Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume Workshop in Amsterdam, November 12–14, 1997

459 Pages·2001·12.26 MB·English

by  Vadim Adamyan

Operator Theory: Advances and
Applications
Vol. 122
Editor:
I. Gohberg
Editorial Office:
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 (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)
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
P. A. Fuhrmann (Beer Sheva) Editorial Board:
S. Goldberg (College Park) C. Foias (Bloomington)
B. Gramsch (Mainz) P. R. Halmos (Santa Clara)
G. Heinig (Chemnitz) T. Kailath (Stanford)
J. A. Helton (La Jolla) P. D. Lax (New York)
M.A. Kaashoek (Amsterdam) M. S. Livsic (Beer Sheva)
Operator Theory
and Analysis
The M.A. Kaashoek Anniversary Volume
Workshop in Amsterdam, November 12-14, 1997
H.Bart
1. Gohberg
A.C.M. Ran
Editors
Springer Basel AG
Editors:
H. Bart A.C.M. Rao
Ecooomctrisch lostituut Divisie Wiskunde eo Informatica
Erasmus Uoiversiteit Rotterdam Faeulteit der Exaete Wctcnschappeo
Postbus 17 3 H Vrije Universiteit
3000 DR Rotterdam De Boelelaan 1081 a
The Netherlands 1081 HV Amsterdam
e-mail: [email protected] The Netherlands
e-mail: [email protected]
I. Gohberg
Departmenl of Mathematical Scienccs
Raymond and Beverly Sackler
Faculty of Exact Sciences
Tel Aviv University
69978 Ramat Aviv
Israel
e-mail: [email protected]
2000 Mathematics Subject Classificalion 47-06
A CIP catalogue record for Ihis book is available from the
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Deutsche Bibliothek Calaloging-in-PlIblication Data
Operator theory and analysis: the M. A. Kaashoek anniversary volllme ;
workshop in Amslerdam. November 12 - 14, 1997 I H. Bart ... cd ..
-Basel; Boston; Bedin : Birkhäuser, 200 1
(Operator theory ; Vol. 122)
ISBN 978-3-0348-9502-6 ISBN 978-3-0348-8283-5 (eBook)
DOI 10.1007/978-3-0348-8283-5
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© 2001 Springer Basel AG
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ISBN 978-3-0348-9502-6
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Preface
On November 12-14, 1997 a workshop was held at the Vrije Universiteit
Amsterdam on the occasion of the sixtieth birthday ofM.A. Kaashoek. The present
volume contains the proceedings of this workshop.
The workshop was attended by 44 participants from all over the world: partici
pants came from Austria, Belgium, Canada, Germany, Ireland, Israel, Italy, The
Netherlands, South Africa, Switzerland, Ukraine and the USA. The atmosphere at
the workshop was very warm and friendly. There where 21 plenary lectures, and
each lecture was followed by a lively discussion.
The workshop was supported by: the Vakgroep Wiskunde of the Vrije Univer
siteit, the department of Mathematics and Computer Science of the Vrije Univer
siteit, the Stichting VU Computer Science & Mathematics Research Centre, the
Thomas Stieltjes Institute for Mathematics, and the department of Economics of
the Erasmus University Rotterdam.
The organizers would like to take this opportunity to express their gratitude for
the support. Without it the workshop would not have been so successful as it was.
Table of Contents
Preface ................................................................ v
Photograph of M.A. Kaashoek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xiii
Curriculum Vitae of M.A. Kaashoek .................................... xv
List of Publications of M.A. Kaashoek ................................. xix
l. Gohberg
Opening Address . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xxxi
H. Bart, A.C.M. Ran and H.I. Woerdeman
Personal Reminiscences ............................................ xxxv
V. Adamyan and R. Mennicken
On the Separation of Certain Spectral Components
of Selfadjoint Operator Matrices ...................................... 1
1. Introduction ............................................. 1
2. Conditions for the Separation of Spectral Components ....... 4
3. Example ................................................ 9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11
D. Alpay, V. Bolotnikov, A. Dijksma and Y. Peretz
A Coisometric Realization for Triangular Integral Operators .......... 13
1. Introduction... ... ... ... . ... . ... ... . . .. . .. . ... ... . .. . ... 13
2. Preliminaries and Notations. . .. . ... . ... . ... .. . .. . ... . .... 15
3. Resolvent Operators and Resolvent Equations . . . . . . . . . . . . .. 23
4. The State Spaces 1i£(S) and 1iR(S) ...................... 26
5. The Coisometric Realization. .. . ... . .. ... .. . .. . ... ... .... 34
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51
D.Z. Arov and H. Dym
Some Remarks on the Inverse Monodromy Problem
for 2 x 2 Canonical Differential Systems .... . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
1. Introduction ............................................ 53
2. Monodromy Matrices with Zero 1Ype
in the Upper Half Plane .................................. 59
3. Monodromy Matrices with Zero 1Ype
in the Lower Half Plane .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66
viii Table of Contents
4. Monodromy Matrices with Nonzero Type
in Both Halfplanes ...................................... 68
5. Reparametrizations ...................................... 71
6. Three Classes of J-inner mvf's and Some Examples ........ 74
7. Another Parametrization. . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. .. 83
References ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86
1.A. Ball, T. T. Trent and V Vinnikov
Interpolation and Commutant Lifting for Multipliers
on Reproducing Kernel Hilbert Spaces ................................ 89
1. Introduction ........................................... 89
2. The Multiplier Space for H(kd) .......................... 93
3. Multipliers for Nevanlinna-Pick-type Kernels. . . . . . . . . . . .. 103
4. Interpolation by Multipliers ............................. 107
5. The Commutant Lifting Theorem for Mk (£, £*) .......... 117
6. Examples and Applications............................. 131
References ............................................... 135
H. Bart, T. Erhardt and B. Silbermann
Sums of Idempotents and Logarithmic Residues
in Matrix Algebras ................................................. 139
1. Introduction ........................................... 139
2. Preliminaries .......................................... 140
3. Matrix Algebras Generated by a Single Matrix. . . . . . . . . . .. 142
4. Rank, Trace and Decomposition of Matrices .............. 147
5. Logarithmic Residues of Matrix and Fredholm
Operator Valued Functions. . . . . . . . . . . . . . . . . . . .. . . . . . . . .. 153
6. The Algebra of Block Upper Triangular Matrices ......... 157
References ............................................... 167
V Derkach, S. Hassi and H. De Snoo
Generalized Nevanlinna Functions
with Polynomial Asymptotic Behaviour
at Infinity and Regular Perturbations ................................ 169
1. Introduction ........................................... 169
2. Preliminaries .......................................... 172
3. Multiplicity of the Generalized Poles for the Sum
of Ncfunctions ........................................ 175
4. Polynomial Behaviour at Infinity ........................ 179
5. The Subclasses Induced via Polynomial
Asymptotics to NK,o .................................... 180
6. Spectral Characterizations of Regular
Rank One Perturbations . .. . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. 184
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 188
Table of Contents IX
R.L. Ellis and I. Gohberg
Extensions of Matrix Valued Inner Products on Modules
and the Inversion Formula for Block Toeplitz Matrices ............... 191
o.
Introduction.......................................... 191
1. The Scalar Case ...................................... 192
1.1 Extension of a Scalar Product .......................... 192
1.2 Connections with Prediction Theory
and the Extension Theorem............................ 196
1.3 Properties of the Extended Scalar Product ............... 198
1.4 The Inversion Formula ................................ 205
2. The Matrix Case ...................................... 208
2.1 Matrix Valued Inner Products on Modules ............... 208
2.2 Extension of a Matrix-valued Inner Product. . . . . . . . . . . . .. 211
2.3 Properties of the Extended Matrix-valued Inner Product ... 217
2.4 The Gohberg-Heinig Formula .......................... 224
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227
K.-H. Forster and B. Nagy
Linear Independence of Jordan Chains .............................. 229
1. Introduction ........................................... 229
2. Linearly Independent Chains with Respect
to a Sequence of Operators .............................. 230
3. Jordan Chains of Holomorphic Operator Functions ........ 234
4. Right (spectral) Roots of a Regular Holomorphic
Matrix Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 242
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 244
A.E. Frazho
Weighted Nevanlinna-Pick Interpolation ...... . . . . . . . . . . . . . . . . . . . . . .. 247
o. Introduction ........................................... 247
1. Preliminaries .......................................... 249
2. Some State Space Existence Results ..................... 254
3. The Outer Spectral Factor Case .......................... 256
4. A State Space Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 261
5. A State Space Computation for Band Bh ................ 266
::s
6. The Case when T QT* Q ............................ 269
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 273
1.K. Hale and S.M. Verduyn Lunel
Effects of Small Delays on Stability and Control . . . . . . . . . . . . . . . . . . . . .. 275
1. Introduction ........................................... 275
2. The Abstract Setting of the Problem ..................... 278
3. Difference Equations ................................... 282
4. Neutral Delay Differential Equations. . . . . . . . . . . . . . . . . . . .. 293
x Table of Contents
5. Delayed Boundary Control in a Hyperbolic Equation ...... 295
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 299
I. Karelin and L. Lerer
Generalized Bezoutian, Factorization of Rational Matrix
Functions and Matrix Quadratic Equations .......................... 303
O. Introduction ........................................... 303
1. Bezoutian of Rational Matrix Functions
and Matrix Quadratic Equations ......................... 305
2. Generalized T-Bezoutian of Rational Matrix Functions .... 311
3. Discrete Quadratic Equation and Factorizations
of Rational Matrix Functions ............................ 314
References ............................................... 320
P. Lancaster and A. Markus
A Note on Factorization of Analytic Matrix Functions ................ 323
1. Introduction ........................................... 323
2. Preliminary Results .................................... 325
3. Spectral Divisors and the Case c(F) = 1 ................. 327
4. When r is a Circle ..................................... 328
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 329
H. Langer and C. Tretter
Diagonalization of certain Block Operator Matrices
and Applications to Dirac Operators ................................. 331
O. Introduction ........................................... 331
1. Basic Propositions... ... ... ... ... ... . . .. . .. ... ... . .. . .. 333
2. Block Operator Matrices .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 337
3. The Self-adjoint Case. ... . .. . ... .. . .. .. . .. . .. .. . .. . . ... 341
4. The Non-self-adjoint Case ............................. 350
5. Dirac Operators with Potential.. . .. . .. . . ... ... .. . ... .... 354
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 357
A. Ran, L. Rodman and D. Temme
Stability of Pseudospectral Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . .. 359
1. Introduction ........................................... 359
2. Dissipative Matrices and their Invariant Subspaces ........ 360
3. Functions of the Form Identity Plus a Contraction ......... 365
4. Positive Real Functions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 377
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 382
L. Rodman, I.M. Spitkovsky and H.Z. Woerdeman
Factorization of Almost Periodic Matrix Functions
of Several Variables and Toeplitz Operators .. . . . . . . . . . . . . . . . . . . . . . . .. 385
1. Introduction ........................................... 385
2. Algebras of Almost Periodic Functions
and Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 387
Table of Contents xi
3. Toeplitz Operators.. . . . . . . . . . . . . . . ... . . .. .. . ... . . . . . ... 392
4. Factorization of Sectorial Matrix Functions ............... 396
5. Factorization of Hermitian Matrix Functions .. . . . . . . . . . . .. 402
6. One-sided Invertibility of Toeplitz Operators .. . . . . . . . . . . .. 405
7. Robustness and Continuity of Factorizations .............. 407
References ............................................... 413
R. Zuidwijk
Simultaneous Similarity of Pairs of Companions to their Transposes 417
1. Introduction ........................................... 417
2. Companion and Bezoutian Matrices ..................... 418
3. Simultaneous Similarity ................................ 422
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 424
Conference Programm ............................................... 427
List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 431