Operator Theory and Its Applications

FIELDS INSTITUTE COMMUNICATIONS

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

A. G. Ramm P. N. Shivakumar A. V. Strauss Editors

American Mathematical Society Selected Titles in This Series

25 A. G. Ramm, P. N. Shivakumar, and A. V. Strauss, Editors, Operator theory and its applications, 2000 24 Edward Bierstone, Boris Khesin, Askold Khovanskii, and Jerrold E. Marsden, Editors, The Arnoldfest: Proceedings of a conference in honour of V. I. Arnold for his sixtieth birthday, 1999 23 Myrna H. Wooders, Editor, Topics in mathematical economics and game theory. Essays in honor of Robert J. Aumann, 1999 22 Graciela Chichilnisky, Editor, Topology and markets, 1999 21 Shigui Ruan, Gail S. K. Wolkowicz, and Jianhong Wu, Editors, Differential equations with applications to biology, 1998 20 Peter A. Fillmore and James A. Mingo, Editors, Operator algebras and their applications II, 1998 19 William G. Dwyer, Stephen Halperin, Richard Kane, Stanley O. Kochman, Mark E. Mahowald, and Paul S. Selick (Editor-in-Chief), Editors, Stable and unstable homotopy, 1998 18 Panos M. Pardalos and Henry Wolkowicz, Editors, Topics in semidefinite and interior-point methods, 1998 17 Joachim J. R. Cuntz and Masoud Khalkhali, Editors, Cyclic cohomology and noncommutative geometry, 1997 16 Victor P. Snaith, Editor, Algebraic K-theory, 1997 15 Stephen P. Braham, Jack D. Gegenberg, and Robert J. McKellar, Editors, Sixth Canadian conference on general relativity and relativistic astrophysics, 1997 14 Mourad E. H. Ismail, David R. Masson, and Mizan Rahman, Editors, Special functions, q-series and related topics, 1997 13 Peter A. Fillmore and James A. Mingo, Editors, Operator algebras and their applications, 1997 12 Dan-Virgil Voiculescu, Editor, Free probability theory, 1997 11 Colleen D. Cutler and Daniel T. Kaplan, Editors, Nonlinear dynamics and time series: Building a bridge between the natural and statistical sciences, 1997 10 Jerrold E. Marsden, George W. Patrick, and William F. Shadwick, Editors, Integration algorithms and classical mechanics, 1996 9 W. H. Kliemann, W. F. Langford, and N. S. Namachchivaya, Editors, Nonlinear dynamics and stochastic mechanics, 1996 8 Larry M. Bates and David L. Rod, Editors, Conservative systems and quantum chaos, 1996 7 William F. Shadwick, Perinkulam Sambamurthy Krishnaprasad, and Tudor Stefan Ratiu, Editors, Mechanics day, 1996 6 Anna T. Lawniczak and Raymond Kapral, Editors, Pattern formation and lattice gas automata, 1996 5 John Chadam, Martin Golubitsky, William Langford, and Brian Wetton, Editors, Pattern formation: Symmetry methods and applications, 1996 4 William F. Langford and Wayne Nagata, Editors, Normal forms and homoclinic chaos, 1995 3 Anthony Bloch, Editor, Hamiltonian and gradient flows, algorithms and control, 1994 2 K. A. Morris, Editor, Control of flexible structures, 1993 1 Michael J. Enos, Editor, Dynamics and control of mechanical systems: The falling cat and related problems, 1993

http://dx.doi.org/10.1090/fic/025

FIELDS INSTITUTE COMMUNICATIONS

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

Operator Theory and Its Applications

A. G. Ramm P. N. Shivakumar A. V. Strauss Editors

American Mathematical Society Providence, Rhode Island The Fields Institute for Research in Mathematical Sciences

The Fields Institute is named in honour of the Canadian mathematician John Charles Fields (1863–1932). Fields was a visionary who received many honours for his scientiﬁc work, including election to the Royal Society of Canada in 1909 and to the Royal Society of London in 1913. Among other accomplishments in the service of the international mathematics community, Fields was responsible for establishing the world’s most prestigious prize for mathematics research—the Fields Medal. The Fields Institute for Research in Mathematical Sciences is supported by grants from the Ontario Ministry of Education and Training and the Natural Sciences and Engineering Research Council of Canada. The Institute is sponsored by McMaster University, the University of Toronto, the University of Waterloo, and York University, and has aﬃliated universities in Ontario and across Canada.

2000 Mathematics Subject Classiﬁcation. Primary 47–02, 35–02, 34–02, 81–02.

Library of Congress Cataloging-in-Publication Data Operator theory and its applications / A. G. Ramm, P. N. Shivakumar, A. V. Strauss, editors. p. cm. — (Fields Institute Communications, ISSN 1069-5265 ; v. 25) Includes bibliographical references. ISBN 0-8218-1990-9 (alk. paper) 1. Operator theory—Congresses. I. Ramm, A. G. (Alexander G.) II. Shivakumar, P. N. III. Strauss, A. V. (Abraham Vilgelmovich), 1920–1999. IV. Series QA329 .O633 2000 515.724—dc21 00-028877

Copying and reprinting. Material in this book may be reproduced by any means for educational and scientiﬁc purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the ﬁrst page of each article.) c 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was prepared by The Fields Institute. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 05 04 03 02 01 00 Contents

Preface ix

Dedication xi

Publications of A.V. Strauss xv

Functional Models of Regular Symmetric Operators 1 Abraham Strauss+

Property C for ODE and Applications to Inverse Problems (Plenary Lecture) 15 A.G. Ramm

Decomposition Theorems in Banach Spaces 77 Ya.I. Alber

On a New Statement of Inverse Problem of Quantum Scattering Theory 95 Ruben Airapetyan

Continuous Methods for Solving Nonlinear Ill-Posed Problems 111 Ruben G. Airapetyan, Alexander G. Ramm and Alexandra B. Smirnova

Quasi-Coisometric Realizations of Upper Triangular Matrices 139 D. Alpay and Y. Peretz

Linear Systems, Operator Model Theory and Scattering: Multivariable Generalizations 151 J.A. Ball

Problems on the Realization of Functions 179 J.A. Ball and N.J. Young

Multiplication Theorems for J-Contractive Operator-Valued Functions 187 Sergey Belyi and Eduard Tsekanovskii

v vi Contents Spectral Theory of Commutative Jacobi Fields: Direct and Inverse Problems 211 Yurij M. Berezansky

The Forward Propagation Method Applied to the Inverse Obstacle Problem of Electromagnetics 225 Giovanni F. Crosta

Spatial Patterning in Reaction-Diﬀusion Systems with Nonstandard Boundary Conditions 239 Jan Eisner and Milan Kuceraˇ

On an Abstract Boundary Value Problem with the Eigenvalue Parameter in the Boundary Condition 257 Anatolii Etkinˇ

Some Applications of the Spectral Shift Operator 267 Fritz Gesztesy and Konstantin A. Makarov

Application of the Hybrid Stochastic-Deterministic Minimization Method to a Surface Data Inverse Scattering Problem 293 S. Gutman and A.G. Ramm

On a Theorem of Mochizuki and Uchiyama About Long Range Oscillating Potentials I 305 W. Jager¨ and P. Rejto

Basic Properties of Linear Fractional Mappings of Operator Balls: Schroeder’s Equation 331 V. Khatskevich and V. Senderov

Homogenization of the Dirichlet Variational Problems in Orlicz-Sobolev Spaces 345 Eugene Ya. Khruslov and Leonid S. Pankratov

Development of M.K. Gavurin’s Pseudoperturbation Method 367 B.V. Loginov, D.G. Rakhimov and N.A. Sidorov

A Bridge Between Operator Theory and Mathematical Biology 383 Julian´ Lopez-Gom´ ez´

Measures on Eﬀects and on Projections in Spaces with Indeﬁnite Metric 399 Marjan Matvejchuk

Concentration Behavior of Solutions to a Chemotaxis System 415 Toshitaka Nagai Contents vii The Solution of Linear Semideﬁnite Ill-Posed Problems by the Conjugate Residual Method 423 R. Plato

Justiﬁcation of the Limiting Absorption Principle in R2 433 A.G. Ramm

Krein’s Method in Inverse Scattering 441 A.G. Ramm

Existence and Uniqueness of the Scattering Solutions in the Exterior of Rough Domains 457 Alexander G. Ramm and Marco Sammartino

Existence and Uniqueness of Solutions of Retarded Quaslilinear Wave Equations 473 Shigui Ruan and John C. Clements

On Solution of Elliptical Interface Crack Problem 485 Efim I. Shifrin and Boˇstjan Brank

Some New Eﬀects for Complete Second Order Linear Diﬀerential Equations in Hilbert Spaces 497 Alexander Shklyar

Abstract Boundary Value Problems for Operator Equations 509 Vladilen A. Trenogin

On Spectral Decompositions of a Restriction of a Diﬀerential Operator 523 A.V. Tsyganov

Approximation of Compactly Supported Functions with Free Phase by Functions with Bounded Spectrum 531 N.N. Voitovich, Yu.P. Topolyuk and O.O. Reshnyak

Sourcewise Representation and A Posteriori Error Estimates for Ill-Posed Problems 543 Anatoly Yagola and Konstantin Dorofeev

Coexistence States for Lotka-Volterra Systems with Cross-Diﬀusion 551 Yoshio Yamada

Nonhomogeneous String Problem with Periodically Moving Boundaries 565 Masaru Yamaguchi and Hiroshi Yoshida

Preface

This volume contains selected papers presented at the International Conference on Operator Theory and Its Applications which was held in Winnipeg, in 1998, Oct. 7-11. The Conference was organized by A.G. Ramm, P.N. Shivakumar and A.V. Strauss. Professor A.V. Strauss has died on Oct. 18, 1999. This volume is dedicated to him and a brief note about his life is included. The Conference was supported by NSF, the Canadian Mathematical Society, the University of Manitoba, the Institute of Industrial Mathematical Sciences, the Fields Institute and other institutions. The organizers and the participants are very thankful to all of the above institutions for their support. The scientiﬁc program of the Conference was rather broad and the applications of the operator theory were strongly emphasized. This is seen in the table of contents of the volume: together with the papers on the abstract operator theory there are many papers on the theory of diﬀerential operators, boundary value problems, and on applications to biology, chemistry, wave propagation, inverse problems and many other areas. The editors believe that applications of operator theory are as important as the theory itself. The operator theory is the language of the modern analysis and its applications and the papers in this volume illustrate this.

A.G. Ramm P.N. Shivakumar A.V. Strauss+ Editors

January 2000

Abraham V. Strauss

Professor Abraham Vilgelmovich Strauss was born on June 15, 1920 in Tashkent, the capital of Uzbekistan, a republic of the former USSR. His father, a citizen of Austria, was sent by the Russian government to Tashkent during World War I. After the war, the Strauss family decided to settle down in Peremyshl, a town in Poland where Abraham’s father was born. Abraham grew up in Peremyshl. In 1939, right before the second World War began, Abraham entered Lvov university as an undergraduate student of mathematics. Abraham attended lectures of Ba- nach, Mazur, Schauder, Orlicz, and M. Krein. During the war, from 1941 till 1944, Abraham studied at the Kuibyshev pedagogical institute (Kuibyshev is now called Samara). After graduation, he began working at the Kuibyshev Aviation institute as an instructor in 1944, and was admitted as a Ph.D. student at the Kuibyshev pedagogical institute in 1945 under the supervision of S.P. Pulkin. Abraham earned both a Ph.D. and a Doctor of Science degree in Mathematics from Moscow State University in 1948 and 1960, respectively. His dissertations have been approved by A. Plessner, M. Naimark, Yu. Berezansky and M. Livsic. The principal areas of his research were spectral theory of linear operators in Hilbert spaces, extension theory for symmetric linear operators, theory of the characteristic functions and functional models of linear operators, boundary value problems with boundary conditions depending on spectral parameter. Professor A.V. Strauss solved difficult mathematical problems: he gave a de- scription of all generalized spectral functions and generalized resolvents of symmetric linear operators, not necessarily densely defined, with possibly infinite defect xi xii Dedication indices; he gave a convenient definition of the characteristic operator-valued functions of unbounded non-selfadjoint operators and considered multiplication and factorization theorems for these functions; he established connections between the characteristic functions and generalized resolvents. His definition of the characteristic function allowed him to compute, for the first time, characteristic functions of partial differential operators. This definition contained the definition of the boundary value spaces which are widely used in spectral theory. Professor Strauss published many results in this area. In the theory of extensions of symmetric operators the results of von Neumann, Friedrichs, Krein, Naimark and Strauss are well known. In 1948, Professor A.V. Strauss began working at the Ul’yanovsk Pedagogical University and from 1954 until his death he worked as a chairman of the Mathemat- ical Analysis Department at this University. More than 35 students received their Ph.D.s in mathematics under his supervision. He was a member of the Presidium of the Scientific and Pedagogical Council of the Ministry of higher education of the USSR, a member of the council on mathematical sciences of the ministry of education of Russian Federation, a member of the Presidium of the regional consortium of the mathematics departments of the pedagogical universities. He was awarded an order of the “Sign of Honour” and some medals. He was a founder and editor-in chief of the Functional Analysis - Inter-Institutional Collection of Scientific Articles. This Journal was published by the Ul’yanovsk Pedagogical University and was popular among specialists in operator theory in the former USSR. In the Mathe- matical Reviews (51#1317) the first issue of this Journal was named “A collection of articles edited by A.V. Strauss”. Professor A.V. Strauss was an Honoured Scientist of the Russian Federation and a member of the International Academy of Informatization associated with the United Nations. Professor A.V.Strauss organized many conferences in Functional Analysis and Operator Theory in Ul’yanovsk. He organized jointly with A.G. Ramm and P.N. Shivakumar the international conference on Operator Theory and Its Ap- plications held in Oct.7-11, 1998, in Winnipeg, Canada. Professor A.V. Strauss was married to Antonina Yakovlevna Oparina in 1944. She was a school teacher who taught Russian language and literature. She is now retired. Professor Strauss and his wife have two sons, Vladimir and Leonid. Both of them earned a Ph.D in mathematics. Vladimir is a professor of mathematics at the Simon Bolivar University in Caracas. Leonid is Dean of Educational Faculty and an associate professor of mathematics of the Algebra Division of the Faculty of Mechanics and Mathematics of the Ul’yanovsk State University. Professor Strauss loved music, visual arts and poetry, especially the poetry of A. Pushkin, S. Esenin and A. Mitzkevich. He liked swimming, ping pong, and chess. Professor Strauss was fluent in English, German, French, Polish and knew Latin very well. Professor A.V. Strauss died on Oct. 18, 1999. He was an outstanding mathematician and teacher, a very gentle and gracious person. His students and colleagues were very fond of him. Many of his colleagues, as well as people who knew him, will remember him as a wonderful person, an outstanding scientist, and a dedicated teacher. Dedication xiii

The editors are very grateful to the wife of A.V. Strauss, his son, Dr V. Strauss, and to Drs. M. Chugunova, A. Vernik and E. Tsekanovsky for the comments and information they provided.

The editors

Publications of Abraham V. Strauss

1. On the theory of Hermitian operators. - Dokl. Akad. Nauk. SSSR, 1949, v. 67, N 4, pp. 611-614 (Russian). 2. A class of regular operator-function. - Dokl. Akad. Nauk. SSSR, 1950, v. 70, N 4, pp. 577-580 (Russian). 3. On generalized resolvents of symmetric operator. Dokl. Akad. Nauk. SSSR, 1950 v. 71, N 2, pp. 241-244 (Russian). 4. On the theory of symmetric operator generalized resolvents. - Dokl. Akad. Nauk. SSSR, 1951, v. 78, N 2, pp. 217-220 (Russian). 5. Spectral functions of symmetric operators with finite defect numbers. - Uchen. zap. Kuibyshev. gos. ped. i uchit. inst., Kuibyshev, 1952, is. 11, pp. 17-66 (Russian). 6. On characteristic properies of generalized resolvents. - Dokl. Akad. Nauk. SSSR, 1952, v. 82, N 2, pp. 209-212 (Russian). 7. Generalized resolvents of symmetric operators. - Izv. Akad. Nauk. SSSR, Ser. Mat., 1954, v. 18, N1, pp. 51-86 (Russian). 8. On spectral functions of differential operators. - Izv. Akad. Nauk. SSSR, Ser. Mat., 1955, v. 19, N4, pp. 201-220 (Russian). 9. A resolution on eigenfunctions a second order boundary problem on semiaxis. Izv. Akad. Nauk. SSSR, Ser. Mat., 1956, v. 20, N6, pp. 783-792 (Russian). 10. Generalized resolvent of even-order differential operator. - Dokl. Akal. Nauk SSSR, 1956, v. 111, N4, pp. 773-776 (Russian). 11. On spectral functions of even-order differential operator. - Dokl. Akad. Nauk SSSR, 1956, v. 111, N1, pp. 67-70 (Russian). 12. Generalized resolvents of symmetric operators and eigenvalue resolution for a class of boundary-value problems. - Uspehi Mat. Nauk, 1957, v. 12, is. 1, pp. 251-253 (Russian). 13. On generalized resolvents and spectral functions of even-order differential operators. Izv. Akad. Nauk. SSSR, Ser. Mat., 1957, v. 21, N6, pp. 785-808 (Russian). 14. On spectral functions of the differential operator. - Uspehi Matem. Nauk, 1958, v.13, is. 6, pp. 185-191 (Russian). 15. On charecteristic functions of linear operators. - Dokl. Akad. Nauk, 1959, v.126, N 3, pp. 514-516 (Russian). 16. On the multiplication theorem for charactheristic functions of linear operators. Dokl. Akad. Nauk, 1959, .126, N 4, 723-726 (Russian). 17. Characteristic functions of linear operators. Izv. Akad. Nauk. SSSR, Ser. Mat., 1960, .24, N 1, p.43-74 (Russian); English transl. in AMS Transl. (2) 40 (1964), pp. 1-37 (Russian). 18. On the theory of characteristic functions of linear operators. - Funkts. Analiz i Pril., Baku, 1961, c.278 (Russian). xv xvi Publications of Abraham V. Strauss

19. On some families of symmetric operator extensions. - Dokl. Akad. Nauk, 1961, v. 139, N 2, pp. 316-319 (Russian). 20. The resolution in eigenfunction som one-dimensional boundary-value problems. - Trudy 1-oi nauch. conf. mat. kaf. ped. inst. Povolzhia. Kuibyshev, 1961, pp. 110-115 (Russian). 21. Some problems of symmetric non-selfadjoint operator extension theory. - - Trudy 2-oi nauch. conf. mat. kaf. ped. inst. Povolzhia. Kuibyshev, 1962, is. 1, pp. 121-124 (Russian). 22. On self-adjoint operators in direct sum of two Hilbert spaces. - Trudy 2-oi nauch. conf. mat. kaf. ped. inst. Povolzhia. Kuibyshev, 1962, is. 1, pp. 125-128 (Russian). 23. On self-adjoint operators in direct sum of Hilbert spaces. - Dokl. Akad. Nauk SSSR, 1962, v.144, N 3, pp. 512-515 (Russian). 24. On symmetric operator spectral theory. - Volzh. mat. sb., Kuibyshev, 1963, is. 1, pp. 221-226 (Russian). 25. On symmetric operators spectral resolutions. - Dokl. Akad. Nauk SSSR, 1963, v.152, N 3, pp. 563-566 (Russian). 26. On the ordinary diﬀerential operator spectra multiplicity. - Dokl. Akad. Nauk SSSR, 1964, v.155, N 4, pp. 771-774 (Russian). 27. On the symmetric operator extensions depending on parameter. - Izv. Akad. Nauk SSSR, Ser. Mat.,1965, v.29, N 6, pp. 1386-1416 (Russian). 28. On one-parameter families of summetric operator extensions. - Izv. Akad. Nauk SSSR, Ser. Mat.,1966, v.30, N 6, pp.1325-1352 (Russian). 29. One-parameter families of symmetric operator extensions. - Tezis kr. nauch. soobchen. Mezdunarodnogo kongressa matematikov, Moscow, Sect. 5, 1966, p. 83. 30. On extensions, characteristic functions and generalized resolvents of symmetric operators. - Dokl. Akad. Nauk SSSR,1968,v.178, N 4, pp. 790-792 (Russian). 31. On extensions and characteristic function of a symmetric operator. - Izv. Akad. Nauk SSSR, Ser. Mat., 1968, v.32, N 1. pp. 186-207 (Russian); (English transl. in Math. USSR - Izvestija, 2(1968), pp. 181-204.) 32. Extensions and generalized resolvents of a symmetric operator which is not densely deﬁned. - Izv. Akad. Nauk SSSR, Ser. Mat., 1970, v.34, N 1. pp. 175-202 (Russian); (English transl. in Math. USSR - Izvestija, 4(1970), pp. 179-208.) 33. On spectral resolutions of a regular symmetric operator. Dokl. Akad. Nauk SSSR, 1972, v. 204, N 1, pp.52-55 (Russian); (English transl. in Sov. Math. Dokl. 13(1972), pp. 614-618.) 34. On extensions of the semi-bounded operator. - Dokl. Akad. Nauk SSSR, 1973, v.211, N 3, pp. 543-546 (Russian). 35. On interconnection between algebra and mathematical analysis courses. - Tezisy vsesojuzn. seminar prepod. algebry i teorii chisel ped. inst., Ulyanovsk, 1973, pp. 17-18 (Russian). (co-author: Basanov B.V.) 36. All-union seminar of algebra and number theory tutors from pedagogical institutes. - Matematika v shcole, 1974, N1, pp.94-95 (Russian). 37. From experience of work according to the new program of mathematical analysis course. - Materialy seminara zav. kafedrami matematiki pedinsti- tutov RSFSR, Tula, 1975, pp.29-33 (Russian). Publications of Abraham V. Strauss xvii

38. On the family of semi-bounded operator extensions. - Funkts. Analiz, Ulyanovsk,1976(6), pp. 155-164 (Russian). 39. On extensions of bounded operator with bound conservation. - Funkts. Analiz, Ulyanovsk, 1976(6), pp. 165-173 (Russian). 40. On self-adjoint extensions of semi-bounded operator. - Funkts. Analiz, Ulyanovsk, 1976(7), pp. 161-171 (Russian). 41. On resolvents of a symmetric operator extensions. - Funkts. Analiz, Ulyanovsk, 1976(8), pp.162-173 (Russian). 42. On semi-bounded operator extension theory. - Funkts. Analiz, Ulyanovsk, 1977(9), pp. 167-173 (Russian). 43. On spectral theory of regular symmetric operators. - Funkts. Analiz, Ulyanovsk, 1978(10), pp.145-153 (Russian). 44. On the spectral representation of symmetric operator. - Funkts. Analiz, Ulyanovsk, 1979(12), pp. 159-166 (Russian). 45. Mark Aronovich Naimark. Obituary.- Uspehi Matem. Nauk, 1980, v.35, is. 4, pp. 135-139 (co-author: Gelfand I.M., Graev M.I.,Zhelobenko D.G., Ismagilov R.S., Krein M.G., Kudriavcev L.D., Nikolskii S.N., Helemskii A. I.) (Russian). 46. On the methodological direction of mathematical analisys course.- Tezisy docladov vsesouz. nauch. conf., v.1, Tashkent, 1982, p. 154. 47. On a theory of bounded positive operator extremal extensions. - Funkts. Analiz, Ulyanovsk, 1982(18), pp. 115-126 (Russian). 48. On the extensions of symmetric operator having regular point on real axis. - Funkts. Analiz, Ulyanovsk, 1985(24), pp. 123-136 (Russian). 49. Analitic models of regular symmetric operators. 11 Vsesouz. shkola po teorii operatorov v funktz. prostr. Tezisy dokladov, v 2, Cheliabinsk, 1986, p. 151. 50. Generalized resolvents of bounded symmetric operators which are not densely deﬁned. - Funkts. Analiz, Ulyanovsk, 1987(27), pp. 187-196 (Russian). 51. The analytic representation of a bounded symmetric operator.- 12 shkola po teorii operatorov v funktz. prostr. Tezisy dokladov, v 2, Tambov, 1987, p.124. 52. On the theory of entire operators.- 13 Vsesouz. shkola po teorii operatorov v funktz. prostr. Tezisy dokladov. Kuibyshev. 1988, pp. 213-214. 53. Pseudo resolvents and representations of linear operators.- Funkts.Analiz, Ulyanovsk, 29 (1989), pp. 141-152 (Russian). 54. Guiding projectors and representations of symmetric operators.- 16 Vsesouz. shkola po teorii operatorov v funktz. prostr. Tezisy dokladov. Nizhnii Novgorod, 1991, p. 251. 55. Guiding projectors and representations of symmetric operators. -INTERN- ATIONAL CONFERENCE dedicated to 100th birthday of STEFAN BA- NACH, May 6-8,1992, L’viv,Ukraine, pp.76-77. 56. Regular and entire operators. - INTERNATIONAL CONFERENCE dedicated to the MEMORY of academician M.P.KRAVCHUC, September 22- 28,Kiev-Lutck, Ukraine, p. 245. 57. On the theory of regular and entire operators. - Funkts. Analiz, Ulyanovsk, 33(1992), pp. 65-83 (Russian). 58. Spectral representations of linear operators. - Funkts. Analiz, Ulyanovsk, 34(1993), 80-93 (Russian). xviii Publications of Abraham V. Strauss

59. Symmetric relations and module space. - Integral Equations Operator The- ory, 1994, 18, 154-165 (Co-authors A. Dijksma, H.S.V. de Snoo) 60. Spectral representations and spectral functions of symmetric operators. - International Conference. Workshop on Applications of Operator Theory. Institute of Industrial Mathematical, Sciences, University of Manitoba, 1994, pp.76-77 61. On Aronszajn’s projection formula.- Funkts. Analiz, Ulyanovsk, 35(1994), pp. 123-136 (Russian). 62. Spectral representations and spectral functions of symmetric operators. Op- erator Theory: Advances and Applications. - Birkh¨auser Verlag, Basel,1996, vol.87, pp. 399-412. 63. The symmetric operator model. - Funkts. Analiz, Ulyanovsk, 36(1995), pp.120-130 (Russian). 64. Functional models of symmetric operators. - Conference on Operator Theory in Honour of Moshe Livsic 80-th Birthday, Ben-Gurion University of the Negev, Beer-Sheva and Weizmann Institute of Science, Rehovot, June 29 -July 4, 1997, Abstracts, p.48. 65. A functional model of symmetric operator and the solution of the corre- sponding moment problem. - Mark Krein International conference. Opera- tor theory and applications. August 18-22, 1997, Odessa, Ukraine. Book of abstracts. Ukrainian National Academy of Science, Institute of Mathematics and Mechanics of Odessa University. Kiev 1997, pp. 115-116. 66. Two classes of regular symmetric operators. - International Conference on the Theory of Characteristic Functions of Linear Operators, Abstracts, Ulyanovsk, pp. 40-42. 67. Functional models and generalized spectral functions of symmetric operators. - Preprint, Ulyanovskij Gosudarst. Ped. Inst.,Ulyanovsk, 1997, pp. 3-70. 68. Functional models of linear operators in Hilbert spaces.- Funktz. prostr., Diﬀeren. operatory. Problemy math. obraz. Tezisy dokladov Mezdunar. konferencii, posviysh. 75-letiu L.D. Kudriyvceva, Moskva. 1998, p. 68. Operator Theory and Its Applications A. G. Ramm, P. N. Shivakumar, and A. B. Strauss, Editors

This volume contains a selection of papers presented at an international conference on operator theory and its applications held in Winnipeg. The papers chosen for this volume are intended to illustrate that operator theory is the language of modern analysis and its applications. Together with the papers on the abstract operator theory are many papers on the theory of differential operators, boundary value problems, inverse scattering and other inverse problems, and on applications to biology, chemistry, wave propagation, and many other areas. The volume is dedicated to the late A. V. Strauss, whose principal areas of research were spectral theory of linear operators in Hilbert spaces, extension theory for symmetric linear operators, theory of the characteristic functions and functional models of linear operators, and boundary value problems with boundary conditions depending on spectral parameter. The bibliography of publications by A. V. Strauss combined with the papers from the conference provide both historical perspective and contemporary research on the ﬁeld of operator theory and its applications.

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