Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space
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285 Pierre de la Harpe University of Warwick, Coventry/England
Classical Banach-LieAlgebras and Banach-Lie Groups of Operators in Hilbert Space
Springer-Verlag Berlin. Heidelberg . NewYork 1972
AMS Subject Classifications (1970): Primary: 17 B65 Secondary: 22E65, 17B20, 17B45, 18H25, 22E60
ISBN 3-540-05984-9 Springer-Verlag Berlin • Heidelberg • New York ISBN 0-387-05984-9 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 1972. Library of Congress Catalog Card Number 72-88729. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
0. I n t r o d u c t i o n
. . . . . . . . . . . . . . . . . . . . . . . . .
D e t a i l e d table
of contents
. . . . . . . . . . . . . . . . . .
Some n o t a t i o n s and c o n v e n t i o n s
Chapter I
Classica~ groups
C h a p t e r III
IV
Examples
Bibliography
21
involutive Banach-Lie
23
a l g e b r a s and
operators . . . . . .
72
of i n f i n i t e d i m e n s i o n a l H i l b e r t spaces
. . . . . . . . . . . . . . . . .
115
On the c o h o m o l o g y of the c l a s s i c a l c o m p l e x Lie algebras
V.
. . . . . . . . . . . . . . . . . .
of bounded and compact
symmetric Chapter
16
C l a s s i c a l i n v o l u t i v e Lie a l g e b r a ~ of finite rank o p e r a t o r s
C h a p t e r II
. . . . . . . . . . . . . . . .
I
of compact
operators . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
134
147
INTRODUCTION
This ,work contains
three
Chapter I is devoted Lie algebras Chapter related
parts of similar
lengths.
to the study of some infinite
dimensional
of linear operators. II to that of Banach-Lie
algebras
and Banach-Lie
groups
to them.
And Chapters symmetric
spaces,
III and IV to applications cobomology
The main results this introduction. respectively
of the stable
are briefly
concerned
with various
and with some indications
classical
described
The first two sections
: infinite
dimensional
groups.
in sections
3 to 4 of
of the introduction
examples
of Banach-Lie
about the general theory
are
groups,
of Banach-Lie
groups.
I have found
it necessary
together the relevant bibliography importance
and helpful to attempt
literature.
do not appear
for our purpose
and draw
Some of the references
in the text. Those are indicated
by a
in the
of particular *
.
0.2
0.1.- Banach-Lie groups
: examples
Banach-Lie algebras and Banach-Lie groups arise naturally in many different contexts. The earliest work devoted to them seems to be one by P~rSs (191
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