The theory of semigroups of operators is one of the most important themes in modern analysis. Not only does it have great intellectual beauty, but also wide-ranging applications. In this book the author first presents the essential elements of the theory, introducing the notions of semigroup, generator and resolvent, and establishes the key theorems of Hille-Yosida and Lumer-Phillips that give conditions for a linear operator to generate a semigroup. He then presents a mixture of applications and further developments of the theory. This includes a description of how semigroups are used to solve parabolic partial differential equations, applications to Levy and Feller-Markov processes, Koopmanism in relation to dynamical systems, quantum dynamical semigroups, and applications to generalisations of the Riemann-Liouville fractional integral. Along the way the reader encounters several important ideas in modern analysis including Sobolev spaces, pseudo-differential operators and the Nash inequality.
Author(s): David Applebaum
Series: London Mathematical Society Student Texts (Book 93)
Edition: 1
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 236
Cover......Page 1
LONDON MATHEMATICAL SOCIETY STUDENT TEXTS......Page 3
Semigroups of Linear Operators:
With Applications to Analysis, Probability and Physics......Page 5
Copyright
......Page 6
Dedication
......Page 7
Epigraph
......Page 8
Contents
......Page 9
Introduction......Page 13
1 Semigroups and Generators......Page 21
2 The Generation of Semigroups......Page 43
3 Convolution Semigroups of Measures......Page 58
4 Self-Adjoint Semigroups and Unitary Groups......Page 95
5 Compact and Trace Class Semigroups......Page 114
6 Perturbation Theory......Page 128
7 Markov and Feller Semigroups......Page 141
8 Semigroups and Dynamics......Page 161
9 Varopoulos Semigroups......Page 178
Notes and Further Reading......Page 191
Appendix A.
The Space C0(Rd)......Page 194
Appendix B.
The Fourier Transform......Page 198
Appendix C.
Sobolev Spaces......Page 202
Appendix D.
Probability Measures and Kolmogorov’s
Theorem on Construction of Stochastic
Processes......Page 206
Appendix E.
Absolute Continuity, Conditional Expectation
and Martingales......Page 209
Appendix F.
Stochastic Integration and Itô’s Formula......Page 217
Appendix G.
Measures on Locally Compact Spaces – Some
Brief Remarks......Page 224
References
......Page 226
Index......Page 231
