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Zbl 0592.47034
Goldstein, Jerome A.
Semigroups of linear operators and applications. (English)
[B] Oxford Mathematical Monographs. New York: Oxford University Press; Oxford: Clarendon Press. X, 245 p. \sterling 40.00 (1985).

The book is subdivided into two chapters: 1. Semigroups of Linear Operators, 2. Linear Cauchy Problems. \par Chapter 1 is mainly devoted to theoretical aspects of $(C\sb 0)$ 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. \par Chapter 2, on the one hand, contains additional theory of $(C\sb 0)$ 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. \par 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. \par The "Historical Notes and Remarks" at the end of each chapter are informative, and in fact essential for finding proofs of unproved results. \par 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.
[J.Voigt]

MSC 2000:: *47D03 (Semi)groups of linear operators
47-02 Research monographs (operator theory)
35G10 Initial value problems for linear higher-order PDE
35K25 Higher order parabolic equations, general
47A40 Scattering theory of linear operators

Keywords: Semigroups of Linear Operators; Hille-Yosida theorem; semigroups in Hilbert space; analytic semigroups; perturbation; approximation; Feynman path formula; mean ergodic theorem; fractional powers; homogeneous and inhomogeneous equations; time dependent equations; cosine functions; parabolic equations; symmetric hyperbolic systems; mixed problems; singular perturbations; scattering theory; initial value problems for parabolic and hyperbolic differential; equations; initial value problems for parabolic and hyperbolic differential equations

Cited in: Zbl 1235.47001 Zbl 1053.47036 Zbl 0981.47010 Zbl 0908.47039

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