The Theory of Fractional Powers of Operators

Celso Martínez Carracedo and Miguel Sanz Alix (Eds.)0444887970, 9780444887979, 9780585474519

This book makes available to researchers and advanced graduates a simple and direct presentation of the fundamental aspects of the theory of fractional powers of non-negative operators, which have important links with partial differential equations and harmonic analysis. For the first time ever, a book deals with this subject monographically, despite the large number of papers written on it during the second half of the century. The first chapters are concerned with the construction of a basic theory of fractional powers and study the classic questions in that respect. A new and distinct feature is that the approach adopted has allowed the extension of this theory to locally convex spaces, thereby including certain differential operators, which appear naturally in distribution spaces. The bulk of the second part of the book is dedicated to powers with pure imaginary exponents, which have been the focus of research in recent years, ever since the publication in 1987 of the now classic paper by G.Dore and A.Venni. Special care has been taken to give versions of the results with more accurate hypotheses, particularly with respect to the density of the domain or the range of the operator. The authors have made a point of making the text clear and self-contained. Accordingly, an extensive appendix contains the material on real and functional analysis used and, at the end of each chapter there are detailed historical and bibliographical notes in order to understand the development and current state of research into the questions dealt with.

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Table of contents :
Content:
Introduction
Pages ix-xii

Chapter 1 Non-negative operators Original Research Article
Pages 1-30

Chapter 2 Differential operators Original Research Article
Pages 31-56

Chapter 3 The balakrishnan operator Original Research Article
Pages 57-72

Chapter 4 An extension of the hirsch functional calculus Original Research Article
Pages 73-104

Chapter 5 Fractional powers of operators Original Research Article
Pages 105-139

Chapter 6 Other questions about fractional powers: Domains, uniqueness and the cauchy problem Original Research Article
Pages 141-170

Chapter 7 Fractional powers with exponents of negative real part. Imaginary powers of operators Original Research Article
Pages 171-189

Chapter 8 The dore-venni theorem Original Research Article
Pages 191-218

Chapter 9 Functional calculus for C0-groups Original Research Article
Pages 219-243

Chapter 10 Imaginary powers on hilbert spaces Original Research Article
Pages 245-256

Chapter 11 Fractional powers and interpolation spaces Original Research Article
Pages 257-278

Chapter 12 Fractional powers of some differential operators Original Research Article
Pages 279-305

Chapter A Appendix
Pages 307-339

Notations
Pages 341-346

Bibliography
Pages 347-360

Index
Pages 361-365