A Short Introduction to Perturbation Theory for Linear Operators

by Tosio Kato

Overview

This book is a slightly expanded reproduction of the first two chapters (plus Introduction) of my book Perturbation Theory tor Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Springer 1980. Ever since, or even before, the publication of the latter, there have been suggestions about separating the first two chapters into a single volume. I have now agreed to follow the suggestions, hoping that it will make the book available to a wider audience. Those two chapters were intended from the outset to be a comprehensive presentation of those parts of perturbation theory that can be treated without the topological complications of infinite-dimensional spaces. In fact, many essential and. even advanced results in the theory have non­ trivial contents in finite-dimensional spaces, although one should not forget that some parts of the theory, such as those pertaining to scatter­ ing. are peculiar to infinite dimensions. I hope that this book may also be used as an introduction to linear algebra. I believe that the analytic approach based on a systematic use of complex functions, by way of the resolvent theory, must have a strong appeal to students of analysis or applied mathematics, who are usually familiar with such analytic tools.

Product Details

ISBN-13:	9781461257028
Publisher:	Springer New York
Publication date:	12/21/2011
Edition description:	Softcover reprint of the original 1st ed. 1982
Pages:	162
Product dimensions:	6.10(w) x 9.25(h) x 0.01(d)

Table of Contents

One Operator theory in finite-dimensional vector spaces.- § 1. Vector spaces and normed vector spaces.- § 2. Linear forms and the adjoint space.- § 3. Linear operators.- § 4. Analysis with operators.- § 5. The eigenvalue problem.- § 6. Operators in unitary spaces.- § 7. Positive matrices.- Two Perturbation theory in a finite-dimensional space.- § 1. Analytic perturbation of eigenvalues.- § 2. Perturbation series.- § 3. Convergence radii and error estimates.- § 4. Similarity transformations of the eigenspaces and eigenvectors.- § 5. Non-analytic perturbations.- § 6. Perturbation of symmetric operators.- § 7. Perturbation of (essentially) nonnegative matrices.- Notation index.- Author index.