One-Parameter Semigroups for Linear Evolution Equations,9780387984636

One-Parameter Semigroups for Linear Evolution Equations

by ; ; ; ; ; ; ; ;
ISBN13: 9780387984636
ISBN10: 0387984631
Format: Hardcover
Pub. Date: 1999-11-01
Publisher(s): Springer Verlag
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Summary

This book gives an up-to-date account of the theory of strongly continuous one-parameter semigroups of linear operators. It includes a systematic discussion of the spectral theory and the long-term behavior of such semigroups. A special feature of the text is an unusually wide range of applications, e.g., to ordinary and partial differential operators, delay and Volterra equations and to control theory, and an emphasis on philosophical motivation and the historical background. The book is written for students, but should also be of value for researchers interested in this field.

Table of Contents

Preface vii
Prelude xvii
Linear Dynamical Systems
1(46)
Cauchy's Functional Equation
2(4)
Finite-Dimensional Systems: Matrix Semigroups
6(8)
Uniformly Continuous Operator Semigroups
14(10)
More Semigroups
24(12)
Multiplication Semigroups on C0(Ω)
24(6)
Multiplication Semigroups on Lp(Ω, μ)
30(3)
Translation Semigroups
33(3)
Strongly Continuous Semigroups
36(11)
Basic Properties
37(5)
Standard Constructions
42(4)
Notes
46(1)
Semigroups, Generators, and Resolvents
47(110)
Generators of Semigroups and Their Resolvents
48(11)
Examples Revisited
59(11)
Standard Constructions
59(6)
Standard Examples
65(5)
Hille--Yosida Generation Theorems
70(26)
Generation of Groups and Semigroups
71(11)
Dissipative Operators and Contraction Semigroups
82(7)
More Examples
89(7)
Special Classes of Semigroups
96(27)
Analytic Semigroups
96(13)
Differentiable Semigroups
109(3)
Eventually Norm-Continuous Semigroups
112(5)
Eventually Compact Semigroups
117(3)
Examples
120(4)
Interpolation and Extrapolation Spaces for Semigroups
123(22)
Simon Brendle
Sobolev Towers
124(5)
Favard and Abstract Holder Spaces
129(8)
Fractional Powers
137(8)
Well-Posedness for Evolution Equations
145(12)
Notes
154(3)
Perturbation and Approximation of Semigroups
157(81)
Bounded Perturbations
157(12)
Perturbations of Contractive and Analytic Semigroups
169(13)
More Perturbations
182(23)
The Perturbation Theorem of Desch--Schappacher
182(10)
Comparison of Semigroups
192(3)
The Perturbation Theorem of Miyadera--Voigt
195(6)
Additive Versus Multiplicative Perturbations
201(4)
Trotter--Kato Approximation Theorems
205(14)
A Technical Tool: Pseudoresolvents
206(3)
The Approximation Theorems
209(5)
Examples
214(5)
Approximation Formulas
219(19)
Chernoff Product Formula
219(12)
Inversion Formulas
231(5)
Notes
236(2)
Spectral Theory for Semigroups and Generators
238(57)
Spectral Theory for Closed Operators
239(11)
Spectrum of Semigroups and Generators
250(20)
Basic Theory
250(9)
Spectrum of Induced Semigroups
259(7)
Spectrum of Periodic Semigroups
266(4)
Spectral Mapping Theorems
270(19)
Examples and Counterexamples
270(5)
Spectral Mapping Theorems for Semigroups
275(8)
Weak Spectral Mapping Theorem for Bounded Groups
283(6)
Spectral Theory and Perturbation
289(6)
Notes
293(2)
Asymptotics of Semigroups
295(52)
Stability and Hyperbolicity for Semigroups
296(12)
Stability Concepts
296(3)
Characterization of Uniform Exponential Stability
299(6)
Hyperbolic Decompositions
305(3)
Compact Semigroups
308(21)
General Semigroups
308(4)
Weakly Compact Semigroups
312(5)
Strongly Compact Semigroups
317(12)
Eventually Compact and Quasi-compact Semigroups
329(8)
Mean Ergodic Semigroups
337(10)
Notes
345(2)
Semigroups Everywhere
347(36)
Semigroups for Population Equations
348(13)
Semigroup Method for the Cell Equation
349(4)
Intermezzo on Positive Semigroups
353(5)
Asymptotics for the Cell Equation
358(3)
Notes
361(1)
Semigroups for the Transport Equation
361(6)
Solution Semigroup for the Reactor Problem
361(3)
Spectral and Asymptotic Behavior
364(3)
Notes
367(1)
Semigroups for Second-Order Cauchy Problems
367(16)
The State Space X = XB1 x X
369(3)
The State Space X = X x X
372(2)
The State Space X XC1 x X
374(8)
Notes
382(1)
Semigroups for ordinary Differential Operators
383(114)
M. Campiti
G. Metafune
D. Pallara
S. Romanelli
Nondegenerate Operators on R and R+
384(4)
Nondegenerate Operators on Bounded Intervals
388(2)
Degenerate Operators
390(10)
Analyticity of Degenerate Semigroups
400(3)
Notes
403(1)
Semigroups for Partial Differential Operators
404(15)
Abdelaziz Rhandi
Notation and Preliminary Results
405(3)
Elliptic Differential Operators with Constant Coefficients
408(3)
Elliptic Differential Operators with Variable Coefficients
411(8)
Notes
419(1)
Semigroups for Delay Differential Equations
419(16)
Well-Posedness of Abstract Delay Differential Equations
420(4)
Regularity and Asymptotics
424(4)
Positivity for Delay Differential Equations
428(7)
Notes
435(1)
Semigroups for Volterra Equations
435(17)
Mild and Classical Solutions
436(6)
Optimal Regularity
442(5)
Integro-Differential Equations
447(5)
Notes
452(1)
Semigroups for Control Theory
452(25)
Controllability
456(10)
Observability
466(2)
Stabilizability and Detectability
468(5)
Transfer Functions and Stability
473(3)
Notes
476(1)
Semigroups for Nonautonomous Cauchy Problems
477(20)
Roland Schnaubelt
Cauchy Problems and Evolution Families
477(4)
Evolution Semigroups
481(6)
Perturbation Theory
487(5)
Hyperbolic Evolution Families in the Parabolic Case
492(4)
Notes
496(1)
A Brief History of the Exponential Function
497(34)
Tanja Hahn
Carla Perazzoli
A Bird's-Eye View
497(3)
The Functional Equation
500(2)
The Differential Equation
502(4)
The Birth of Semigroup Theory
506(3)
Appendix
A. A Reminder of Some Functional Analysis
509(6)
B. A Reminder of Some Operator Theory
515(7)
C. Vector-Valued Integration
522(9)
The Bochner Integral
522(4)
The Fourier Transform
526(4)
The Laplace Transform
530(1)
Epilogue Determinism: Scenes from the Interplay Between Metaphysics and Mathematics 531(24)
Gregor Nickel
1. The Mathematical Structure
533(3)
2. Are Relativity, Quantum Mechanics, and Chaos Deterministic?
536(2)
3. Determinism in Mathematical Science from Newton to Einstein
538(8)
4. Developments in the Concept of Object from Leibniz to Kant
546(3)
5. Back to Some Roots of Our Problem: Motion in History
549(4)
6. Bibliography and Further Reading
553(2)
References 555(22)
List of Symbols and Abbreviations 577(3)
Index 580

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