Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
693
trebliH Space srotarepO
Proceedings, California State University Long Beach
Long Beach, California, 20-24 June, 1977
Edited yb
.J .M Bachar, .rJ and D.W. Hadwin
galreV-regnirpS
nilreB Heidelberg New kroY 1978
Editors
nhoJ .M Bachar, .rJ
Department of Mathematics
California State University Long Beach
Long Beach, CA 90840/USA
Donald W. Hadwin
Department of Mathematics
University of New Hampshire
Durham, NH 03824/USA
AMS Subject Classifications (1970): 28A65, 46L ,51 47-02, 47A10,
47 A15, 47 A35,47 B05, 47 B10,47 B 20,47 B35,47 B40,47 B 99,47 C05,
47C10,47C15, 47D05, 47E05, 47G05
ISBN 3-540-09097-5 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-09097-5 Springer-Verlag New York Heidelberg Berlin
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PREFACE
This volume contains the contributions to the Conference on Hilbert Space
Operators, held at California State University Long Beach during the week of 20-24
June 1977. The purpose of the conference was to present some recent develol~nents
and some problems in Hilbert Space Operator Theory which are likely to be of
importance for further advances in the field.
Three main lecturers each delivered three lectures on the main topic of
concrete representations of Hilbert space operators:
1. .P .R Halmos, Integral Operators (f(x) ~ / k(x,y)f(y)dy).
.2 B. Abrahamse, Multiplication Operators (f(x) ~ ~(x)f(x)).
.3 .E Nordgren, Composition Operators (f(x) ~ f(T(x))).
Professor Halmos has included a description of the main topic in the introduction
to his paper. Additionally, other lectures were given in the theory of Hilbert
space operators, some of which are related to concrete representations of operators.
The 21 papers in this volume contain, in varying degrees, historical background,
expository accounts, the develol~nent and presentation of new ideas and results~ and
the posing of new problems for research.
The conference was funded jointly by the National Science Foundation (Grant
number MCS 77-15176) and by the host, California State University Long Beach.
We express much gratitude to them for making the conference possible~ to the
authors for their manuscripts~ to Elaine Barth for her excellent typing~ to the
participants3 and to Springer-Verlag for publishing this volume.
John Baehar
Donald Hadwin
Canference Participants
Bruce Abr ahamse Darrell J. Johnson
Brian Amr ine Gerhard Kalisch
Richard Arens Robert Kelly
Sheldon Axler Jerry Koliha
John M. Baehar, Jr. Ray Kunze
Jose Barria Alan Lambert
Estelle Basor Tan Yu ~e
Brad Beaver Tu~ig Po Lin
I. D. Berg Arthur Lubin
Charles A. Berger Carl Maltz
George Biriuk William Margulie s
Richard Bouldin John McDonald
James R. Brown C.R. Miers
Alice Chang Paul Muhly
Jen- chung Chuan Eric Nordgren
Wai-Fong Chuan Catherine 01sen
Floyd Cohen Boon-Hua 0ng
Carl Cowen Joseph Oppenheim
James Deddens Effrem Ost ~ow
Charles DePrima Barbara Rentzseh
Des Deut sch Wglliam G. Rosen
Henry Dye Mel Rosenfeld
Brent Ellerbroek Peter Rosenthal
J. M. Erdman James Rovnyak
John Ernest Norberto Salinas
John T. Fu,:?~"ason Bonnie Sannder s
Herb Gindler Howard Schwartz
Nell Gret sky Nien-Tsu Shen
Ted Guinn Allan Shields
Donald W. Hadwin A. R. Sourour
Paul ~{almo s Joseph St amp fli
Bernard Harvey James D. Stein~ Jr.
Thomas Haskell John B. Stubblebine
William Helton Barbara Turner
Domingo A. Herrero Larry ~ Wallen
Michael Hoffman Kenneth Warner
Richard B. Holmes Steven Weinstein
Thomas Hoover Gary Weiss
Donald H. Hyers Joel We stman
Nicholas Jewell Robert Whitley
Harold Widom
CONTENTS
MAIN LECTURES
P.R. HALMOS
Integral Operators . . . . . . . . . . . . . . . . . .
M.B. ABRAHAMSE
Multiplication Operators . . . . . . . . . . . . . . . 17
E.A. NORDGREN
Composition Operators in Hilbert Spaces . . . . . . . . 37
ADDITIONAL LECTURES
J. BARRIA and D.A. HERRERO
Closure of Similarity Orbits of Nilpotent Operators.
J.R. BROWN
Ergodic Groups of Substitution Operators Associated
with Algebraically Monothetic Groups . . . . . . . . . 65
C.C. COWEN
Commutants of Analytic Toeplitz Operators with
Automorphic Symbol . . . . . . . . . . . . . . . . . . 17
J.A. DEDDENS
Another Description of Nest Algebras . . . . . . . . . 77
M. EMBRY-WARDROP and A. LAMBERT
Weighted Translation Semigroups on L2(O,~) ...... 87
J. ERNEST
Concrete Representations and the von Neumann Type
Classification of Operators . . . . . . . . . . . . .
D.W. HADWIN and T.B. HOOVER
Weighted Translation and Weighted Shift Operators 93
B.N. HARVEY
An Operator Not a Shift, Integral, Nor Multiplication. 101
D.A. HERRERO
Strictly Cyclic Operator Algebras and Approximation
of Operators . . . . . . . . . . . . . . . . . . . . . 103
G.K. KALISCH
On Singular Self-Adjoint Sturm-Liouville Operators 109
A° LUBIN
Extensions of Commuting Subnormal Operators ..... 115
M. McASEY, P. MUHLY and K.-S. SAITO
Non-Self-Adjoint Crossed Products . . . . . . . . . . 121
This paper presented at the conference is not included
in this volume - the results will be published elsewhere.
lllV
J.N. McDONALD
Some Operators on L2(dm) Associated with Finite
Blaschke Products . . . . . . . . . . . . . . . . . . 125
C.L. OLSEN
A Concrete Representation of Index Theory in
von Neumann Algebras . . . . . . . . . . . . . . . . 133
N. SALINAS
A Classification Problem for Essentially n-normal
Operators . . . . . . . . . . . . . . . . . . . . . . 145
A.L. SHIELDS (notes by M.J. HOFFMAN)
Some Problems in Operator Theory . . . . . . . . . . 157
J.G. STAMPFLI
On a Question of Deddens . . . . . . . . . . . . . . 169
G. WEISS
The Fuglede Commutativity Theorem Modulo the Hilbert-
Schmidt Class and Generating Functions for Matrix
Operators . . . . . . . . . . . . . . . . . . . . . . 175
INTEGRAL OPERATORS
P. R. Halmos
PREFACE
The following report on integral operators, and the introduction to the three
principal themes of the conference that precedes it, should be accompanied by an
apology. They are not~ obviously not~ polished exposition; they are what at best
they might seem to be~ namely lecture notes. They are the notes I prepared before
the conference and kept peering at as I was lecturing. In the original hand-written
version there were three errors (that I know of). I corrected them, but that is
the only change I made.
The result is a compressed summary for those who were not at the lectures~
and a reasonably representative reminder for those who were. I have agreed to its
publication now~ so that the proceedings of the conference may have some claim to
completeness. A more detailed exposition of the part of the theory of integral
operators that I am interested in will be contained in a research monograph that is
now in preparation.
INTRODUCTION -
The major obstacle to progress in operator theory is the dearth of concrete
examples whose properties can be explicitly determined.
All known (and perhaps all conceivable) examples belong to one of three species.
The reason for tha~ is that the only concrete example of Hilbert space is L 2 (over
some measure space), and there isn't much one can do to the functions in L 2.
The simplest thing to do is to fix a function ~ and multiply every f in
L 2 by that ~. (In order for this operation to turn out to be boundedj the multi-
plier ~ must, of course~ belong to L~.) The multiplication operators so obtained,
and their immediate family~ are the best known and most extensively studied examples.
The spectral theorem assures us that every normal operator is of this kind. The
special case of diagonal matrices is too easy to teach us much but is~ nevertheless,
too important to be neglected. Multiplicity theory, unitary equivalence theory, and
the effective calculability of invariant subspaces of diagonal matrices can more or
less be extended to all multiplication operators~ and a large part of operator
theory is directed toward making it more instead of less.
Dilation theory began wi~h the observation that (to within unitary equivalence)
every operator can be obtained by compressing multiplication operators to suitable
subspaces. Certain special compressions (for example~ the restrictions to invariant
subspaces, which yield the subnormal operators# and, for another example, the ones
suggested by the passage from certain groups to their most important subsemigroups,
which yield the Toeplitz operators) are amenable to study. M. B. Abrahamse has made
substantial contributions to several aspects of the theory of multiplication oper-
ators, broadly interpreted, and in his lectures will present a part of that theory.
Next to multiplication the simplest thing to do to a function is substitution:
to get a new function from an old one, calculate the value of the old function at a
new place. In symbols: map f(x) to f(Tx). Simplest instance: map a sequence
If(n) in ~2 to the translated sequence f(n +l)}, thus getting a shift (uni-
lateral or bilateral). More sophisticated: let T be a measure-preserving trans-
formation on the underlying measure space, and thus make contact with ergodic
theory.
Members of the same species can interbreed; combinations of multiplication
operators and substitution operators yield weighted shifts and5 more generally~
weighted translation operators, studied by Parrott and others.
If the underlying space has additional structure (e.g., analytic structure),
and the substitutions permitted are correspondingly richer, the theory makes con-
tact with classical analysis. Much of this circle of ideas has been studied by
E. A. Nordgren.
In a sense the most natural, but, as it turns out, the least helpful way to
try to construct operators is via infinite matrices -- with rare exceptions
(diagonal, Toeplitz, Hankel), which can usually be subsumed under multiplications
and substitutions, matrices have not been a rich source of examples. Integral
kernels are generalizations of matrices, and, incidentally~ are the source of almost
all modern analysis. I turned to them a few years ago in the hope of finding
rewarding examples~ and found that they have quite an extensive theory that is not
yet completely worked out -- there are still reasons to maintain the hope of reward.
I shall try to tell you something about the present state of the theory of integral
operators.
LECTURE i. CONCEPTS
Definitions.
An integral operator is induced by a measurable function k on the Cartesian
product X×Y of two measure spaces by an equation such as
f(x) =/k(x,y)g(y)dy.
Known isomorphism theorems in measure theory make it possible to ignore all but the
pleasantest of classical measure spaces with no essential loss of generality: the
only spaces that need to be considered are the finite ~ (={l,...3n}) and
n
( = O,1), and the infinite ~ (or ~+ ) and ~ (or JR+ .) I shall refer to
and ~Z as the atomic cases and to ~ and ~ as the divisible cases.
n
In addition to the isomorphism theorems that leave measures as they find them~
there are some important ones that change measures. Their use makes it possible to
pass back and forth between finite and infinite spaces; the only really important
distinction is the one between atomic and divisible spaces. The theory I am hint-
ing at is perfectly illustrated by the effect of the mapping ~: Z~ ~+,
x
~xj - i- x
The transformation U: L2(~+) ~L2(~) defined by
Nf(x)
- l f(~(x)),
where 5 is the (Radon-Nikodym) derivative, 5(x) - i is unitary, it
(1 +x)2 '
sends integral operators to integral operators, and it preserves all properties of
kernels pertinent to the category (operators on Hilbert space) under study.
At the heart of the theory is the finite divisible case~ i.e.3 the unit
interval. In the systematic search for examples~ however, it is unwise to ignore
the atomic case (~) and the infinite case ( ~ .) There is a standard "inflation"
process for converting a matrix a (on ~+ × ~+) into a kernel k (on ~+ × ~+):
write k =a(i,j) on the unit square with diagonal from (i,j) to (i +i, j +i).
The change of measure described before can be used to get examples over Z from
examples over ~+; combined with matrix inflation, it can be used to get examples
over Z from examples over ~+. In what follows I shall be interested in Z
only, and when I say "kernel" I shall mean "kernel on E × Z"; in view of the
preceding comments, however~ when I need an example of a kernel, I shall feel free
to produce one elsewhere and expect that inflation and change of measure will be
applied automatically so as to re-establish contact with Z.
An integral operator is one induced by a kernel; in order for that expression
to make semse the kernel has to satisfy the following three conditions:
(1) If g ~ L 2, then k(x,.)g c L I almost everywhere.
(2) If g ~ L ,2 then lk(',y)g(y)dy c L 2.
(3) There exists a constant c such that if g ~ L 2, then
t/k(.,y)g(y)dyll ~ .llgNc
Banach knew 45 years ago that (i) and (2) imply (3); this useful fact follows
from some non-trivial measure theory (and not just from straightforward application
of the closed graph theorem).
What are some typical examples of integral operators? The best known ones
are the Hilbert-Schmidt operators induced by kernels in L 2. They are compact.
Among the simplest kernels in L 2 are the ones of the form
k(x,y) = u(x)v(y)
(with u and v in L2); the corresponding operators have rank 1. One of the
simplest of them is given by
k(x,y) ~ i;
the corresponding operator is a projection of rank i.
