Recent Developments in Operator Theory and Its Applications: International Conference in Winnipeg, October 2–6, 1994

447 Pages·1996·12.79 MB·English

by  Daniel Alpay, Israel Gohberg, Lev Sakhnovich (auth.), I. Gohberg, P. Lancaster, P. N. Shivakumar (eds.)

Operator Theory
Advances and Applications
Vol. 87
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)
E. Meister (Darmstadt)
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 (Blackburg) 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)
R.G. Douglas (Stony Brook) D. Yafaev (Rennes)
H. Dym (Rehovot)
A. Dynin (Columbus)
P.A. Fillmore (Halifax) Honorary and Advisory
C. Foias (Bloomington) Editorial Board:
P.A. Fuhrmann (Beer Sheva) 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)
Recent Developments
in Operator Theory
and Its Applications
International Conference in Winnipeg, October 2-6, 1994
Edited by
I. Gohberg
P. Lancaster
P.N. Shivakumar
Birkhauser Verlag
Basel· Boston· Berlin
Editors' addresses:
1. Gohberg
School of Mathematical Sciences
Tel Aviv University
69978 Tel Aviv
Israel
P. Lancaster
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta T2N INA
Canada
P.N. Shivakumar
Department of Applied Mathematics and
Institute of Industrial Mathematical Sciences
University of Manitoba
Winnipeg, Manitoba
Canada R3T 2N2
1991 Mathematics Subject Classification 47-XX
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Recent developments in operator theory and its applications:
international conference in Winnipeg, October 2-6, 1994/ ed.
by I. Gohherg ... -Basel; Boston; Berlin: Birkhauser, 1996
(Operator theory; Vol. 87)
ISBN-13:978-3-0348-9878-2 e-ISBN-13:978-3-0348-9035-9
001: 10.1007/978-3-0348-9035-9
NE: Gochberg, Izrail' C. [Hrsg.l; GT
This work is subject to copyright. All rights are reserved, whcther the whole or part of the material is concerned, specifically the
rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind of use the permission of the copyright holder must be obtained.
© 1996 Birkhauser Verlag, P.O. Box 133, CHA01 0 Basel, Switzerland
Softcover reprint of the hardcover 1st edition 1996
Printed on acid-free paper produced from chlorine-free pulp. TCF co
Cover design: Heinz Hiltbrunner, Basel
ISBN-13 :978-3-0348-9878-2
e-ISBN-13:978-3-0348-9035-9
987654321
Table of Contents
Editorial Introduction ..................... x
D. Alpay, 1. Gohberg, L. Sakhnovich
Inverse scattering problem for continuous transmission lines with
rational reflection coefficient function . . 1
1. Introduction . . . . . . . . . . . . 1
2. Some results on differential expressions 2
3. The reflection coefficient function 6
4. The rational case 10
References . . . . . . . . . . 15
J.A. Ball, 1. Gohberg, M.A. Kaashoek
The band method and the Grassmannian approach for completion
and extension problems. . . . 17
1. Introduction . . . . . . . . 17
2. The Grassmannian approach . 24
3. Proofs of theorems 2.2 and 2.3 32
4. The Caratheodory and Nehari extension problems . 45
5. Operator matrix extension problems . 48
References . . . . . . . . . . . . . . . . . . . . 59
Y. Boishakov, C. V.M. van der Mee, A.C.M. Ran, B. Reichstein, L. Rodman
Polar decompositions in finite dimensional indeflnite scalar product
spaces: Special cases and applications. 61
1. Introduction . . . . . 61
2. H -contractive matrices 64
3. H -plus matrices 67
4. Indefinite scalar products with only one positive square 76
5. Polar decompositions with special unitary factors 82
6. Applications: Linear optics 88
References . . . . . . . . . . . . . . . . . . 92
VI
B. Curgus, B. Najman
Positive differential operators in Krein space L2(R) 95
1. Abstract results . . . . . . . . . . . . . 96
2. Differential operators with constant coefficients . 99
3. Half-range completeness 103
References . 103
M. Faierman, H. Langer
Elliptic problems involving an indefinite weight function 105
1. Introduction . 105
2. Preliminaries. 107
3. Main results 114
4. Examples 123
References . . 124
F. Gesztesy, R. Ratnaseelan, G. Teschl
The KDV hierarchy and associated trace formulas 125
1. Introduction . . . . . . . . . . . . . . . . . 125
2. The KDV hierarchy, recursion relations, and hyperelliptic curves 127
3. The stationary formalism . . 130
4. The time-dependent formalism . . . . . . . . . . . . . . . 140
5. General trace formulas . . . . . . . . . . . . . . . . . . 149
Appendix A. Hyperelliptic curves of the KDV-type and theta functions 154
Appendix B. An explicit illustration of the Riemann-Roch theorem 161
References . . . . . . . . . . . . . . . . . . . . . . . . . 162
E. Grinshpun
On spectral properties of Schrodinger-type operator with complex
potential . . . . . . . . . . . . . . . . . . . 164
1. General perturbation results . . . . . . . . 165
2. Application to the Schrodinger-type operator 167
Appendix 1. Proof of Theorem 1 . . . . 169
Appendix 2. Proof of Theorem 6 . . . . 171
Appendix 3. The subordination condition 174
References . . . . . . . . . . 174
L. Grunen/elder, T. Kosir
Coalgebras and spectral theory in one and several parameters 177
1. Introduction . . . . . . . . . . . . . . 177
2. Coalgebras and comodules . . . . . . . . 178
3. The coalgebra dual of a polynomial algebra 182
4. The primary decomposition theorem 184
5. Monic matrix polynomials 186
6. Several commuting maps 187
7. Multiparameter systems 189
References . . . . . . . . 191
VII
B. Jacob
Destabilization of infinite-dimensional time-varying systems via
dynamical output feedback . . . . . 193
1. Introduction . . . . . . . . . . 193
2. Notation and fundamental results. 194
3. System description . . 195
4. Problem formulation 199
5. Destabilization results . 200
References . 206
P. Lancaster, A. Markus, V. Matsaev
Perturbations of G-selfadjoint operators and operator polynomials
with real spectrum . . . . . . 207
1. Introduction . . . . . . . . . 207
2. Perturbations of finite rank 209
3. Small and compact perturbations 210
4. Applications to operator polynomials 211
5. A factorization theorem in the monic case 213
.6. Differential equations with stably bounded solutions 214
7. The case of noninvertible leading coefficient 217
References . . . . . . . . . . . . . . . . 220
P. Lancaster, A. Markus, V. Matsaev
Definitizable G-unitary operators and their applications to operator
polynomials. . . . . . . . . . . . . 222
1. Introduction . . . . . . . . . . 222
2. Preliminary definitions and results 223
3. Compact perturbations . . . . . 227
4. Operator polynomials quasihyperbolic on T 227
5. Other charcterizations of QHP on T 228
References . . . . . . . . . . . . . 231
R.J. Ober
System theoretic aspects of completely symmetric systems 233
1. Introduction . . . . . . 233
2. Discrete time systems . . 234
3. Continuous-time systems 241
References . . . . . . . . 261
VITI
L. Qiu, T. Chen
Contractive completion of block matrices and its
application to Hoo control of periodic systems. 263
1. Introduction . . . . . . . . . 263
2. H periodic control and lifting. . . . 265
00
3. Matrix contractive completion . . . . 268
4. All Hoo suboptimal periodic controllers 272
5. Concluding remarks. . . . 275
Appendix: Proof of Theorem 1 275
References . . . . . . . . . 279
S. Roch
Spline approximation methods for Wiener-Hopf operators 282
1. Introduction . . . . . . . . . . . . . . . . . . . . . 282
2. Technical preliminaries . . . . . . . . . . . . . . . . 285
3. An algebra of approximation sequences for Wiener-Hopf operators 295
4. Approximation methods for composed operators 304
References . . . . . . . . . . . . . . . . . . . . . . . . . . 307
A.H. Sayed, B. Ha&&ibi, T. Kailath
Inertia conditions for the minimization of quadratic forms in indefinite
metric spaces . . . . . . . . . . . . . . . . 309
1. Introduction . . . . . . . . . . . . . . . 309
2. An inertia result for linear transformations 311
3. The indefinite-weighted least-squares problem 313
4. The equivalent estimation problem . . . . . 315
5. Relations between the IWLS and EE problems 317
6. Incorporating state-space structure . . . . . 320
7. A recursive IWLS problem in the presence of state-space structure 329
8. An application to Hoo-filtering . . . . . . . 336
9. An application to robust adaptive filters. . . 340
10. An application to total least-squares methods 342
11. Concluding remarks 345
References . . . . . . . . . . . . . . . . . 345
P.N. Shivakumar, Q. Ye
Bounds for the width of the instability intervals in the Mathieu
equation 348
1. Introduction . 348
2. Preliminaries . 349
References . . . 357
IX
A.A. Shkalikov
Operator pencils arising in elasticity and hydrodynamics:
The instability index formula . . . . . . 358
Introduction . . . . . . . . . . . . . . . . . . . . . 358
1. Classes of unbounded operator pencils ....... . 360
2. Root subspaces of linear dissipative pencils and their properties 363
3. Quadratic dissipative pencils and the instability index formula 373
4. Applications 381
References 384
B. Silbermann
Toeplitz-like operators and their finite sections 386
1. Introduction . . 386
2. The main results 389
3. Proofs 392
References 397
A. V. Strauss
Spectral representations and spectral functions of symmetric operators 399
O. Introduction . . . . . . . . . . . . . . . 399
1. Spectral representations of linear operators 400
2. Selfadjoint extension in a larger Hilbert space 402
3. The generalized spectral function of a symmetric operator and the
corresponding spectral transformation . . . . 404
4. Generalized resolvents of a symmetric operator 405
References . . . . . . . . . . . . . . . . . 411
T. Tonev, K. Yale
Hankel type operators, Bourgain algebras, and isometries 413
1. Hankel type operators and Bourgain algebras ...... 413
2. Complete continuity of Hankel type operators and isometries . 414
3. Biholomorphic equivalence and Bourgain algebras 416
References . . . . . . . . . . . . . . . . . . . 418
E. Venturino
Effective computation of operators defined by line integrals 419
1. Introduction . . . . . . . . . 419
2. Numerical procedure . . . . . 420
3. Modified Gaussian quadratures . 422
4. Piecewise modified Guassian quadrature on an interval 425
5. The calculation of line integrals .......... 426
6. Extension for the calculation of operators defined by Cauchy principal
value integrals 430
References 431
Appendix 432
EDITORIAL INTRODUCTION
The present volume contains the proceedings of the International Conference on Ap
plications of Operator Theory held in Winnipeg, Canada (October 2nd to 6th, 1994),
which was organized by the Institute of Industrial Mathematical Sciences (IIMS) of the
University of Manitoba. At this conference 92 participants representing 15 countries par
ticipated, and 64 papers were presented. This meeting was the second of a linked pair. The
first was a program of advanced instruction held at the Fields Institute, Ontario, followed
by a research conference. The first of these events gave rise to the volume "Lectures on
Operator Theory and its Applications", published by the American Mathematical Society
for the Fields Institute in 1995.
These two events were the creation of the following Program Committee:
M. A. Dahleh (M.I.T.)
P. A. Fillmore (Dalhousie)
B. A. Francis (Toronto)
F. Ghahramani (Manitoba)
K. Glover (Cambridge)
I. Gohberg (Tel Aviv)
T. Kailath (Stanford)
P. Lancaster (Calgary), Chair
H. Langer (Vienna)
P. N. Shivakumar (Manitoba)
A. A. Shkalikov (Moscow)
B. Simon (Cal. Tech.)
H. Widom (Santa Cruz)
Both events focused on the following main topics: Infinite matrices and projection
methods, linear operators on indefinite scalar product spaces, differential operators and
mathematical systems theory and control.