Texts and Readings in Mathematics

This book combines the spirit of a textbook and of a monograph on the topic of Semigroups and their applications. It is expected to have potential users across a broad spectrum including operator theory, partial differential equations, harmonic analysis, probability and statistics and classical and quantum mechanics. A reasonable amount of familiarity with real analysis, including the Lebesgue-integration theory, basic functional analysis and bounded linear operators is assumed. However, any discourse on a theory of semigroups needs an introduction to unbounded linear operators, some elements of which have been included in the Appendix, along with the basic ideas of the Fourier transform and of Sobolev spaces. The chapters 4 through 6 contain advanced material, not often found in textbooks, but which have many interesting applications such as the Feynman-Kac formula, the central limit theorem and the construction of Markov semigroups. The exercises are given in the text as the topics are developed, so that the interested reader can be persuaded to solve these as a part of learning that topic.
1 Vector-valued functions
2 C0-semigroups
3 Dissipative operators and holomorphic semigroups
4 Perturbation and convergence of semigroups
5 Chernoff's Theorem and its applications
6 Markov semigroups
7 Applications to partial differential equations

Appendix
A.1 Unbounded operators
A.2 Fourier transforms
A.3 Sobolev spaces
References
Index