Analytic Perturbation Theory for Matrices and Operators - ISBN:3764316640

Analytic Perturbation Theory for Matrices and Operators

Date: 15 April 2011, 12:14

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This book is an extended and partly modified English version of the author's book "Endlichdimensionale analytische Storungstheorie", which appeared in 1972. A systematic presentation of analytic perturbation theory for matrices (i.e., linear operators on finite-dimensional linear spaces) is given and the central problems are solved completely.
The implications of this theory for analytic perturbations of isolated eigenvalues of finite algebraic multiplicity of closed operators on Banach spaces are investigated.
Essential contributions to the perturbation theory for isolated eigenvalues of finite algebraic multiplicity are contained in the works of F. RELLICH, B. v. SZ.-NAGY, T. KATO and F. WOLF. In 1967, the author realized that this theory could be extended and brought to completion. The results are presented in this book with detailed proofs. The book may be considered as a supplement to KATO'S monograph on perturbation theory for linear operators, in particular Ch. II.
The main part of this book is devoted to the analytic perturbation theory for one complex variable. A supplement deals with the transfer of the essential results to the case of analytic perturbations depending on several complex variables.
Applications of perturbation theory receive minor emphasis in this book.
Of course, problems of perturbation theory occur in several parts of mathematics (for example, in the theory of matrices, ordinary and partial differential equations, and integral equations) and perturbation theory strongly influences areas beyond mathematics, for example, in theoretical physics.
For these reasons, the presentation is also intended for researchers and advanced students in the physical sciences. It is the author's hope that the book will be useful for those interested - for whatever reason - in a detailed and precise foundation of their ideas and knowledge of analytic perturbation theory.
The reader is assumed to have a basic knowledge of linear algebra and real and complex analysis. For the convenience of the reader, the basic concepts and propositions are presented, without proofs, in Chapter I and in the Appendix.

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