytic Semigroups 236
Approximation of Co- semigroups by Uniformly Continuous Semigroups 237
Stability in the u.o.t 238
Semigroups on Hilbert Space 239
Stability in the u.o.t. on Hilbert Space 239
Hille–Yosida Space, Semi-Simplicity Space, etc. 241
Approximation Formula for the Integrated Semigroup 245
Semigroup Induced on Quotient Space 246
Semigroup Induced on l8(X) 246
Semigroup Induced on a Tensor Space 247
Infinite Product of Semigroups 247
Perturbation of Generator by B B([D(A)]) 248
Intertwining and Spectrum 249
Mining Lemma 2.16 250
The Eberlein and Schoenberg Criteria for Fourier– Stieltjes Transforms 251
Notes and References 253
Part I. General Theory 253
A. Basic Theory 253
B. The Semi-simplicity Space for Groups 254
C. Analyticity 254
D. The Semigroup as a Function of its Generator 254
E. Large Parameter 254
F. Boundary Values 254
G. Pre-Semigroups 254
Part II. Integral Representations 255
A. The Semi-Simplicity Space 255
B. The Laplace–Stieltjes Space 255
C. Families of Unbounded Symmetric Operators 255
Part III. A Taste of Applications 255
A. Dependence on Parameters 255
B. Similarity (etc.) 256
Bibliography 257
Index 266