Topics in Operator Semigroups (Progress in Mathematics, 281) - Hardcover

Kantorovitz, Shmuel

Hardcover

ISBN 10: 081764931X ISBN 13: 9780817649319

Publisher: Birkhäuser, 2009

Synopsis

This monograph is concerned with the interplay between the theory of operator semigroups and spectral theory. The basics on operator semigroups are concisely covered in this self-contained text. Part I deals with the Hille--Yosida and Lumer--Phillips characterizations of semigroup generators, the Trotter--Kato approximation theorem, Kato’s unified treatment of the exponential formula and the Trotter product formula, the Hille--Phillips perturbation theorem, and Stone’s representation of unitary semigroups. Part II explores generalizations of spectral theory’s connection to operator semigroups.

About this title

From the Back Cover

The theory of operator semigroups was essentially discovered in the early 1930s.  Since then, the theory has developed into a rich and exciting area of functional analysis and has been applied to various mathematical topics such as Markov processes, the abstract Cauchy problem, evolution equations, and mathematical physics.

This self-contained monograph focuses primarily on the theoretical connection between the theory of operator semigroups and spectral theory. Divided into three parts with a total of twelve distinct chapters, this book gives an in-depth account of the subject with numerous examples, detailed proofs, and a brief look at a few applications.

Topics include:

* The Hille–Yosida and Lumer–Phillips characterizations of semigroup generators

* The Trotter–Kato approximation theorem

* Kato’s unified treatment of the exponential formula and the Trotter product formula

* The Hille–Phillips perturbation theorem, and Stone’s representation of unitary semigroups

*  Generalizations of spectral theory’s connection to operator semigroups

* A natural generalization of Stone’s spectral integral representation to a Banach space setting

With a collection of miscellaneous exercises at the end of the book and an introductory chapter examining the basic theory involved, this monograph is suitable for second-year graduate students interested in operator semigroups.

Review

From the book reviews:

“This monograph is suitable for second-year graduate students, but it can be recommended also to any researcher interested in operator semigroups.” (László Kérchy, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2), 2012)

“The present graduate level text expands the previous lecture notes from the same author, Semigroups of operators and spectral theory ... . It begins with a succinct introduction to operator semigroups covering classical topics such as generators, the Hille-Yosida theorem, dissipative operators and the Lumer–Phillips theorem, the Trotter convergence theorem, exponential formulas, perturbation theory, Stone’s theorem, and analytic semigroups. ... The text is also intended for second-year graduate students ... . it will be a valuable source for researchers working in this area.” (G. Teschl, Monatshefte für Mathematik, Vol. 162 (4), April, 2011)

“This book is based on the author’s lecture notes ... in which the more advanced parts concentrated on spectral representations. ... There is also a presentation of a well-known stability theorem for semigroups under countable spectral conditions. ... The increased variety of topics covered will make the book more useful ... . Other advantages are the inclusion of an index and some exercises, considerable extensions of the bibliography and the list of contents, and more attractive typesetting.”­­­ (C. J. K. Batty, Mathematical Reviews, Issue 2010 k)

Edition details

• Publisher: Birkhäuser
• Publication date: 2009
• ISBN 10: 081764931X
• ISBN 13: 9780817649319
• Binding: Hardcover
• Language: English
• Number of pages: 280