The Adjoint of a Semigroup of Linear Operators
This monograph provides a systematic treatment of the abstract theory of adjoint semigroups. After presenting the basic elementary results, the following topics are treated in detail: The sigma (X, X )-topology, -reflexivity, the Favard class, Hille-Yosid
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Jan van Neerven
The Adjoint of a Semigroup of Linear Operators
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Autor Jan van Neerven California Institute of Technology Department of Mathematics Pasadena, CA 91125, USA
Mathematics Subject Classification (1991): 47D03, 47D06, 46A20, 46B22, 47 A80, 47B65
ISBN 3-540-56260-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56260-5 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author 46/3140-543210 - Printed on acid-free paper
Preface
This lecture note is an extended version of the author's Ph.D. thesis "The adjoint of a semigroup of linear operators" (Leiden, 1992). The main difference consists of two new chapters (3 and 4) dealing with Hille-Yosida operators, extra- and interpolation and perturbation theory. Also, the sections 7.4 and 8.2 are new. The general theory of adjoint semigroups was initiated by Phillips [Ph2], whose results are presented in somewhat more generality in the book of Hille and Phillips [HPh], and was taken up a little later by de Leeuw [dL]. Before that, Feller [Fe] had already used adjoint semigroups in the theory of partial differential equations. After these papers almost no new results on adjoint semigroups were published, although the theory of strongly continuous semigroups continued to develop rapidly. Recently the interest in adjoint semi groups revived however, due to many applications that were found to, e.g., elliptic partial differential equations [Am], population dynamics [Cea1-6], [DGT], [GH], [GW], [In], control theory [Heij], approximation theory [Ti], and delay equations [D], [DV], [HV], [V]. This stimulated also renewed interest in the abstract theory of adjoint semigroups, e.g. [Pa1-3]' [GNa] and [DGH]. The aim of the present lecture note is to give a systematic exposition of the abstract theory of adjoint semigroups. Although we illustrate many results with concrete examples, we do not give applications of the theory. An exposition of the various fields where adjoint semigroups have found fruitful application would require a volume of at least comparable size. Rather, this lecture note should provide the interested reader with sufficient background material in order to make these application
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