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 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 »Verwertungsgesellschaft Wort«, Munich. © 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 I>uUum. 1985 CoIU>tLJn.t..in. Apo~.t becaJre ~ .U/. .the fVlOb&m 0-6 .e.oca.t and g.l!olxLt ~ equA..va.€.ence. He ~n ~ a. ~ wUh ~~. W10 coue.d. ~ 1iuLt .t;hM, ~ !A.Il.-6 :to be a. ~ 0-6 hU £.tu.:t pape!t, and .u: .u. wUh .t;hM, pape!t 1iuLt .t;hM, ~ ~ beg-i.nA. The ~ :t.ouche..6 .U/. .the ~n 0-6 :t:hA.-6 pape!t -604. pubUca;Uon t..eJIe d.onJ2. by DoITKn,go 1-IeNrvto; .the ~ !A.Il.-6 abtuJdy .U/. ha~. CoIU>tLJn.t..in. ~ w<.U be ~ a.6 an ou.t.6tan.cU.n. ~ .U/. 0~"1.. .t11eoJIY, .U/. w-w::.h a..orea. he rrrJ.d2. VlVIY ..iJrpoJLtan.:t and deep c.on:t/Ub..L- u.oM>. H.u. ~ and c.o.tee.ag./e-6 ~ .t;hM, vo.eume. :to hU rtrzm.JJIY. HeaA;ty :.thanJv., aile ciu£ :to DoITKn,go I-/eJvuv!o -&0"1.. hU ~ .U/. ~ .t;hM, vo.fune. 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