Semigroups of Linear Operators: With Applications to Analysis, Probability and Physics

Description

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.

David Applebaum is Professor of Mathematics at the University of Sheffield. His specialist research area is stochastic analysis, with particular emphasis on analytic and probabilistic aspects of processes with jumps on Lie groups, symmetric spaces and manifolds.

'... Applebaum has written a book that provides substantial depth and rigor, with a plethora of references. A notable feature of the text that increases its appeal is the author's inclusion of applications of the theory of semigroups to partial differential equations, dynamical systems, physics, and probability. This book also includes several advanced topics-such as measure spaces, spectral decompositions, and fractional calculus-but Applebaum offers motivating examples for readers to consider, interesting exercises to increase their comprehension, and additional resources to help them find complete details, so that a student could successfully navigate through this material independently if need be.' M. Clay, Choice 'Overall, this book is an interesting contribution to the semigroup literature which does not follow a standard route.' Eric Stachura, MAA Reviews 'Experts can quickly browse through any of the chapters, and get nicely acquainted with examples they are not yet fully aware of. Students can read this book fairly casually, and gain great motivation to study functional, stochastic, and/or harmonic analysis further. Last but not least, teachers of graduate courses can design several great courses by elaborating on one of the many threads running through the book under review and using the referred sources to turn them into self-contained stories. All will appreciate the book's excellent mix of erudition and pedagogy.' Pierre Portal, MathSciNet 'Some readers will enjoy the topic for its inherent attraction as a means of presenting results in a simple and widely applicable way. A masters student who is interested in researching in analysis but not in technical details of PDEs may nd this text particularly useful for finding a research topic in one of the related areas. In these respects the book achieves the aims declared in its introduction, in a way which is not found in earlier texts.' Charles Batty, The Mathematical Gazette