Spectral Theory of Banach Space Operators Ck-classification, abstrac
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1012 Shmuel Kantorovitz
Spectral Theory of Banach Space Operators C k-c1assification, abstract Volterra operators, similarity, spectrality, local spectral analysis.
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Shmuel Kantorovitz Oepartment of Mathematics, Bar-llan-University Ramat-Gan, Israel
AMS Subject Classifications (1980): 47-02, 46H30, 4'i' A60, 47 A65, 47 A55, 47005,47010,47040,47 B47, 47 A 10 ISBN 3-540-12673-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12673-2 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All nghts 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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
To Ita, Bracha, Peninah, Pinchas, and Ruth.
Table of Content O.
I ntroduct ion.
1.
Ope rat ional calculus.
6
2.
Examples.
8
3.
Fi rst reduct ion.
15
4.
Second reduct ion.
20
5.
Volterra elements.
25
6.
The fami I y
7.
Convolution operators in LP•
49
8.
Some regula r semi groups.
59
9.
Simi larity.
65
10.
Spect ra I analysis
73
11.
The fami ly
12.
Simi larity (corrt i nued) •
99
13. Singular Cn-ope rators.
123
14.
Local ana lvs l s .
146
Notes and references.
171
Bibl iography.
174
Index.
177
S + r,V.
S + r,V,
38
S unbounded.
82
O.
Introduction.
\4e may view sel fadjoint operators in Hi 1bert space as the best understood properly infinite dimensional abstract operators.
If we desire to recuperate some
of their nice properties without the stringent selfadjointness hypothesis, we are led to a "non-selfadjoint theory" such as Dunford's theory of spectral operators [5; Part III] or Foias' theory of general ized spectral operators [9,4], to mention only a few, and it is not our purpose to describe here anyone of these.
Our basic
concept, as in Foias' theory or distribution theory (as opposed to Dunford's), wi 11 be the operational calculus (and not the resolution of the identity). will be very I ittle overlapping between
However there
[4] and the present exposition.
Indeed, we
shall go in an entirely different di rection: starting in an abstract setting, we shall reduce the general situation to a very concrete one, and we shall then concentrate on various problems within this latter frameworl< or its abstract lifting. These notes are based on lectures given at various universities in 1981, and present in a unified (and often simpl ified) way results scattered through our papers since 1964. We proceed now with a more specific description of the main features of this exposition. Let
K
be a compact subset of the real line
R,
and denote by
HR
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