One-Parameter Semigroups for Linear Evolution Equations

by Engel, Klaus-Jochen; Nagel, Rainer; Brendle, S.; Campiti, M.; Hahn, T.; Metafune, G.; Nickel, G.; Pallara, D.; Perazzoli, C.

• ISBN13: 9780387984636
• ISBN10: 0387984631
• Format: Hardcover
• Copyright: 1999-11-01
• Publisher: Springer Verlag

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