Calkin Algebras and Algebras of Operators on Banach Spaces

159 Pages·1974·6.435 MB·English

by  S. R. Caradus, W. E. Pfaffenberger

calkin a lg e b ra s and
a lg e b ra s of
on b a n a ch s p a c e s
S. R. Canadus
W. E. Pfaffenberger
Bertram Yood
Calkin Algebras and Algebras
of Operators on BanachSpaces
Lecture Notes in Pure and Applied Mathematics
Executive Editor
Earl J. Taft
RUTGERS UNIVERSITY, NEW BRUNSWICK, NEW JERSEY
Chairman of the Editorial Board
S. Kobayashi
UNIVERSITY OF CALIFORNU AT BERKLEY
1. N. Jacobson, Exceptional Lie Algebras
O
2. L.-A. Lindahl and F. Poulsen, Thin Sets in Harmonic Analysis
3. /. Satake, Classification Theory of Semi-Simple Algebraic Groups
4. F. Hirzebruch, W. D. Neumann, and S. S. Koh, Differentiable
Manifolds and Quadratic Forms
5. /. Chavel, Riemannian Symmetric Spaces of Rank One
6. R. B. Burckel, Characterization of C(X) among Its
Subalgebras
7. B. R, McDonald, A. R. Magid, and K, C. Smith, Ring Theory:
Proceedings of the Oklahoma Conference
8. Yum-Tong Siu, Techniques of Extension of Analytic Objects
9. 5. R, Caradus, W. E. Pfaffenberger, and Bertram Yood, Calkin
Algebras and Algebras of Operators on Banach Spaces
Other volumes in preparation
Calkin Algebras and Algebras
of Operators on Banach Spaces
S. R. Caradas
queen’s university
W. E. Pfaffenberger
UNIVERSITY OF VICTORIA
Bertram Yood
PENNSYLVANIA STATE UNIVERSITY
MARCEL DEKKER, INC. NewYoik 1974
COPYRIGHT © 1974 by MARCEL DEKKER, INC. ALL RIGHTS RESERVED.
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ISBN: 0-8247-6246-0
Current printing (last digit) :
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PRINTED IN THE UNITED STATES OF AMERICA
This book is dedicated to
I. C. Gohberg
for his important and lasting
contributions to functional analysis.

PREFACE
The interaction between the theories of Banach algebras with involution
and that of bounded linear operators on a Hilbert space has been extensively
developed ever since the birth of Banach algebra theory. In the meantime
there have evolved, in a natural way but at a much slower pace, interesting
connections of Banach algebras with the theory of bounded linear operators
on a Banach space. These notes are intended to provide an introduction to
the latter set of ideas. Here central items of interest include Fredholm
operators, semi-Fredholm operators, Riesz operators and Calkin algebras.
We begin with a treatment of the classical Riesz-Schauder theory which
takes advantage of more recent developments. Some of this material (Riesz-
Schauder operators) appears here for the first time. In order to make our
exposition suitable for readers with a modest background, we have included
an introductory chapter on Banach algebras. With admirable restraint we
have not tried to give a short course in Banach algebras but have included
only material rather directly relevant to our aims. This is followed by
chapters on Riesz and semi-Fredholm operators.
Let B(X) be the Banach algebra of all bounded linear operators on a
Banach space X. Chapter 5 features the remarkable result of Gohberg,
Markus and Feldman (1960) that, for X = l_<p<a> and X = c^, the
compact operators on X furnish the only proper closed two-sided ideal in
B(X). (The case X = ¿2 Calkin (1941)). Finally in Chapter 6
we indicate relations of our subject matter to a variety of recent
developments.
The authors are indebted to the National Science Foundation, the
National Research Council of Canada and the University of Victoria (faculty
research grant) for financial support. We are also very grateful to the
typist, Mrs. D. J. Leeming, and to the publishers for their assistance and
patience.

CONTENTS
PREFACE
Chapter I INTRODUCTION
1.1 Introduction I
1.2 On duality 4
1.3 Semi-Fredholm operators 7
1.4 On ascent and descent 10
Chapter 2 BANACH ALGEBRAS
2.1 Introduction 19
2.2 On spectra 22
2.3 Quotient algebra and ideals 26
2.4 Topological divisors of zero 28
2.5 On B(X) as a Banach algebra 29
Chapter 3 RIESZ OPERATORS
3.1 Introduction 35
3.2 Characterization in terms of Fredholm region 36
3.3 Characterization in terms of Calkin algebra 43
3.4 Characterization in terms of the resolvent operator 44
3.5 The West decomposition 50
3.6 Extensions and generalizations 54
Chapter 4 SEMI-FREDHOLM OPERATORS
4.1 Introduction 61
4.2 Semi-Fredholm operators as open semi groups in B(X) 61
4.3 Semi-Fredholm operators and the mapping tt:B(X) ^ B(X)/K(X) 63
4.4 Perturbations and restrictions of semi-Fredholm and
Fredholm operators 66
Vii