Semigroups of linear operators and applications.

*(English)*Zbl 0592.47034
Oxford Mathematical Monographs. New York: Oxford University Press; Oxford: Clarendon Press. X, 245 p. £40.00 (1985).

The book is subdivided into two chapters: 1. Semigroups of Linear Operators, 2. Linear Cauchy Problems.

Chapter 1 is mainly devoted to theoretical aspects of \((C_ 0)\) semigroups: Hille-Yosida theorem, semigroups in Hilbert space, analytic semigroups, perturbation and approximation theory, just to give the main topics. In sections on applications and further developments one fineds the Feynman path formula, the mean ergodic theorem, fractional powers of generators, and further topics.

Chapter 2, on the one hand, contains additional theory of \((C_ 0)\) semigroups and related topics: homogeneous and inhomogeneous equations, time dependent equations, cosine functions and others. Further, Chapter 2 contains applications: parabolic equations, symmetric hyperbolic systems, mixed problems, singular perturbations, scattering theory, and others.

From this incomplete list of topics the wide scope of the book may already be apparent. The author hops ”that this work will be of use to graduate students in science and engineering as well as in mathematics.... An effort was made to solve some nontrivial initial value problems for parabolic and hyperbolic differential equations without doing the hard work associated with elliptic theory.” In fact, the book will be valuable for the advanced reader as well as for the expert. In Chapter 2, in particular, the author has included numerous results for which no proofs are given in the book. In this way, several topics are covered which otherwise would exceed the framework of the treatise.

The ”Historical Notes and Remarks” at the end of each chapter are informative, and in fact essential for finding proofs of unproved results.

It remains to mention the extensive list of references (51 pages) which definitely should be extremely interesting and useful for anybody working in this area.

Chapter 1 is mainly devoted to theoretical aspects of \((C_ 0)\) semigroups: Hille-Yosida theorem, semigroups in Hilbert space, analytic semigroups, perturbation and approximation theory, just to give the main topics. In sections on applications and further developments one fineds the Feynman path formula, the mean ergodic theorem, fractional powers of generators, and further topics.

Chapter 2, on the one hand, contains additional theory of \((C_ 0)\) semigroups and related topics: homogeneous and inhomogeneous equations, time dependent equations, cosine functions and others. Further, Chapter 2 contains applications: parabolic equations, symmetric hyperbolic systems, mixed problems, singular perturbations, scattering theory, and others.

From this incomplete list of topics the wide scope of the book may already be apparent. The author hops ”that this work will be of use to graduate students in science and engineering as well as in mathematics.... An effort was made to solve some nontrivial initial value problems for parabolic and hyperbolic differential equations without doing the hard work associated with elliptic theory.” In fact, the book will be valuable for the advanced reader as well as for the expert. In Chapter 2, in particular, the author has included numerous results for which no proofs are given in the book. In this way, several topics are covered which otherwise would exceed the framework of the treatise.

The ”Historical Notes and Remarks” at the end of each chapter are informative, and in fact essential for finding proofs of unproved results.

It remains to mention the extensive list of references (51 pages) which definitely should be extremely interesting and useful for anybody working in this area.

Reviewer: J.Voigt

##### MSC:

47D03 | Groups and semigroups of linear operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

47A40 | Scattering theory of linear operators |