Topics in Operator Theory: Constantin Apostol Memorial Issue

277 Pages·1988·7.785 MB·English

by  Domingo A. Herrero (auth.), I. Gohberg (eds.)

OT32
Operator Theory: Advances and Applications
VoI.32
Editor:
I. Gohberg
Tel Aviv University
Ramat Aviv, Israel
Editorial Office:
School of Mathematical Sciences
Tel Aviv University
RamatAviv, Israel
Editorial Board:
A. Atzmon (TeIAviv) M. A. Kaashoek (Amsterdam)
J. A. Ball (Blacksburg) T. Kailath (Stanford)
L. de Branges (West Lafayette) H. G. Kaper (Argonne)
K. Clancey (Athens, USA) S. T. Kuroda (Tokyo)
L. A. Coburn (Buffalo) P. Lancaster (Calgary)
R. G. Douglas (Stony Brook) L. E. Lerer (Haifa)
H. Dym (Rehovot) E. Meister (Darmstadt)
A. Dynin (Columbus) B. Mityagin (Columbus)
P. A. Fillmore (Halifax) J. D. Pincus (Stony Brook)
C. Foias (Bloomington) M. Rosenblum (Charlottesville)
P. A. Fuhrmann (Beer Sheva) J. Rovnyak (Charlottesville)
S. Goldberg (College Park) D. E. Sarason (Berkeley)
B. Gramsch (Mainz) H. Widom (Santa Cruz)
J. A. Helton (La Jolla) D. Xia (Nashville)
D. Herrero (Tempe)
Honorary and Advisory
Editorial Board:
P. R. Halmos (Santa Clara) S. G. Mikhlin (Leningrad)
T. Kato (Berkeley) R. Phillips (Stanford)
P. D. Lax (New York) B. Sz.-Nagy (Szeged)
M. S. Livsic (Beer Sheva)
Springer Basel AG
lbpics in
Operator Theory
Constantin Apostol
Memorial Issue
Edited by
I. Gobberg
1988 Springer Basel AG
Volume Editorial Office:
Raymond and Beverly Sackler Faculty of Exact Sciences
School of Mathematical Sciences
Tel Aviv University
Tel Aviv, Israel
CIP-Titelaufnahme der Deutschen Bibliothek
Topics in operator theory: Constantin Apostol memorial issue
/ed. by I. Gohberg.-Basel ; Boston ; Berlin : Birkhäuser, 1988
(Operator theory ;Vol. 32)
NE: Gochberg, Izrail' [Hrsg.]; Apostol, Constantin: Festschrift; GT
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© Springer Basel AG 1988
Originally published by Birkhäuser Verlag Basel in 1988
Softcover reprint of the hardcover 1st edition 1988
ISBN 978-3-0348-5477-1 ISBN 978-3-0348-5475-7 (eBook)
DOI 10.1007/978-3-0348-5475-7
v
TABLE OF OONTENTS
From the Editor....................... VII
D.A. Herrero: In Memory of
Constantin Apostol ....••..•......••.••
List of Publications of
Constantin Apostol. . . . • . . • • • . . . • • . • • . • 7
C. Apostol: On the Spectral
Equivalence of Operators.............. 15
BEN-ARTZI, A. Fredholm properties of band
GOHBERG, I. matrices and dichotomy ..••••••.••....• 37
BERCOVICI, H. A note on the algebra generated
CONWAY, J.B. by a subnormal operator ..••••.....•..• 53
CEAU~ESCU, Z. Extreme points in the set of
sucru, I. contractive intertwining dilations •.•. 57
DAVIDSON, K.R. Extreme points in quotients
FEEHAN, T.G. of operator algebras ....•..••........• 67
SHIELDS, A.L.
FOIAS, C. On the four block problem, I ......... . 93
TANNENBAUM, A.
HERRERO, D.A. Variation of the point spectrum
TAYLOR, T.J. under compact perturbations ••••....... 113
WANG, Z.Y.
LARSON, D.R. Bimodules of nest subalgebras
SOLEL, B. of von Neumann algebras ••.••••.•.••.. 159
RAN, A.C.M. Stability of invariant
RODMAN, L. Lagrangian subspaces I ...•••.•.......• 181
SALINAS, N. Products of kernel functions
and module tensor products............ 219
VASILESCU, F.-H. Spectral capacities in
quotient Frechet spaces •...••....•.••• 243
VOICULESCU, D. A note on quasidiagonal operators ..... 265
CONSTANTIN APOSTOL
(1936-1987)
VII
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I. GohbeJra
IN MEMORY OF CONSTANTIN APOSTOL
Domingo A. Herrero
Constantin Apostol was born on February 1st, 1936, in
Ia§i, in the region of Moldavia, Romania. His father died when
he was only one year old, and he was raised by his mother, in pov
erty conditions bordering hunger, in a Romania ravaged by the war.
In spite of the material difficulties, his mother gave hima strict
education; the best she could afford. He was a gifted child,
wi th a passion for reading, especially Geography, History, travels,
novels.
In 1961, he received his Master in Arts degree from the
University of Bucharest, and in 1968, he received his Ph. D. de
gree from the same institution. The same year, he was awarded
the Romanian Academy Prize "Gheorghe Ti1;eica" for research in op
erator Theory. By this time, Bucharest was already a well-known
center in that area. Constantin was part of the Romanian School
of Operator Theory of the 70's (and a very important part, in
deed!) Some day, some historian of Mathematics will analyze the
rising and evolution of this outstanding group of mathematicians
that suddenly appear in an Eastern European country, among socio
economico - political conditions far from optimal for the develop
ment of any kind of science.
We live in an age of massification. People go to a bas
ketball game to see a group of ten individuals whose average
height surpasses the two meter mark, and everybody considers
that there is nothing special about this group. Sometimes, I
think that in the same way, the Operator Theory community (all
around the World) lived throughout the 70's expecting every year
2 Herrero
a flush of exceptionally good results coming from Romania. That
was the "natural", the "obvious" thing to expect ...
In 1968, Paul R. Halmos introduced the notion of "quasi
triangular" (Hilbert space) operator, and two years later he pub
lished that wonderful survey article that has influenced so much
the research in this area of Mathematics: "Ten problems in Hil
bert space" (Bull. Amer. Math. Soc. 76 (1970), 887-933).
Around 1972, Constantin began to work on the spectral
characterization of quasitriangularity, and wrote his article
"Quasitriangularity in Hilbert space". It was not the solution
of the problem but, clearly, it was pointing in the right di
rection. Shortly afterwards, the joint work of Constantin Apos
tol, Ciprian Foia§ and Dan Voiculescu produced the desired result:
an operator T is quasitriangular if and only if index(z -T) isnon
negative for all z such that z -T is semi-Fredholm.
The series of articles leading to this fundamental re
sult contain an incredible number of "approximation tricks", that
will strongly contribute to produce a whole body of work in the
area of "non-commutative approximation" in the years to come.
1973 was a key year in Operator Theory: it was the year
of the Apostol-Foia§-Voiculescu Theorem, and the year of the
Brown-Douglas-Fillmore Theorem, two cornerstones of the theory!
A year later, in 1974, the joint work of C. Apostol, C.
Foia§ and D. voiculescu solved one of the most difficult problems
of Halmos: the spectral characterization of the norm-closure of
the set of all nilpotent operators. In 1975, Apostol and Voicu
lescu extended the spectral characterization of quasitrianguarity
to operators on c , IP (1 ~ p ~ 00), and several other Banach
o
spaces.
1976 was another exceptional year. Apostol introduced
his "triangular model" of a Hilbert space operator, and showed
that this model is a universal tool to solve problems related to
the modification of the behavior by means of compact perturbatbns.
In 1976, voiculescu's Theorem was recent news; Apostol made one
of the first nontrivial applications of this powerful tool in or
der to characterize the inner derivations with closed range: the
inner derivation induced by an operator T has closed range if and