Stability of Operators and Operator Semigroups

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ISBN: 9783034601948 | 3034601948
Cover: Hardcover
Copyright: 10/30/2010

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The asymptotic behaviour, in particular "stability" in some sense, is studied systematically for discrete and for continuous linear dynamical systems on Banach spaces. Of particular concern is convergence to an equilibrium with respect to various topologies. Parallels and differences between the discrete and the continuous situation are emphasised.

Introduction	p. 1
Functional analytic tools	p. 5
Compact semigroups	p. 5
Preliminaries	p. 5
Relatively compact sets in L¿(X)	p. 7
Compact semitopological semigroups	p. 8
Abstract Jacobs-Glicksberg-de Leeuw decomposition for operator semigroups	p. 9
Operators with relatively weakly compact orbits	p. 12
Jacobs-Glicksberg-de Leeuw decomposition for C₀-semigroups	p. 12
Mean ergodicity	p. 13
Mean ergodic operators	p. 13
Uniformly mean ergodic operators	p. 18
Mean ergodic C₀-semigroups	p. 20
Uniformly mean ergodic C₀-semigroups	p. 22
Specific concepts from semigroup theory	p. 23
Cogenerator	p. 23
Inverse Laplace transform for C₀-semigroups	p. 26
Positivity in L(H)	p. 32
Preliminaries	p. 32
Spectral properties of positive operators	p. 33
Implemented operators	p. 34
Implemented semigroups	p. 35
Stability of linear operators	p. 37
Power boundedness	p. 37
Preliminaries	p. 37
Characterisation via resolvent	p. 40
Polynomial boundedness	p. 43
Strong stability	p. 45
Preliminaries	p. 46
Representation as shift operators	p. 48
Spectral conditions	p. 49
Characterisation via resolvent	p. 53
Weak stability	p. 55
Preliminaries	p. 55
Contractions on Hilbert spaces	p. 57
Characterisation via resolvent	p. 60
Almost weak stability	p. 61
Characterisation	p. 61
A concrete example	p. 66
Category theorems	p. 67
Unitary operators	p. 67
Isometries	p. 71
Contractions	p. 72
Stability via Lyapunov's equation	p. 74
Uniform exponential and strong stability	p. 74
Power boundedness	p. 76
Stability of C₀-semigroups	p. 79
Boundedness	p. 79
Preliminaries	p. 79
Representation as shift semigroups	p. 81
Characterisation via resolvent	p. 82
Polynomial boundedness	p. 85
Uniform exponential stability	p. 91
Spectral characterisation	p. 91
Spectral mapping theorems	p. 92
Theorem of Datko-Pazy	p. 94
Theorem of Gearhart	p. 96
Strong stability	p. 99
Preliminaries	p. 99
Representation as shift semigroups	p. 102
Spectral conditions	p. 103
Characterisation via resolvent	p. 105
Weak stability	p. 108
Preliminaries	p. 108
Contraction semigroups on Hilbert spaces	p. 109
Characterisation via resolvent	p. 112
Almost weak stability	p. 114
Characterisation	p. 114
Concrete example	p. 120
Category theorems	p. 122
Unitary case	p. 123
Isometric case	p. 126
Contractive case: remarks	p. 127
Stability via Lyapunov's equation	p. 129
Uniform exponential stability	p. 129
Strong stability	p. 130
Boundedness	p. 132
Connections to ergodic and measure theory	p. 133
Stability and the Rajchman property	p. 133
From stability to the Rajchman property and back	p. 133
Category result, discrete case	p. 137
Category result, continuous case	p. 141
Stability and mixing	p. 141
Ergodicity	p. 142
Strong and weak mixing	p. 144
Rigidity phenomena	p. 148
Preliminaries	p. 148
Discrete case: powers of operators	p. 150
Continuous case: C₀-semigroups	p. 156
Further remarks	p. 161
Discrete vs. continuous	p. 163
Embedding operators into C₀-semigroups	p. 163
Sufficient conditions via functional calculus	p. 164
A necessary condition	p. 165
Normal operators and projections on Hilbert spaces	p. 167
Isometries and co-isometries on Hilbert spaces	p. 168
Abstract examples: a residuality result	p. 170
Cogenerators	p. 172
(Power) boundedness	p. 172
Characterisation via cogenerators of the rescaled semigroups	p. 175
Examples	p. 178
Polynomial boundedness	p. 180
Strong stability	p. 181
Weak and almost weak stability	p. 182
Bibliography	p. 183
List of Symbols	p. 201
Index	p. 203
Table of Contents provided by Ingram. All Rights Reserved.

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Stability of Operators and Operator Semigroups

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