Semigroups of Operators and Spectral Theory (Research Notes in Mathematics Series)

This book presents some aspects of the theory of semigroups of operators, mostly from the point of view of its interaction withspectral theory. In order to make it self-contained, a concise description of the basic theory of semigroups, with complete proofs, is included in Part I. Some of the author's recent results, such as the construction of the Hille-Yosida space for general operators, the semi-simplicity manifold, and a Taylor formula for semigroups as functions of their generator, are also included in Part I.Part II describes recent generalizations (most of them in bookform for the first time), including pre-semigroups, semi-simplicity manifolds in situations more general than that considered in Part I, semigroups of unbounded symmetric operators, and an analogous result on "local cosine families" and semi-analytic vectors. It is hoped that this book will inspire more research in this field. This book will be of particular interest to graduate students and researchers working operator theory and its applications. Cover......Page 1 Title Page......Page 8 Copyright Page......Page 9 Introduction......Page 10 Contents......Page 13 PART I. GENERAL THEORY......Page 14 A. THE HILLE-YOSIDA THEORY......Page 16 B. THE HILLE-YOSIDA SPACE......Page 32 C. DISSIPATIVITY......Page 36 D. THE TROTTER-KATO CONVERGENCE THEOREM......Page 40 E. EXPONENTIAL FORMULAS......Page 44 F. THE HILLE-PHILLIPS PERTURBATION THEOREM......Page 47 G. GROUPS AND SEMI-SIMPLICITY MANIFOLD......Page 53 H. ANALYTICITY......Page 71 K. NON-COMMUTATIVE TAYLOR FORMULA......Page 77 PART II. GENERALIZATIONS......Page 86 A. PRE-SEMIGROUPS......Page 88 B. SEMI-SIMPLICITY MANIFOLD (real spectrum case)......Page 96 C. SEMI-SIMPLICITY MANIFOLD (case R+ C p(-A))......Page 109 D. LAPLACE-STIELTJES SPACE......Page 117 E. SEMIGROUPS OF UNBOUNDED SYMMETRIC OPERATORS......Page 130 F. LOCAL COSINE FAMILIES OF SYMMETRIC OPERATORS......Page 136 Notes and References......Page 143 Bibliography......Page 145 Back Cover......Page 149 Shmuel Kantorovitz. Includes Bibliographical References (p. 132-135).