Semigroups of operators and spectral theory.

*(English)*Zbl 0865.47028
Pitman Research Notes in Mathematics Series. 330. Harlow: Longman Scientific & Technical. 135 p. (1995).

This well organized book consists of two parts. Part I introduces the reader to the theory of semigroups from the point of view of its application to spectral theory. In particular, the Hille-Yosida theory, Trotter-Kato convergence theorem, exponential formulas and the Hille-Phillips perturbation theorem have been presented. This part includes also more recent results, due mainly to the author. Among them is a Banach space version of the Stone theorem on the existence of a spectral measure, the notion of a semi-simplicity manifold, the analyticity problem for semigroups and a non-commutative Taylor formula for strongly continuous semigroups as functions of their generators.

Part II describes some recent generalizations of the theory presented in the first part of the book. Firstly, by weakening the elementary properties characterizing semigroups, the notion of a pre-semigroup is introduced and studied similarly as in the case of semigroups. Here the abstract Cauchy problem has the following form: \[ u'=Au, \quad u(0)=S(0)x \qquad(\text{on }[0,\infty)), \] where \(x\in D(A)\) and \(A\) generates the pre-semigroup \(S(.)\).

Semi-simplicity manifolds are next carefully discussed both for operators \(A\) with real spectrum for which \(iA\) is not assumed to generate a \(C_0\)-group and for operators \(A\) with spectrum in a half-plane. The operational calculus on the semi-simplicity manifold is presented in the case of reflexive Banach spaces. One section is concerned with the Laplace-Stieltjes space and the integrated Laplace space for a family of closed operators, with application to the spectral integral representation of semigroups of closed operators.

Generalizing Stone’s theorem the Klein-Laundau theory of semigroups of unbounded symmetric operators is developed. The second order abstract Cauchy problem: \[ u''=Au, \qquad u(0)=x, \qquad u'(0)=0, \] in a Banach space is associated in a natural manner to the so-called “cosine operator functions”. The theory of local cosine families of operators is presented in the last section. Bibliographical notes and comments on references close the book. This research notes volume will certainly be of interest to graduate students and researchers working in operator theory. It should also appeal to researchers in mathematical physics applying spectral theory and semigroups of operators.

Part II describes some recent generalizations of the theory presented in the first part of the book. Firstly, by weakening the elementary properties characterizing semigroups, the notion of a pre-semigroup is introduced and studied similarly as in the case of semigroups. Here the abstract Cauchy problem has the following form: \[ u'=Au, \quad u(0)=S(0)x \qquad(\text{on }[0,\infty)), \] where \(x\in D(A)\) and \(A\) generates the pre-semigroup \(S(.)\).

Semi-simplicity manifolds are next carefully discussed both for operators \(A\) with real spectrum for which \(iA\) is not assumed to generate a \(C_0\)-group and for operators \(A\) with spectrum in a half-plane. The operational calculus on the semi-simplicity manifold is presented in the case of reflexive Banach spaces. One section is concerned with the Laplace-Stieltjes space and the integrated Laplace space for a family of closed operators, with application to the spectral integral representation of semigroups of closed operators.

Generalizing Stone’s theorem the Klein-Laundau theory of semigroups of unbounded symmetric operators is developed. The second order abstract Cauchy problem: \[ u''=Au, \qquad u(0)=x, \qquad u'(0)=0, \] in a Banach space is associated in a natural manner to the so-called “cosine operator functions”. The theory of local cosine families of operators is presented in the last section. Bibliographical notes and comments on references close the book. This research notes volume will certainly be of interest to graduate students and researchers working in operator theory. It should also appeal to researchers in mathematical physics applying spectral theory and semigroups of operators.

Reviewer: J.J.Telega (Warszawa)

##### MSC:

47D06 | One-parameter semigroups and linear evolution equations |

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

47D09 | Operator sine and cosine functions and higher-order Cauchy problems |

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |