Spectral Decompositions on Banach Spaces
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623 Ivan Erdelyi Ridgley Lange
Spectral Decompositions on Banach Spaces
Springer-Verlag Berlin Heidelberg New York 1977
Authors Ivan Erdelyi Department of Mathematics Temple University Philadelphia, PA 19122/USA
Ridgley Lange Department of Mathematics University of New Orleans New Orleans, LA 70122/USA
AMS Subject Classifications (1970): 47 A 10, 47 A 15, 47 A60, 47 A65, 47B99 ISBN 3-540-08525-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08525-4 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1977 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210
FOREWORD There is a new trend developing in the spectral theory of linear operators. In contrast to the classical spectral theory of linear operators and to the Dunford-type spectral operators which depend on some algebraic and topological structures outside their domains of definition, the contemporary spectral decomposition is defined only in regard to the operators invariant subspaces.
In
this way, the spectral theory can be conceived as an axiomatic system functioning within the underlying Banach space with possible extensions to more general topological vector spaces. Our purpose in this work is to extend and unify the intrinsic axiomatic perspective on spectral decompositions.
In such extension we wish to consider the
widest feasible generalization of the notion "spectral decomposition" in order to learn more about the special cases.
In this spirit we start in Chapter II the
study of the most abstract form of spectral decomposition so that when we come to the more special theory of "decomposable operators" (Chapter IV) we find that many of the known results of the latter theory are easy consequences of the preceding material.
More importantly, however, we obtain solutions to deep
problems which have been open and vigorously studied (e.g. the dual theory). Chapter I presents various classes of invariant subspaces a given operator may have.
Special attention is devoted to the single-valued extension property
as an essential tool in the study of spectral decompositions.
Chapter II is the
foundation of our axiomatic attack in the general problem of spectral decomposition.
We show that the single-valued extension property is an intrinsic element
of the spectral decomposition.
The more recent theories of "asymptotic spectral
decompositions" are treated in Chapter III.
Chapter IV brings the full power
of the spectral decomposition to bear in the theory of duality. The Appendix is aimed to
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