Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics: The Carl M. Pearcy Anniversary Volume

212 Pages·1998·16.58 MB·English

by  Edward A. Azoff

OperatorTheory
Advances and Applications
Vol. 104
Editor:
I. Gohberg
Editorial Office: T. Kailath (Stanford)
School of Mathematical H.G. Kaper (Argonne)
Sciences S.T. Kuroda (Tokyo)
Tel Aviv University P. Lancaster (Calgary)
Ramat Aviv, Israel L.E. Lerer (Haifa)
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Editorial Board: B. Mityagin (Columbus)
J. Arazy (Haifa) V. V. Peller (Manhattan, Kansas)
A. Atzmon (Tel Aviv) J. D. Pincus (Stony Brook)
J. A. Ball (Blacksburg) M. Rosenblum (Charlottesville)
A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville)
H. Bercovici (Bloomington) D. E. Sarason (Berkeley)
A. Bottcher (Chemnitz) H. Upmeier (Marburg)
L. de Branges (West Lafayette) S. M. Verduyn-Lunel (Amsterdam)
K. Clancey (Athens, USA) D. Voiculescu (Berkeley)
L. A. Coburn (Buffalo) H. Widom (Santa Cruz)
K. R. Davidson (Waterloo, Ontario) D. Xia (Nashville)
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H. Dym (Rehovot)
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P. A. Fillmore (Halifax) Honorary and Advisory
C. Foias (Bloomington) Editorial Board:
P.A.Fuhrmann(BeerShev~ P. R. Halmos (Santa Clara)
S. Goldberg (College Park) T. Kato (Berkeley)
B. Gramsch (Mainz) P. D. Lax (New York)
G. Heinig (Chemnitz) M. S. Livsic (Beer Sheva)
J. A. Helton (La Jolla) R. Phillips (Stanford)
M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged)
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Editors:
Hari Bercovici and Ciprian L Foias
Department of Mathematics
Indiana University
Bloomington, IN 47405-4301
USA
1991 Mathematics Subject Classification 47-xx, 46L05, 46L30
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Nonselfadjoint operator algebras, operator theory, and related topics : the
Cari M. Pearcy anniversary volume / Hari Bercovici ; Ciprian Foias ed. -
Basel ; Boston; Berlin: Birkhiiuser, 1998
(Operator theory ; VoI. 104)
ISBN 978-3-0348-9771-6 ISBN 978-3-0348-8779-3 (eBook)
DOI 10.1007/978-3-0348-8779-3
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned,
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© 1998 Springer Basel AG
Originally published by Birkh!luser Verlag Basel Switzerland in 1998
Softcover reprint ofthe hardcover Ist edition 1998
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9771-6
987654321
CONTENTS
EDITORIAL PREFACE VII
PORTRAIT OF CARL PEARCy........................................ viii
CARL M. PEARCY: A BIOGRAPHICAL SKETCH ix
PUBLICATIONS OF CARL M. PEARCY xiii
E. A. AZOFF f3 L. DING,
A good side to non-reflexive transformations 1
H. BERCOVICI, C. FOIAS, f3 A. TANNENBAUM,
On skew Toeplitz operators, 'II ,.......... 23
H. BERCOVICI f3 D. VOICULESCU,
Regularity questions for free convolution 37
S. W. BROWN f3 E. KO,
Operators ofPutinar type 49
R. E. CURTO f3 L. A. FIALKOW, Flat extensions ofpositive
moment matrices: Relations in analytic or conjugate terms 59
R. G. DOUGLAS f3 G. MISRA,
Geometric invariants for resolutions ofHilbert modules 83
G. R. EXNER f3I. B. JUNG,
Some multplicities for contractions with Hilbert-Schmidt defect 113
D W. HADWIN f3 D. R. LARSON, Strong limits ofsimilarities......... 139
A. LAMBERT, LP multipliers and nested Sigma-algebras 147
W. S. LI f3 D. TIMOTIN, On isometric intertwining liftings 155
M. MARSALLI, The predual ofa type I Von Neumann algebra. .......... 169
M. MARTIN f3 N. SALINAS, The canonical complex structure of
flag manifolds in aC*-algebra 173
P. S. MUHLY f3 B. SOLEL,
An algebraic chamcterization ofboundary representations. ........... 189
A. OCTAVIO f3 S. PETROVIC,
Joint spectrum and nonisometric functional calculus 197
EDITORIAL PREFACE
This volume is dedicated to Carl Pearcy on his 60th
birthday. It collects recent contributions to operator
theory, nonselfadjoint operator algebras, measure the
ory, and the theory ofmoments by several ofthe lead
ingspecialists in thoseareas. Manyofthe contributors
are collaborators or former students of Carl Pearcy,
and the variety ofthe topics bears witness to the wide
range of his work and interests.
The editors were helped by many in the compi
lation of this volume. Srdjan Petrovic helped com
pile Carl's list ofpublications, while Arlen Brown and
George Exner helped in writing the biographical and
mathematical sketch. The work ofmany referees, who
must remain anonymous, was very valuable. Israel
Gohberg suggested that we publish this volume in the
distinguished series Operator Theory: Advances and
Applications.
The whole volume was expertly typeset by Elena
Fraboschi. Wewishtoextendto allofthese peopleour
heartfelt thanks.
CARL M. PEARCY
Carl M. Pearcy:
A Biographical Sketch
H. BERCOVICI fj C. FOIAS
Carl Mark Pearcy, Jr. was born on August 23, 1935 in Beaumont, Texas. He was
theeldest oftwosonsofCarl Mark Pearcy, Sr., and CarrieEdith (Tilbury) Pearcy.
His family moved to Galveston in 1940, and that is where Carl resided until he left
home to attend the university. Carl entered Texas A.&M. University in 1951 at
age sixteen. He graduated (B.S.) in 1954 and, again (M.S.) in 1956. Carl entered
Rice Institute (later Rice University) as a candidate for the Ph.D. in Mathematics
in 1956. The degree was conferred in 1960 under the direction of Arlen Brown.
In 1957-58 Carl was a fellow in the mathematics department at the University of
Chicago, and in 1959-60 he held an appointment as Assistant Professor at Texas
A.&M. He was then appointed a post-doctoral fellow at Rice for the year 1960-61.
From1961to 1963CarlwasemployedattheHoustonresearchcenterofHumbleOil
Co. In 1963hewasappointed Assistant ProfessorofMathematicsat the University
ofMichigan, where heremained untilhis "retirement" inthe rank ofFullProfessor
in 1990, at which time he accepted reappointment in the mathematics department
of Texas A.&M. Carl was an A.P. Sloan foundation fellow from 1966 to 1968,
and was the main speaker at two conferences sposored by the Conference Board
of Mathematical Sciences of the National Science Foundation. These conferences
were at Bucknell University in 1975, and Arizona State University in 1984.
While Carl appreciates good mathematics regardless of the field, most of
his mathematical career was closely intertwined with the development of modern
operatortheory. Thereis practicallyno areaofoperatortheory whereCarl did not
contribute. Manyofthese contributionswerefundamental ortrendsetting. We will
highlight just some ofthe most influential ofthese contributions. The numbers in
brackets refer to Carl's publication list.
The characterization ofcommutators, i.e., operators ofthe form T = AB
BA on a Hilbert space was posed as a problem by P. Halmos. It was known that
operators ofthe form zI+C, with z a scalar and C a compact operator, are not
commutators. CarlPearcyand Arlen Brownprovedin [15] that allotheroperators
are in fact commutators. This is one of the earliest deep results pertaining to
arbitraryoperators on a Hilbert space. The methods used in the proofcontain the
germs of the development of the general approximation theory of Hilbert space
operators. In a difft~ent direction, this work led to developments in the theory
of operator algebras. Indeed, it became possible to characterize commutators in
various kinds ofC* andvon Neumannalgebras, andthis also yielded results about
the radical structure ofsuch algebras.
IX
x H. BERCOVICI €3 C. FOIAS
Another circle of ideas introduced by Halrnos is the study of quasitrangular
operators. Theseoperatorswereintroducedin relationwith the invariantsubspace
problem. Quasitriangularity is a geometric condition related with the behavior of
an operator on a chain of finite dimensional subspaces. Quite surprizingly, Carl
Pearcy and Ronald Douglas found in [35] a necessary condition for quasitriangu
larity expressed purely in spectral (and Fredholm index) terms. This was again
a very general result, applying to a wide class of Hilbert space operators subject
only to mild conditions, and it raised the possibility that quasitriangularity might
be entirely characterized in spectral terms. This characterization was indeed real
ized (by Apostol, Foias, and Voiculescu) when the condition discovered by Pearcy
and Douglas was shown to be sufficient as well. These developments led to the
search for other relations between geometric and spectral properties ofoperators,
and there is now a vast body ofworkon this subject. It should be mentioned that
techniques from Carl's work on commutators also turned out to be inspiring in
the study ofquasitriangularity and related questions.
The invariant subspace problem, already present in quasitriangularity, was
one of Carl's long lasting preoccupations. One line of research is illustrated by
his work on the Lomonosov technique (see [50], [55], and [56]). Another line was
inaugurated by his joint work with S. Brown and B. Chevreau [65] in which the
existence ofinvariant subspaces is deduced from a richness condition on the spec
trum. Carl recognized very early that the methodology of this work can yield
much more. In fact it was seen through the work of Carl (and his collaborators
and students) that a great variety of results about the structure of the invariant
subspaces of an operator can be derived from factorization methods. To give an
interestingexample, it is well known that the invariant subspacesofthe usualshift
operator 3 (multiplication by the variable on the Hardy space H2) are classified
by the inner functions defined in the unit disk. Thus, for instance, if M is one
ofthese invariant subspaces then 31M is unitarily equivalent to 3. It was known
that the corresponding multiplication operator T on the Bergman space of the
unit disk is not as tractable, and very few facts were known about the invariant
subspaces of T. As an application of the factorization techniques, it was shown
in [82] and [83] that T EElT EElT EEl··· can be realized as the compression of T to
a semiinvariant subspace. This result has amazing consequences for the structure
of invariant subspaces of T. The result has redirected the work of researchers in
function theory, who wanted to approach it from a more classical point of view,
and were in fact able to derive independently some ofits consequences. The num
ber of results in this general area was such that it required a separate entry in
the 1991 Mathematics Subject Classification: 47D27 Dual Algebras. We cannot
conclude the discussion of dual algebras without mentioning the definitive result
of [95]: every contraction whose spectrum contains the unit circle has nontrivial
invariant subspaces.
Besides these areas, to which Carl Pearcy contributed in a major way, there
are many areas which he kept alive by getting his students and colleagues inter-
Carl M. Pearcy: A Biographical Sketch xi
ested. Onesucharea is thesimilarity problemfor polynomially bounded operators
on a Hilbert space (which was eventually solved in the negative by Gilles Pisier).
CarlhasinfluencedresearchinOperatorTheory notonlydirectlythroughhis
work, but also indirectly through his numerous students, many ofwhom became
leadersin thefield. Carl's courses introduced hisstudents to allsignificant aspects
of modern operator theory. Much of the material in these courses is contained in
the - still unpublished - Part II ofhis book on operator theory written jointly
with A. Brown.
Carl's contributions to mathematics were not limited to his research and
teaching. For many years Carl organized sessions at the annual meetings of the
American Mathematical Society. He helped launchsuccessfullythe Journal ofOp
erator Theory, and he supervised for several years the operations of Mathematical
Reviews. He maintained close relations with his colleagues in Eastern Europe at a
time when they had little access to current mathematical publications, and only
sporadicoccasions to travel out$ide their countries. Carl ultimately encouraged or
helped many of his Romanian colleagues to relocate in the United States; some
came as seniormathematicians (C. Foias, C. Apostol, D. Voiculescu), some as stu
dents (H. Bercovici, G. Popescu, R. Gadidov, A. Ionescu). All ofthem owe a debt
ofgratitude to Carl.
For more than thirty years, Carl Pearcy was one ofthe most influential per
sonalities in operator theory. His strength as a mathematician, his dedication as a
teacher, and his warmth as a friend helped him achieve this elevated status. Be
sides that, he was one ofthe few who had the vision ofwhat the most important
feasible problems at each stage in pure operator theory were. He not only worked
on those problems, but he always succeded in making other people work on them
as well. For that reason, the whole operator theory community wishes him the
best for many more fruitful years in the profession.