International Workshop

on Operator Theory and Its Applications

June 13 – June 16, 2000 Université Bordeaux I

Program Abstracts List of Participants

IWOTA-2000

International Workshop Operator Theory and Its Applications

June 13 - June 16, 2000 Université Bordeaux 1

Welcome to IWOTA 2000 in Bordeaux.

The International Workshop on Operator Theory and its Applications (IWOTA) is a satellite meeting of the international symposium on the Mathematical Theory of Networks and Systems (MNTS). This year the MNTS is held in Perpignan, France, June 19-23. IWOTA 2000 is the eleventh workshop of this kind.

The previous IWOTA meetings were held in

Santa Monica, California, USA (1981) Rehovot, Israel (1983) Amsterdam, The Netherlands (1985) Mesa, Arizona, USA (1987) Rotterdam, The Netherlands (1989) Sapporo, Japan (1991) Regensburg, Germany (1995) Bloomington, Indiana, USA (1996) Groningen, The Netherlands (1998).

The international Steering Committee of IWOTA consists of I. Gohberg (President, Tel Aviv), T. Ando (Sapporo), H. Bart (Rotterdam), H. Bercovici (Bloomington), R. Dijksma (Groningen), H. Dym (Rehovot), C. Foias (College Station), J. W. Helton (Vice-President, La Jolla), M. A. Kaashoek (Vice-President, Amsterdam), H. Langer (Vienna), R. Mennicken (Regensburg), L. Rodman (Williamsburg), J. G. Stampfli (Bloomington).

The purpose of IWOTA is to bring together mathematicians and engineers interested in operator theory and its applications to related fields, from differential equations, harmonic and complex analysis, to mathematical systems and control theory and signal processing. In fact, every two years IWOTA gathers leading experts from over the world

for an intensive exchange of information and opinions, and for tracing the future developments in the field. The large participation of researchers from Eastern European and CIS countries is also an important feature of IWOTA meetings.

The program consists of 17 plenary lectures of 50 minutes including 5 minutes for questions and remarks, and 90 section lectures of 45 and 30 minutes also including 5 minutes for questions and remarks. The response to the invitation was rather overwhelming. Therefore, to give most of the participants willing to speak the opportunity to present their results, we have introduced five parallel sessions in the afternoon. Despite our efforts we did not succeed in avoiding Poster Session, which will be organized on Thursday, June 15, in the afternoon. The organizers understand, the responsibility they undertook squeezing the participants proposals into this strict framework. Abstracts of all proposals received by June 5 are published in the Book of Abstracts. We hope for your understanding of the necessity this decision.

We are registered more than 170 participants from 29 countries of all continents (except Antarctica): Argentina - 1, Armenia -1, Austria - 1, Azerbaijan - 1, Canada - 1, Finland - 4, France - 37, Germany - 5, Hungary - 1, Iran - 1, Israel - 14, Japan - 1, Maroc - 2, Mexico - 1, The Netherlands - 6, New Zealand - 2, Norway - 2, Poland - 3, Portugal - 2, Romania - 6, Russia - 10, Spain - 3, Sweden - 2, Turkey - 1, Ukraine - 20, United Kingdom - 4, USA - 32, Uzbekistan - 1, Venezuela - 4.

Some financial support can be provided for some participants, especially from the Eastern countries. This has been made possible through the generous sponsoring by international and French institutions listed below. We are very grateful for their help.

Special thanks

We thank the City of Bordeaux for the welcome reception on Tuesday, June 13.

Warmest thanks are given to an informal team of our closest collaborators working hard for this large meeting,

Geneviève Castaignede, Emmanuel Fricain, Stanislav Kupin, Cathy Metivier, Ludmila Nikolskaia, and Pascale Vitse,

as well as to many others whose timely advice was so important, namely to

Eric Amar, Hakim Bachir, Robert Deville, Iban Harlouchet, Andréas Hartmann, Chantal Menini, Macha Nikolskaia, David Sherman, Elisabeth Strouse, and Fouad Zarouf.

The Organizers Talence, June 2000

Acknowledgements

As the majority of its homologues, the IWOTA conferences have no permanent funding. Their existence and function rely on financial aid from international and national organizations supporting the development of fundamental sciences.

The organizers of the International Workshop on Operator Theory and its Applications 2000 gratefully acknowledge the following for their financial support:

INTAS (International Association for the Promotion of Cooperation with Scientists from the Independent States of the former Soviet Union), European Union Ministère de l’Education Nationale, de la Recherche et de la Technologie, France INRIA (Institut National de Recherche en Informatique et en Automatique, Sophia Antipolis), France CNRS (Centre National de Recherche Scientifique), France Ministère des Affaires Etrangères, France Région Aquitaine Groupe d’Analyse Mathématique de Bordeaux Université Bordeaux I Université Lyon I Université Lille I Laboratoire de Mathématiques Pures de Bordeaux Institut de Mathématiques de Bordeaux Ville de Bordeaux

Organizing Committee: Prof. Nikolai Nikolski (Chairman, Université Bordeaux-I), Prof. Laurent Baratchart (INRIA, Sophia Antipolis), Dr. Aleksandr Borichev (Université Bordeaux I), Dr. Gilles Cassier (Université Lyon I), Prof. Jean Esterle (Université Bordeaux I), Prof. V.-H.Vasilescu (Université Lille I)

Contents

Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Contents  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Program  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Abstracts  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
List of participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Table de matières

Préface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Table de matières  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Programme  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Résumés  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Liste de participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Program Programme

Location:

Plenary lectures: Grand Amphitéâtre de Mathématiques (G.A.M.), building A33

Session lectures: building A29

Section topics:

Section A: Operators
Section B: RKHS, Constrained Interpolation
Section C: SIO, Toeplitz, Hankel and Hardy
Section D: Differential Operators, Scattering
Section E: Systems, Frames, Operators

Books:

On Wednesday, June 14, 2000, 10.25–13.00, and Thursday, June 15, 10.25–13.00, 18.05–19.00, Springer-Verlag, Birkhäuser-Verlag and Mollat Book Store (Bordeaux) will have a book display in the entrance hall of G.A.M., A33.

IWOTA 2000, Université Bordeaux I (buildings A29, A33)

PROGRAM (short version)

Tuesday, June 13, 2000

time  Section A Section B Section C Section D Section E
8.30–9.30  Registration
9.35–9.45  Opening
Chair  Kaashoek
9.45–10.35  Meyer

Coffee/tea

Chair  Sadosky
11.00–11.50  Curtain
11.55–12.45  Treil

Lunch

Chair  Amar  Alpay  Spitkovsky Tretter  Yger
14.35–15.20 Douglas  Kheifets  Leblond  Mityagin  Staffans
15.25–16.10 Lomonosov Ball  Dym  Naboko  Putinar

Coffee/tea

Chair  Volberg
16.45–17.35  de Branges
17.45  Departure to the reception

Wednesday, June 14, 2000

time  Section A Section B Section C  Section D Section E
Chair  Dym
8.40–9.30  Foias
9.35–10.25  Alpay

Coffee/tea

Chair  H. Langer
10.50–11.40  Birman
11.45–12.35  Megretski

Lunch

Chair  Douglas  Seip  Ferguson  Tkachenko Pavlov
14.25–15.10 Le Merdy McCarty  Arazy  Solomyak Latushkin
15.15–15.45 Li  Kapustin Yu. Karlovich Mennicken Minkin
15.50–16.20 Abakumov Bakonyi Dritschel Petkov Belishev

Coffee/tea

Chair  Le Merdy McCarthy Böttcher  Shkalikov Putinar
16.45–17.15 Amar  Bruzual  Brudnyi  Gasimov  Jaming
17.20–17.50 Tomilov  Kerchy  Nakazi  Berezanski Yger
17.55–18.25 Zemanek  Dehghan  Lyubarskii  Olshevski Xia
18.30–19.00 Kurasov  Kosiek  Pott  Hasanov  de Snoo

Thursday, June 15, 2000

time  Section A Section B Section C  Section D Section E
Chair  Partington
8.40–9.30  Böttcher
9.35–10.25  Sadosky

Coffee/tea

Chair  Staffans
10.50–11.40  Volberg
11.45–12.35  Saff

Lunch

Chair  Rodman  Vasilescu  Treil  Megretski Vinnikov
14.25–15.10 Kaashoek Shimorin  Spitkovsky  Shkalikov Pavlov
15.15–16.00 Nevanlinna Partington Ferguson  Tkachenko Yafaev

Coffee/tea

Chair  Peller  Baratchart Verdyun Lunel Naboko  Yafaev
16.25–16.55 Azizov  Dyakonov Litvinchuk  Malamud Ivanov
17.00–17.30 Markus  Badea  Matsaev  Janas  Nudelman
17.35–18.05 Tikhonov Kupin  Paneah  van Shagen Gubreev
18.15–19.40  Poster session & Coffee/tea
20.00–23.00  Dinner

Friday, June 16, 2000

time  Section A Section B  Section C Section D Section E
Chair  Helton
8.40–9.30  Verduyn Lunel
9.35–10.25  Seip

Coffee/tea

Chair  Mennicken
10.50–11.40  Pisier
11.45–12.35  Davidson

Lunch

Chair  Cassier  Saff  Davidson  Mityagin  Curtain
14.25–15.10 Peller  Yuditskii  Coburn  Arov  Vinnikov
15.15–16.00 Adamyan  H. Langer  Poltoratskii Zworski  Larson
Chair  Nevanlinna Deville  Coburn  Petkov  Solomyak
16.05–16.35 Belitskii  Tsekanovskii Xu  Tretter  Fedorov
16.40–17.10 Rodman  Marcantognini Pedersen  Zwart  Avdonin

Coffee/tea

Chair  Kaashoek
17.45–18.35  Helton
18.35  Closing

IWOTA 2000, Université Bordeaux I (buildings A33, A29)

PROGRAM (full version)

Tuesday, June 13, 2000

Time  Section  Lecture

(location)

8.30–9.30  Hall/A33  Registration

9.35–9.45 G.A.M./A33 Opening Prof. Francis Hardouin, Vice-President of the University

Bordeaux 1

G.A.M./A33 Welcome to IWOTA 2000 by Prof. M. A. Kaashoek, Vice-

President of the IWOTA Steering Committee

9.45–10.35 G.A.M./A33 Meyer, Y., Paris, Real analysis, complex analysis, partial differential equations and operator theory

Hall/A33  Coffee/tea

11.00–11.50 G.A.M./A33 Curtain, R., Groningen, Stabilization of disspative systems by col-

located feedback

11.55–12.45 G.A.M./A33 Treil, S., Providence, Recent advances in similarity problem

Lunch

14.35–15.20 A, A29 Douglas, R., College Station, Quotient Hilbert modules and mul-

tivariate canonical models

B, A29 Kheifets, A., Ariel, Couplings, scales and wave operators in inter-

polation problems

C, A29 Leblond, J., Sophia-Antipolis, Diagonalization of Toeplitz operators, convergence properties and asymptotic estimates for H^{2} approximation on part of the boundary

D, A29 Mityagin, B., Columbus, Smoothness of solutions of nonlinear

ODEs

E, A29 Staffans, O., Åbo, Lax-Phillips scattering and well-posed linear systems

15.25–16.10 A, A29 Lomonosov, V., Kent, Positive functionals on general operator al-

gebras

B, A29 Ball, J., Virginia, Multivariable linear systems, scattering, unitary

dilations, and operator model theory for row contractions

C, A29 Dym, H., Rehovot, On the role of the Muckenhoupt condition in

the theory of J-inner matrix functions

D, A29 Naboko, S., St. Petersburg, Asymptotics of generalized eigenvectors and the spectral analysis of unbounded Jacobi matrices

E, A29 Putinar, M., Santa Barbara, The Friedrichs operator of a planar domain

Hall/A33  Coffee/tea

16.45–17.35 G.A.M./A33 de Branges, L., Nevanlinna factorization and invariant sub-

spaces

17.45  Departure to the reception

Wednesday, June 14, 2000

Time  Section  Lecture

(location)

8.40–9.30 G.A.M./A33 Foias, C., Bloomington, Unitary invariants for commuting isome-

tries

9.35–10.25 G.A.M./A33 Alpay, D., Beer Sheva, Unbounded operators in the theory of nonstationary linear systems

Hall/A33  Coffee/tea

10.50–11.40 G.A.M./A33 Birman, M., St. Petersburg, Perturbation theory and the spectral

properties of the periodic operators near the edge of the spectrum

11.45–12.35 G.A.M./A33 Megretskii, A., Massachusetts, Relaxations of quadratic programs in Operator Theory and System Analysis

Lunch

14.25–15.10 A, A29 Le Merdy, C., Besancon, Sums of commuting operators with L^{p}-

maximal regularity

B, A29 McCarthy, J., St. Louis, QMcCarualitative properties of Pick in-

terpolants

C, A29  Arazy, J., Haifa, Asymptotic expansions of Berezin transforms

D, A29 Solomyak, M., Rehovot, On a class of elliptic operators with non-

Weylian spectral asymptotics

E, A29 Latushkin, Y., Columbia, Spectral mapping theorems in control theory and for nonlinear Schrödinger and Euler equations

15.15–15.45 A, A29 Li, W. S., Atlanta, Inequalities of eigenvalues of sums of self-adjoint

operators

B, A29 Kapustin, V., St. Petersburg, One-dimensional perturbations of

unitary operators and exposed points in H^{1}

C, A29 Karlovich, Yu ., Mexico and Odessa, Toeplitz operators with matrix oscillating symbols on weighted Hardy spaces D, A29 Mennicken, R., Regensburg, On the spectrum of the product of

closed operators

E, A29 Minkin, A ., Haifa, Unconditional basicity of dissipative differential

operators

15.50–16.20 A, A29 Abakumov, E., Marne-la-Valee, Shift invariant subspaces of arbi-

trary index in ℓ^{p} spaces

B, A29 Bakonyi, M., Atlanta, On extensions of positive definite operator-

valued functions on ordered groups

C, A29 Dritschel, M., Newcastle upon Tyne, An elementary proof of the

operator Fejer Riesz theorem and multivariate generalizations

D, A29 Petkov, V., Bordeaux, Breit-Wigner approximations for the scat-

tering phase

E, A29 Belishev, M., St. Petersburg, Triangular models in boundary value dynamical inverse problems

Hall/A29 Coffee/tea
16.45–17.15 A, A29  Amar, E., Bordeaux, On the corona problem for operators
B, A29  Bruzual, R., Caracas, On the local operator commutation relations

C, A29 Brudnyi, A., Beer Sheva, Polynomial type inequalities for analytic

functions

D, A29 Gasimov, Z., Baku, The singular Sturm-Liouville operators and

their applications in inverse problems

E, A29  Jaming, P., Orleans, The discrete radar ambiguity problem
17.20–17.50 A, A29  Tomilov, Yu., Kiev, On convergence of operator families
B, A29  Kerchy, L., Szeged, Invariant subspace lattices of isometries

C, A29 Nakazi, T., Hokkaido, Rudin’s orthogonality problem and isomet-

ric operators between two weighted Hardy spaces

D, A29 Berezansky, Yu., Kiev, The applications of spectral theory of Ja-

cobi fields to integration of nonlinear equations

E, A29 Yger, A., Bordeaux, About the direct realisation of algebraic identities and some relations with operator theory

17.55–18.25 A, A29  Zemanek, J., Warsaw, Recent advances in operator ergodic theory
B, A29  Dehghan, Y. Tabriz, Weighted composition operatoprs

C, A29 Lyubarskii, Y., Trondheim, Approximation of subharmonic func-

tions

D, A29 Olshevsky, V., Atlanta, Fast and numerically accurate algorithms

for structured matrices: a survey

E, A29 Xia, J., Buffalo, Diagonalization and unitary equivalence modulo Schatten p-classes

18.30–19.00 A, A29 Kurasov, P., Stokholm, Krein’s formula for supersingular pertur-

bations of self-adjoint operators

B, A29 Kosiek, M., Krakow, Lebesgue-type decompositions for n-tuples of

contractions without von Neumann inequlity

C, A29 Pott, S., York, Hilbert transform and martingale transforms on operator-weighted L^{2} spaces D, A29 Hasanov, M., Istanbul, On the operator-valued functions of quasi-

hyperbolic type

E, A29 de Snoo, H., Groningen, Rank one perturbations in a Pontryagin space with one negative square

Thursday, June 15, 2000

Time  Section  Lecture

(location)

8.40–9.30 G.A.M./A33 Böttcher, A., Chemnitz, The Cauchy singular integral operator

and its beautiful spectrum

9.35–10.25 G.A.M./A33 Sadosky, C., Washington, Almost orthogonality and block diagonal operators in perturbation theory

Hall/A33  Coffee/tea

10.50–11.40 G.A.M./A33 Volberg, A., East Lansing, Carleson measures for K_{θ} cannot be

checked on reproducing kernels

11.45–12.35 G.A.M./A33 Saff, E., Tampa, Best meromorphic approximation of Markov functions

Lunch

14.25–15.10 A, A29 Kaashoek, M., Amsterdam, The Perelson determinant and limit

theorems of Kac-Achiezer type

B, A29 Shimorin, S., Lund, Wold-type decompositions and wandering sub-

spaces of operators close to isometries

C, A29 Spitkovsky, I., Williamsburg, Those evil Toeplitz operators with

semi almost periodic symbols

D, A29 Shkalikov, A., Moscow, Schrödinger operators with singular poten-

tials from the space of multipliers

E, A29 Pavlov, B., Auckland, The Poincaré–Hardy inequality on the complement of a Cantor set

15.15–16.00 A, A29  Nevanlinna, O., Helsinki, Resolvent as a meromorphic function

B, A29 Partington, J., Leeds, Approximation and interpolation in Hilbert

spaces of analytic functions

C, A29  Ferguson, S., Detroit, Higher-order Hankel forms on the n-ball

D, A29 Tkachenko, V., Beer Sheva, Non-selfadjoint Sturm-Liouville oper-

ators with multiple spectra

E, A29 Yafaev, D., Rennes, Singular Friedrichs model and Hardy-Rellich inequalities

Hall/A29  Coffee/tea

16.25–16.55 A, A29 Azizov, T ., Voronezh, An indefinite approach to the diagonaliza-

tion of a nonnegative Hamiltonian operator function

B, A29 Dyakonov, K., Barcelona and St. Petersburg, Compact Bernstein

inequalities in star-invariant subspaces

C, A29 Litvinchuk, G., Madeira, On a Hilbert value problem with shift

D, A29 Malamud, M., Donetsk, Spectral analysis of some tensor products

and applications

E, A29 Ivanov, S., St. Petersburg, Interpolation of subspaces and exponential bases

17.00–17.30 A, A29 Markus, A., Beer Sheva, Quadratic numerical range of block oper-

ator matrix. Properties and applications

B, A29 Badea, C., Lille, Completely polynomially dominated operators

and similarity problems

C, A29 Matsaev, V., Tel Aviv, Division and estimates of resolvents

D, A29 Janas, J., Krakow, Jacobi matrices with periodically modulated

entries

E, A29 Nudelman, A., Odessa, Interpolation problems of Nevanlinna-Pick type with spectral restrictions

17.35–18.05 A, A29 Tikhonov, A., Simferopol, Spectral theory for operators with spec-

trum on a curve

B, A29 Kupin, S., Bordeaux and Kharkov, Some criteria for similarity of

integral dissipative operator to a normal operator

C, A29 Paneah, B., Haifa, Support dependent estimates for Fourier trans-

forms related to uncertainty principles

D, A29 van Schagen, F., Amstredam, Finite section method for linear or-

dinary differential equations

E, A29 Gubreev, G., Odessa, The structure of Volterra operators and biorthogonal decompositions

18.15–19.40 A33  Poster sesion & Coffee/tea (1st floor)
20.00–23.00  Dinner at the Passerelle Restaurant

Friday, June 16, 2000

Time  Section  Lecture

(location)

8.40–9.30 G.A.M./A33 Verduyn Lunel, S. M., Leiden, Periodic differential equations with

time lags and their spectral properties

9.35–10.25 G.A.M./A33 Seip, K., Trondheim, Frames of complex exponentials and problems of norm equivalence for Hilbert spaces of entire functions

Hall/A33  Coffee/tea

10.50–11.40 G.A.M./A33 Pisier, G., Paris, On the space H^{1} with values in the trace class,

and related non-commutative analysis

11.45–12.35 G.A.M./A33 Davidson, K., Waterloo, The structure of free semigroup algebras

Lunch

14.25–15.10 A, A29 Peller, V., Kansas, Balanced unitary-valued functions and new fac-

torization theorems

B, A29 Yuditskii, P., East Lansing and Kharkov, Character–automorphic scattering function, functional model and asymptotic behavior of polynomials orthonormal on a homogeneous set

C, A29  Coburn, L., Buffalo, On the Berezin-Toeplitz calculus

D, A29 Arov, D., Odessa, Bitangential direct and inverse input scattering

problems

E, A29 Vinnikov, V., Beer Sheva, Commuting isometries and vector bundles on algebraic curves

15.15–16.00 A, A29 Adamyan, V., Odessa, Contractive solutions of some operator Ric-

cati equations

B, A29 Langer, H., Veinna, The Schur algorithm for generalized Schur

functions

C, A29 Poltoratski, A., College Station, Maximal properties of the nor-

malized Cauchy transform

D, A29  Zworski

E, A29 Larson, D., College Station, Some connections between wavelets, frames and operator theory

16.05–16.35 A, A29 Belitskii, G., Beer Sheva, Multidimensional functional equations

generated by affine transformations

B, A29 Tsekanovskii, E., New York, Interpolation theory in sectorial Stielt-

jes classes and explicit system solutions

C, A29  Xu, Q., Besancon, Non-commutative inequalities

D, A29 Tretter, C., Regendburg, Diagonalization of block operator matri-

ces and applications to Dirac operators

E, A29 Fedorov, S., Auckland, On invertibility of Toeplitz operators in the unit disk and in a multiply connected domain

16.40–17.10 A, A29 Rodman, L., Williamsburg, Jordan forms of nilpotent matrices

having prescribed generic triangular part

B, A29 Marcantognini, S., Caracas, Weighted Nehari-Dym-Gohberg prob-

lem in the Wiener class

C, A29 Pedersen, T., Bordeaux, Translation invariant subspaces of

weighted Hilbert spaces on the real line

D, A29  Zwart, H., Enschede, Conjectures on exact observability

E, A29 Avdonin, S., Knoxville, Riesz bases of exponential divided differences and control problems for partial differential equations

Hall/A33  Coffee/tea

17.45–18.35 Helton, W., San Diego, Operator theory and noncommutative

computer algebra

18.35  Closing

IWOTA 2000, Université Bordeaux I (bâtiments A29, A33)

PROGRAMME (version abrégée)

Mardi, le 13 juin, 2000

heure  Section A Section B Section C Section D Section E
8.30–9.30  Inscription
9.35–9.45  Ouverture
Président  Kaashoek
9.45–10.35  Meyer

Café/thé

Président  Sadosky
11.00–11.50  Curtain
11.55–12.45  Treil

Déjeuner

Président  Amar  Alpay  Spitkovsky Tretter  Yger
14.35–15.20 Douglas  Kheifets  Leblond  Mityagin  Staffans
15.25–16.10 Lomonosov Ball  Dym  Naboko  Putinar

Café/thé

Chair  Volberg
16.45–17.35  de Branges
17.45  Départ à la reception

Mercredi, le 14 juin, 2000

heure  Section A Section B Section C  Section D Section E
Président  Dym
8.40–9.30  Foias
9.35–10.25  Alpay

Café/thé

Président  H. Langer
10.50–11.40  Birman
11.45–12.35  Megretski

Déjeuner

Président  Douglas  Seip  Ferguson  Tkachenko Pavlov
14.25–15.10 Le Merdy McCarty  Arazy  Solomyak Latushkin
15.15–15.45 Li  Kapustin Yu. Karlovich Mennicken Minkin
15.50–16.20 Abakumov Bakonyi Dritschel Petkov Belishev

Café/thé

Président  Le Merdy McCarthy Böttcher  Shkalikov Putinar
16.45–17.15 Amar  Bruzual  Brudnyi  Gasimov  Jaming
17.20–17.50 Tomilov  Kerchy  Nakazi  Berezanski Yger
17.55–18.25 Zemanek  Dehghan  Lyubarskii  Olshevski Xia
18.30–19.00 Kurasov  Kosiek  Pott  Hasanov  de Snoo

Jeudi, le 15 juin, 2000

heure  Section A Section B Section C  Section D Section E
Président  Partington
8.40–9.30  Böttcher
9.35–10.25  Sadosky

Café/thé

Président  Staffans
10.50–11.40  Volberg
11.45–12.35  Saff

Déjeuner

Président  Rodman  Vasilescu  Treil  Megretski Vinnikov
14.25–15.10 Kaashoek Shimorin  Spitkovsky  Shkalikov Pavlov
15.15–16.00 Nevanlinna Partington Ferguson  Tkachenko Yafaev

Café/thé

Peller  Baratchart Verdyun Lunel Naboko  Yafaev
16.25–16.55 Azizov  Dyakonov Litvinchuk  Malamud Ivanov
17.00–17.30 Markus  Badea  Matsaev  Janas  Nudelman
17.35–18.05 Tikhonov Kupin  Paneah  van Shagen Gubreev
18.15–19.40  Séance de posters & café/thé
20.00–23.00  Diner

Vendredi, le 16 juin, 2000

heure  Section A Section B  Section C Section D Section E
Président  Helton
8.40–9.30  Verduyn Lunel
9.35–10.25  Seip

Café/thé

Président  Mennicken
10.50–11.40  Pisier
11.45–12.35  Davidson

Déjeuner

Président  Cassier  Saff  Davidson  Mityagin  Curtain
14.25–15.10 Peller  Yuditskii  Coburn  Arov  Vinnikov
15.15–16.00 Adamyan  H. Langer  Poltoratskii Zworski  Larson
Président  Nevanlinna Deville  Coburn  Petkov  Solomyak
16.05–16.35 Belitskii  Tsekanovskii Xu  Tretter  Fedorov
16.40–17.10 Rodman  Marcantognini Pedersen  Zwart  Avdonin

Café/thé

Président  Kaashoek
17.45–18.35  Helton
18.35  Fermeture

IWOTA 2000, Université Bordeaux I (bâtiments A29, A33)

PROGRAMME (version complète)

Mardi, le 13 juin, 2000

Heure  Section  Conférence

(localisation)

8.30–9.30  Hall/A33  Inscription
9.35–9.45  G.A.M./A33  Ouverture par Prof. Francis Hardouin, Vice-Président de l’Univer-

sité Bordeaux 1

G.A.M./A33 Bienvenue à IWOTA 2000 par Prof. M. A. Kaashoek, Vice-Prési-

dent du Comité International d’IWOTA

9.45–10.35 G.A.M./A33 Meyer, Y., Paris, Real analysis, complex analysis, partial differential equations and operator theory

Hall/A33  Café/thé

11.00–11.50 G.A.M./A33 Curtain, R., Groningen, Stabilization of dissipative systems by col-

located feedback

11.55–12.45 G.A.M./A33  Treil, S., Providence, Recent advances in similarity problem

Déjeuner

14.35–15.20 A, A29 Douglas, R., College Station, Quotient Hilbert modules and mul-

tivariate canonical models

B, A29 Kheifets, A., Ariel, Couplings, scales and wave operators in inter-

polation problems

C, A29 Leblond, J., Sophia-Antipolis, Diagonalization of Toeplitz operators, convergence properties and asymptotic estimates for H^{2} approximation on part of the boundary

D, A29 Mityagin, B., Columbus, Smoothness of solutions of nonlinear

ODEs

E, A29 Staffans, O., Åbo, Lax-Phillips scattering and well-posed linear systems

15.25–16.10 A, A29 Lomonosov, V., Kent, Positive functionals on general operator al-

gebras

B, A29 Ball, J., Virginia, Multivariable linear systems, scattering, unitary

dilations, and operator model theory for row contractions

C, A29 Dym, H., Rehovot, On the role of the Muckenhoupt condition in

the theory of J-inner matrix functions

D, A29 Naboko, S., St. Petersburg, Asymptotics of generalized eigenvectors and the spectral analysis of unbounded Jacobi matrices

Hall/A33  Café/thé

16.45–17.35 G.A.M./A33 de Branges, L., Nevanlinna factorization and invariant sub-

spaces

17.45  Départ à la réception

Mercredi, le 14 juin, 2000

Heure  Section  Conférence

(localisation)

8.40–9.30  G.A.M./A33  Foias, C., Bloomington, Unitary invariants for commuting isome-

tries

9.35–10.25 G.A.M./A33 Alpay, D., Beer Sheva, Unbounded operators in the theory of nonstationary linear systems

Hall/A33  Café/thé

10.50–11.40 G.A.M./A33 Birman, M., St. Petersburg, Perturbation theory and the spectral

properties of the periodic operators near the edge of the spectrum

11.45–12.35 G.A.M./A33 Megretskii, A., Massachusetts, Relaxations of quadratic programs in Operator Theory and System Analysis

Déjeuner

14.25–15.10 A, A29 Le Merdy, C., Besancon, Sums of commuting operators with L^{p}-

maximal regularity

B, A29 McCarthy, J., St. Louis, QMcCarualitative properties of Pick in-

terpolants

C, A29  Arazy, J., Haifa, Asymptotic expansions of Berezin transforms

D, A29 Solomyak, M., Rehovot, On a class of elliptic operators with non-

Weylian spectral asymptotics

E, A29 Latushkin, Y., Columbia, Spectral mapping theorems in control theory and for nonlinear Schrödinger and Euler equations

15.15–15.45 A, A29 Li, W. S., Atlanta, Inequalities of eigenvalues of sums of self-adjoint

operators

B, A29 Kapustin, V., St. Petersburg, One-dimensional perturbations of

unitary operators and exposed points in H^{1}

C, A29 Karlovich, Yu ., Mexico and Odessa, Toeplitz operators with matrix oscillating symbols on weighted Hardy spaces D, A29 Mennicken, R., Regensburg, On the spectrum of the product of

closed operators

E, A29 Minkin, A ., Haifa, Unconditional basicity of dissipative differential

operators

15.50–16.20 A, A29 Abakumov, E., Marne-la-Valee, Shift invariant subspaces of arbi-

trary index in ℓ^{p} spaces

B, A29 Bakonyi, M., Atlanta, On extensions of positive definite operator-

valued functions on ordered groups

C, A29 Dritschel, M., Newcastle upon Tyne, An elementary proof of the

operator Fejer Riesz theorem and multivariate generalizations

D, A29 Petkov, V., Bordeaux, Breit-Wigner approximations for the scat-

tering phase

E, A29 Belishev, M., St. Petersburg, Triangular models in boundary value dynamical inverse problems

Hall/A29 Café/thé
16.45–17.15 A, A29  Amar, E., Bordeaux, On the corona problem for operators
B, A29  Bruzual, R., Caracas, On the local operator commutation relations

C, A29 Brudnyi, A., Beer Sheva, Polynomial type inequalities for analytic

functions

D, A29 Gasimov, Z., Baku, The singular Sturm-Liouville operators and

their applications in inverse problems

E, A29  Jaming, P., Orleans, The discrete radar ambiguity problem
17.20–17.50 A, A29  Tomilov, Yu., Kiev, On convergence of operator families
B, A29  Kerchy, L., Szeged, Invariant subspace lattices of isometries

C, A29 Nakazi, T., Hokkaido, Rudin’s orthogonality problem and isomet-

ric operators between two weighted Hardy spaces

D, A29 Berezansky, Yu., Kiev, The applications of spectral theory of Ja-

cobi fields to integration of nonlinear equations

E, A29 Yger, A., Bordeaux, About the direct realisation of algebraic identities and some relations with operator theory

17.55–18.25 A, A29  Zemanek, J., Warsaw, Recent advances in operator ergodic theory
B, A29  Dehghan, Y. Tabriz, Weighted composition operatoprs

C, A29 Lyubarskii, Y., Trondheim, Approximation of subharmonic func-

tions

D, A29 Olshevsky, V., Atlanta, Fast and numerically accurate algorithms

for structured matrices: a survey

E, A29 Xia, J., Buffalo, Diagonalization and unitary equivalence modulo Schatten p-classes

18.30–19.00 A, A29 Kurasov, P., Stokholm, Krein’s formula for supersingular pertur-

bations of self-adjoint operators

B, A29 Kosiek, M., Krakow, Lebesgue-type decompositions for n-tuples of

contractions without von Neumann inequlity

C, A29 Pott, S., York, Hilbert transform and martingale transforms on operator-weighted L^{2} spaces D, A29 Hasanov, M., Istanbul, On the operator-valued functions of quasi-

hyperbolic type

E, A29 de Snoo, H., Groningen, Rank one perturbations in a Pontryagin space with one negative square

Jeudi, le 15 juin, 2000

Heure  Section  Conférence

(localisation)

8.40–9.30  G.A.M./A33  Böttcher, A., Chemnitz, The Cauchy singular integral operator

and its beautiful spectrum

9.35–10.25 G.A.M./A33 Sadosky, C., Washington, Almost orthogonality and block diagonal operators in perturbation theory

Hall/A33  Café/thé

10.50–11.40 G.A.M./A33 Volberg, A., East Lansing, Carleson measures for K_{θ} cannot be

checked on reproducing kernels

11.45–12.35 G.A.M./A33 Saff, E., Tampa, Best meromorphic approximation of Markov functions

Déjeuner

14.25–15.10 A, A29 Kaashoek, M., Amsterdam, The Perelson determinant and limit

theorems of Kac-Achiezer type

B, A29 Shimorin, S., Lund, Wold-type decompositions and wandering sub-

spaces of operators close to isometries

C, A29 Spitkovsky, I., Williamsburg, Those evil Toeplitz operators with

semi almost periodic symbols

D, A29 Shkalikov, A., Moscow, Schrödinger operators with singular poten-

tials from the space of multipliers

E, A29 Pavlov, B., Auckland, The Poincaré–Hardy inequality on the complement of a Cantor set

15.15–16.00 A, A29  Nevanlinna, O., Helsinki, Resolvent as a meromorphic function

B, A29 Partington, J., Leeds, Approximation and interpolation in Hilbert

spaces of analytic functions

C, A29  Ferguson, S., Detroit, Higher-order Hankel forms on the n-ball

D, A29 Tkachenko, V., Beer Sheva, Non-selfadjoint Sturm-Liouville oper-

ators with multiple spectra

E, A29 Yafaev, D., Rennes, Singular Friedrichs model and Hardy-Rellich inequalities

Hall/A29  Café/thé

16.25–16.55 A, A29 Azizov, T ., Voronezh, An indefinite approach to the diagonaliza-

tion of a nonnegative Hamiltonian operator function

B, A29 Dyakonov, K., Barcelona and St. Petersburg, Compact Bernstein

inequalities in star-invariant subspaces

C, A29 Litvinchuk, G., Madeira, On a Hilbert value problem with shift

D, A29 Malamud, M., Donetsk, Spectral analysis of some tensor products

and applications

E, A29 Ivanov, S., St. Petersburg, Interpolation of subspaces and exponential bases

17.00–17.30 A, A29 Markus, A., Beer Sheva, Quadratic numerical range of block oper-

ator matrix. Properties and applications

B, A29 Badea, C., Lille, Completely polynomially dominated operators

and similarity problems

C, A29 Matsaev, V., Tel Aviv, Division and estimates of resolvents

D, A29 Janas, J., Krakow, Jacobi matrices with periodically modulated

entries

E, A29 Nudelman, A., Odessa, Interpolation problems of Nevanlinna-Pick type with spectral restrictions

17.35–18.05 A, A29 Tikhonov, A., Simferopol, Spectral theory for operators with spec-

trum on a curve

B, A29 Kupin, S., Bordeaux and Kharkov, Some criteria for similarity of

integral dissipative operator to a normal operator

C, A29 Paneah, B., Haifa, Support dependent estimates for Fourier trans-

forms related to uncertainty principles

D, A29 van Schagen, F., Amstredam, Finite section method for linear or-

dinary differential equations

E, A29 Gubreev, G., Odessa, The structure of Volterra operators and biorthogonal decompositions

18.15–19.40 A33  Séance de posters & Café/thé (premier étage)
20.00–23.00  Diner au Restaurant La Passerelle

Vendredi, le 16 juin, 2000

Heure  Section  Conférence

(localisation)

8.40–9.30 G.A.M./A33 Verduyn Lunel, S. M., Leiden, Periodic differential equations with

time lags and their spectral properties

9.35–10.25 G.A.M./A33 Seip, K., Trondheim, Frames of complex exponentials and problems of norm equivalence for Hilbert spaces of entire functions

Hall/A33  Café/thé

Déjeuner

14.25–15.10 A, A29 Peller, V., Kansas, Balanced unitary-valued functions and new fac-

torization theorems

B, A29 Yuditskii, P., East Lansing and Kharkov, Character–automorphic scattering function, functional model and asymptotic behavior of polynomials orthonormal on a homogeneous set

C, A29  Coburn, L., Buffalo, On the Berezin-Toeplitz calculus

D, A29 Arov, D., Odessa, Bitangential direct and inverse input scattering

problems

E, A29 Vinnikov, V., Beer Sheva, Commuting isometries and vector bundles on algebraic curves

15.15–16.00 A, A29 Adamyan, V., Odessa, Contractive solutions of some operator Ric-

cati equations

B, A29 Langer, H., Veinna, The Schur algorithm for generalized Schur

functions

C, A29 Poltoratski, A., College Station, Maximal properties of the nor-

malized Cauchy transform

D, A29  Zworski

E, A29 Larson, D., College Station, Some connections between wavelets, frames and operator theory

16.05–16.35 A, A29 Belitskii, G., Beer Sheva, Multidimensional functional equations

generated by affine transformations

B, A29 Tsekanovskii, E., New York, Interpolation theory in sectorial Stielt-

jes classes and explicit system solutions

C, A29  Xu, Q., Besancon, Non-commutative inequalities

D, A29 Tretter, C., Regendburg, Diagonalization of block operator matri-

ces and applications to Dirac operators

E, A29 Fedorov, S., Auckland, On invertibility of Toeplitz operators in the unit disk and in a multiply connected domain

16.40–17.10 A, A29 Rodman, L., Williamsburg, Jordan forms of nilpotent matrices

having prescribed generic triangular part

B, A29 Marcantognini, S., Caracas, Weighted Nehari-Dym-Gohberg prob-

lem in the Wiener class

C, A29 Pedersen, T., Bordeaux, Translation invariant subspaces of

weighted Hilbert spaces on the real line

D, A29  Zwart, H., Enschede, Conjectures on exact observability

E, A29 Avdonin, S., Knoxville, Riesz bases of exponential divided differences and control problems for partial differential equations

Hall/A33  Café/thé

17.45–18.35 Helton, W., San Diego, Operator theory and noncommutative

computer algebra

18.35  Fermeture

Abstracts Résumés

Abakumov, Evgeny

Marne-la-Valee, France

Shift invariant subspaces of arbitrary index in ℓ^{p} spaces

In 1949 Acta Mathematica paper Arne Beurling gave a complete description of the (right) shift invariant subspaces in ℓ^{2} = ℓ^{2}(ℤ_{+}). Beurling’s Theorem states that every such subspace is either zero or is generated by the sequence of the Taylor coefficients of an inner analytic function. At the end of the paper Beurling mentions that it would be desirable to have a similar understanding of the lattice of shift invariant subspaces of the spaces ℓ^{p}, for p other than 2. No complete characterization is known up to this moment, and this problem seems to be very difficult. The main goal of the talk is to demonstrate that for 2 < p < ∞, the lattice of invariant subspaces in ℓ^{p} looks quite different from that for p = 2.

Let us introduce some terminology. For 1 ≤ p < ∞, let S be the forward shift operator acting on ℓ^{p}, S(a_{0}, a_{1}, a_{2}, …) = (0, a_{0}, a_{1}, …). Let M be an S-invariant subspace in ℓ^{p}. The operator S being an isometry on ℓ^{p}, the image SM of M under S is a closed subspace of M. The dimension of the quotient subspace M/SM is called the index of M. It is clear from Beurling’s Theorem that each nonzero shift invariant subspace of ℓ^{2} has index 1. The same is true for p = 1 (as follows from a standard Banach algebras argument).

Our main result states that for 2 < p < ∞, and for a given number n = 2, 3, …, ∞, there exists a shift invariant subspace M of ℓ^{p} having index n. As a corollary, we obtain that all ℓ^{p}, 2 < p < ∞, contain shift invariant subspaces which are not finitely generated.

This is joint work with Alexander Borichev.

Adamyan, Vadim

Odessa, Ukraine

Contractive solutions of some operator Riccati equations

Let A be a closed linear operator A acting in some Hilbert space H_{1} and B, C, D be bounded linear operators acting from a Hilbert space H_{2} into H_{1}, from H_{1} into H_{2} and in H_{2}, respectively. Under certain assumptions we establish criteria and suficient conditions of the existence and uniqueness of contractive solutions K of the operator Riccati equation

KBK + KA − DK − C = 0

and apply obtained results to the block diagonalization of the operator

L = ( \begin{matrix}A & B \\ C & D \end{matrix} )

in H_{2} ⊕ H_{1} and the factorization of the Schur complement

S_{A}(z) = A − z − B(D − z)^{−1}C, z ∈ ρ(D).

By means of the Cayley transformation, the existance and uniqueness of an accreative solution of a Riccati equation is considered. A special attention is given to the case where L becomes selfadjoint.

This talk is based on work done with Heinz Langer and Christiane Tretter.

Alpay, Daniel

Beer Sheva, Israel

Unbounded operators in the theory of nonstationary linear systems

In the theory of nonstationary discrete linear systems, the role of bounded analytic functions is played by upper triangular bounded operators between ℓ_{2}(Z) spaces. On the other hand, unbounded analytic functions play also an important role in system theory. For instance,

• Brune sections (also called Blaschke–Potapov factors of the third kind) are not bounded in the open unit disk, and have an important role in boundary interpolation.

• Carathéodory functions (functions analytic in the open unit disk and with a positive real part there) need not be bounded, and play a central role in the theory of stationary second–order stochastic processes.

In the talk, we consider the nonstationary analogs of these two examples.

The talk is based on the papers [1] and [2].

References

1. D. Alpay, V. Bolotnikov, P. Dewilde, and A. Dijksma, Sections de Brune en théorie des systèmes non stationaires, to appear in Comptes–Rendus de l’Académie des Sciences de Paris, 2000.

2. D. Alpay, V. Bolotnikov, A. Dijksma, and B. Freydin, Non stationary analogs of the Herglotz representation theorem: the discrete unbounded case, preprint, 2000.

Amar, Eric

Bordeaux, France

On the corona problem for operators

This is joint work with C. Menini.

Let 𝔻 be the unit disk in the complex plane and H^{∞}(𝔻^{n}) the algebra of bounded holomorphic functions in 𝔻^{n}, we prove that:

if g_{1}, ..., g_{N} ∈ H^{∞}(𝔻^{n}) verify ∑_{i=1}^{N} |g_{i}|^{2} ≥ δ^{2} > 0, i.e. the Corona hypothesis for functions, then the associated operators T_{g_{i}}(·) ∈ H^{∞}(L(H^{2}

(𝔻^{n−1}))) verify the Corona for operators:

∃F_{1}(·), ..., F_{N}(·) ∈ H^{∞}(L(H^{2}(𝔻^{n−1}))) such that ∀z ∈ 𝔻, ∑_{i=1}^{N} T_{g_{i}}(z)F_{i}(z) = I, where I is the identity in L(H^{2}(𝔻^{n−1})). The proof is based on the H^{2}(𝔻^{n}) Corona theorem

and on the lifting of the commutant theorem.

On the other hand we also prove that the lifting of the commutant theorem cannot be true for more than two contractions for any constant.

This is the obstruction to get a proof of the Corona for functions in H^{∞}(𝔻^{n}), n ≥ 2 from our previous result.

In the same spirit we built a counter example to the Corona for operators in H^{∞}(L(H^{2} (𝔻^{n−1}))), n ≥ 4 which is close to the Corona for functions in the sense that the associated subspaces are invariant by all the shifts. Precisely we built G_{1}, G_{2} ∈ H^{∞}(L(H^{2}(𝔻^{n−1}))) such that

∃δ > 0, ∀z ∈ 𝔻, ∀e ∈ H^{2}(𝔻^{n−1}), ∥G_{1}(z)e∥^{2} + ∥G_{2}(z)e∥^{2} ≥ δ^{2}∥e∥^{2},

but there are

NO F_{1}, F_{2} ∈ H^{∞}(L(H^{2}(𝔻^{n−1}))) such that ∀z ∈ 𝔻, ∑_{i=1}^{2} G_{i}(z)F_{i}(z) = I and the subspaces E_{i} := H^{2}(𝔻^{n}) ⊖ 𝔾_{i}H^{2}(𝔻^{n}), i = 1, 2 are co-invariant by the multipli-

cation by all the z_{k}, k = 1, ..., n.

In fact these subspaces are spanned by the reproducing kernels in H^{2}(𝔻^{n}).

The proof is inspired by Treil’s counter example, but is quite different, and it uses deeply the characterization of interpolating sequences for the weighted Bergman spaces in 𝔻 done by K. Seip.

Amosov, Grigori

Moscow, Russia

Approximation of the flow of shift operators in a Hilbert space and cocycle conjugacy class of the flow of Powers shifts on the CAR algebra

Let (R, V) be a pair consisting of a one-parameter semigroup of contractions R in a Hilbert space H and its unitary dilation V in a Hilbert space H′, H ⊂ H′. We consider two parameter sets E = R_{+} and E = Z_{+} and call a pair (R, V) to be approximating a pair (T, S) if V_{t} − S_{t} ∈ s_{2}, V_{t}f = S_{t}f, f ∈ H′ ⊖ H, t ∈ E. Every (R, V) can be lifted up to a pair (A(R), A(V)) consisting of a dynamical semigroup A(R) and its automorphic dilation A(V) on the algebra of the canonical anticommutation relations (CAR). Our definition of an approximation for pairs (R, V) is associated with the cocycle conjugacy conditions for semigroups included in pairs (A(R), A(V)). We described the class of all (R, V) approximating the fixing pair (T, S), where T^{∗} consists of shifts in the Hardy space and S is a flow of shift operators in H′ that is characterized by the property of every S_{t} to act by the formula S_{t}e_{i}^{t} = e_{i+1}^{t}, i ∈ Z, in the appropriate basis (e_{i}^{t})_{i∈Z}. The semigroup A(T^{∗}) defines flows of Powers shifts on hyperfinite factors generated by the CAR algebra. We described the cocycle conjugacy class of these flows.

References

1. G.G. Amosov, On approximation of continuous semigroups of isometries in a Hilbert space, Izv. Vysch. Uchebn. Zaved. Matem. 2 (2000), 3–8 (Russian).

Arazy, Jonathan

Haifa, Israel

Asymptotic expansions of Berezin transforms

Let D ⊂ ℂ^{d} be a Cartan domain.The group G := Aut(D) of all biholomorphic automorphisms of D acts transitively on D, and D ≡ G/K, where K := {g ∈ G; g(0) = 0} is the maximal compact subgroup of G. It is known that the Bergman kernel of D has the form h(z, w)^{−p}, where h(z, w) is an irreducible polynomial which is holomorphic in z and anti-holomorphic in w, and p is a certain positive integer (called the genus of D). In the most important case where D consists of all n × m complex matrices z with ∥z∥ < 1 we have h(z, w) = det(I_{n} − zw^{∗}). It is also known that if ν > p − 1 then h(z, w)^{−ν} is the reproducing kernel of the weighted Berman space L_{a}^{2}(D, μ_{ν}) := L^{2}(D, μ_{ν}) ∩ {holomorphic functions}, where dμ_{ν}(z) := c_{ν}h(z, z)^{ν−p} dm(z), c_{ν} is a normalization constant and dm(z) is the Lebesgue measures.

The corresponding Berezin transform B_{ν} is the convolution operator B_{ν}f = f ∗ μ_{ν}, i.e.

B_{ν}(f)(z) := ∫_{D} f(g(w)) dμ_{ν}(w),

where g ∈ G is any element for which g(0) = z. It follows that B_{ν} is G-invariant, i.e. B_{ν}(f ∘ φ) = (B_{ν}(f)) ∘ φ for all φ ∈ G and f ∈ dom(B_{ν}). Also,

B_{ν}(f)(z) = ∫_{D} f(w)|h(z, w)|^{−2ν}h(z, z)^{ν} dμ_{ν}(w),

Namely, B_{ν}f is the Berezin symbol of the Töplitz operator T_{f}^{ν} on L_{a}^{2}(D, μ_{ν}). The study of the Töplitz quantization f ↦ T_{f}^{ν} and the associated operator B_{ν} was initiated by F. Berezin as a part of his research on quantization on symmetric spaces. One of the main themes is the study the asymptotic behavior of the operators T_{f}^{ν} and B_{ν} as the parameter ν (which is interpreted as the reciprocal of Planck’s constant) tends to ∞.

The reproducing kernel admits an expansion

h(z, w)^{−ν} = ∑_{m}(ν)_{m} K_{m}(z, w),

where m range over all partitions, K_{m}(z, w) are reproducing kernels for the irreducible Kinvariant subspaces P_{m} of L_{a}^{2}(D, μ_{ν}), and (ν)_{m} := ∏_{j=1}^{r} ∏_{ℓ=0}^{m_{j}−1}(ν + ℓ − (j − 1)a/2). Here r and a are two integral invariants of D, called the rank and the characteristic multiplicity respectively. Using the expansion of h(z, w)^{−ν} we show in [1] that the measure μ_{ν} has the asymptotic expansion

μ_{ν} = ∑_{m} \frac{L_{m}}{(ν)_{m}} , where L_{m} := K_{m}(∂, ∂) δ_{0}.

Moreover, defining L_{m}(f)(z) := L_{m}(f ∘ φ), where g ∈ G, g(0) = z, we obtain the asymptotic expansion of the Berezin transforms

B_{ν} = ∑_{m} \frac{L_{m}}{(ν)_{m}} .

This expansion is very natural, since the operators {L_{m}} form a linear basis for the ring Diff(D)^{G} of G-invariant differential operators on D, and {L_{1_{j}}}_{j=1}^{r}, 1_{j} := (1, 1, … 1, 0, …, 0) (with j “1’s”), are algebraically independent generators of Diff(D)^{G}. From this we can deduce in particular the known results that B_{ν} → I strongly as ν → ∞ (namely, the correspondence principle holds), and that the coefficient of 1/ν is the Laplace-Beltrami operator Δ̃ = L_{(1,0,0,... ,0)}.

Our asymptotic expansion of B_{ν} is related to the expansion B_{ν} = ∑_{k=0}^{∞} \frac{Q_{k}}{ν^{k}} obtained earlier by M. Englis [2], in which the operators Q_{k} have more complicated form.

We obtain also the asymptotic expansions of the weighted Bergman projections P_{ν} : L^{2}(D, μ_{ν}) → L_{a}^{2}(D, μ_{ν}), and the Berezin transforms associated with the weighted Fock spaces

References

1. J. Arazy and B. Orsted, Asymptotic expansions of Berezin transforms, to appear.

2. M. Englis, Invariant operators and the Berezin transform on Cartan domains, Math. Nachr. 195 (1998), 61-75.

Arlinskii, Yury

Lugansk, Ukraine

On sectorial block operator matrices

Consider a linear operator in a Hilbert space H = H_{1} ⊕ H_{2} given by the matrix

T = [\begin{matrix}A & B \\ C & D \end{matrix}] ,

where blocks satisfy the following conditions:

(a) the operator A is a closed m-sectorial and coercive in a Hilbert H_{1} with the domain D(A),

(b) the operator C is a closed from H_{1} into Hilbert space H_{2} and D(C) ⊇ D(A),

(c) the operator D is a closed in H_{2},

(d) the operator B is a closed from H_{2} into H_{1} and D(B) ⊆ D(D).

We give necessary and sufficient conditions in terms of entries for T to be a sectorial operator in H and describe all its m-acretive and m-sectorial extensions.

Arov, Damir

Odessa, Ukraine

Bitangential direct and inverse input scattering problems

This talk is based on joint work with Harry Dym.

Every p × q matrix valued function that is holomorphic and strictly contractive in the open upper half plane will be considered as the input scattering matrix of a canonical system of differential or integral equations. There are infinitely many such systems. In this talk we shall show that each such system can be uniquely specified by two chains of entire inner matrix valued functions, of sizes p×p and q ×q, respectively. Thus, the full set of data for the bitangential inverse input scattering problem under consideration is the given input scattering matrix and the two chains of the kind mentioned above. The classes of regular and singular (including the limit point case) canonical systems and the corresponding Weyl property will be discussed. Formulas for the solution of the bitangential inverse input scattering problem will also be presented.

References

1. D. Arov, H. Dym, J-inner matrix functions interpolation and inverse problems for canonical systems, I: Foundations, Integral Equations Operator Theory, 29(1997), pp. 373–454.

2. D. Arov, H. Dym, J-inner matrix functions interpolation and inverse problems for canonical systems, II: The inverse monodromy problem, Integral Equations Operator Theory, in press.

3. D. Arov, H. Dym, J-inner matrix functions interpolation and inverse problems for canonical systems, III: More on the inverse monodromy problem, Integral Equations Operator Theory, in press.

4. D. Arov, H. Dym, J-inner matrix functions interpolation and inverse problems for canonical systems, IV: The input scattering problem, in preparation.

Avdonin, Sergei

Knoxville, USA

Riesz bases of exponential divided differences and control problems for partial differential equations

The method of moments in control theory of distributed parameter systems has been widely used since the classical papers of H.O. Fattorini and D.L. Russell in the end of 60-ies – beginning of 70-ies. The method is based on properties of exponential families (usually in space L^{2}(0, T)), the most important from which for control theory are minimality and the Riesz basis property. The challenging subject concerning simultaneous controllability of several systems, proposed by D.L. Russell (1986) and J.–L. Lions (1988), has raised a number of new difficult problems in the theory of exponential families.

The principal questions which we consider in this talk are connected with the basis property of linear combinations from exponentials e^{iλ_{n}t} in the case when the distance between some points λ_{n} tends to zero. D. Ullrich (1980) considered the case when {λ_{n}} = ⋃ Λ_{k} and the groups Λ_{k} consist of equal number of points all of them are close enough to k, k ∈ ℤ. S. Jaffard, M. Tucsnak and E. Zuazua (1997) and C. Baiocchi, V. Komornik and P. Loreti (1998) extended the classical Ingham inequality to the case when groups may contain one or two points. We have generalized these results for groups with arbitrary number of close points. Moreover, we obtained a full description of Riesz bases of special kinds linear combinations of exponentials — generalized divided differences — and applied our results to simultaneous control problems of several elastic strings and beams.

Azizov, Tomas

Voronezh, Russia

An indefinite approach to the diagonalization of a nonnegative Hamiltonian operator function

Let G be a complex or a real Hilbert space, let

H = G ⊕ G.

A continuous 1-periodic operator function (o.f.)

A(t) = [ \begin{matrix}A(t) & B(t) \\ C(t) & −A^{∗}(t) \end{matrix} ] , t ∈ [0, 1]

is called a nonnegative Hamiltonian, if B(t) ≥ 0, C(t) ≥ 0. We say that A(t) is dichotomous, if σ(A(t)) ∩ iℝ = ∅. A continuous dichotomous 1-periodic function A(t) is called conditionally reducible if there is a continuous and continuously invertible 1-periodic o.f. V (t) such that A_{1}(t) = V^{−1}(t)A(t)V (t) is a diagonal block-matrix and σ(A_{1}(t)|_{G⊕{0}}) ⊂ ℂ_{r} and σ(A_{1}(t)|_{{0}⊕G}) ⊂ ℂ_{l}.

Theorem 1 Let A(t) be a continuous dichotomous nonnegative Hamiltonian. Then it is conditionally reducible, moreover, the operator function A_{1}(t) can be chosen as Hamiltonian.

This research is supported by the grant RFBR 99-01-00391.

Badea, Catalin

Lille, France

Completely polynomially dominated operators and similarity problems

We say that T_{1} ∈ B(H_{1}) is completely polynomially dominated by T_{2} ∈ B(H_{2}) if there exists a constant M such that

∥[p_{ij}(T_{1})]_{1≤i,j≤n}∥ ≤ M∥[p_{ij}(T_{2})]_{1≤i,j≤n}∥,

for all positive integers n and all n×n matrices [p_{ij}] with polynomial entries. Here [p_{ij}(T)]_{1≤i,j≤n} is identified with an operator acting on the direct sum of n copies of the corresponding Hilbert space in a natural way.

Vern Paulsen’s criterion for similarity to contractions can be rephrased in the following way : A Hilbert space operator T ∈ B(H) is similar to an operator C ∈ B(H) satisfying I − C^{∗}C ≥ 0 (i.e. to a contraction) if and only if it is completely polynomially dominated by the unilateral shift S ∈ B(ℓ_{2}) (i.e. it is completely polynomially bounded).

The aim of my talk is to present several generalizations of this assertion, the contractions being replaced by operators satisfying some positivity conditions and the unilateral shift by other "model" operators.

Bakonyi, Mihaly

Atlanta, USA

On extensions of positive definite operator-valued functions on ordered groups

Let G be an abelian group and let Λ be a subset of G. A function k : S = Λ − Λ → L(H) (the algebra of bounded linear operators on a Hilbert space H) is called positive (semi)definite with respect to Λ if, for every finite subset λ_{1}, λ_{2}, ⋯, λ_{n} ∈ Λ, the operator matrix {k(λ_{i} − λ_{j})}_{i,j=1}^{n} is positive (semi)definite. M.G. Krein [Kre] proved that every positive definite continuous scalar function on a real interval (−a, a) admits a continuous positive definite extension to ℝ. A.P. Artjomenko [Art] provided a new proof for Krein’s Extension Theorem without the continuity requirement. J. Friedrich and L. Klotz [FK] proved that, given 0 < a < ∞ and a topological group G, then any strongly continuous positive definite function k : (−a, a) × G → L(H) admits a positive definite extension to ℝ × G. We generalize the latter result by omitting the continuity requirement and by substituting ℝ with an ordered abelian group. The proof is based upon Artjomenko’s approach. Several extension results for positive definite functions are presented as corollaries of our main result.

References

1. A. Artjomenko, Hermitian positive functions and positive functionals, Teor. Funkcii, Funk. Anal. i Priloz̆en, 41(1983), 1–16; 42(1984), 1–21 (Russian).

2. J. Friedrich and L. Klotz, On extensions of positive definite operator-valued functions, Rep. Math. Phys., 26(1988), no. 1, 45–65.

3. M. Krein, Sur le probléme de prolongement des functions hermitiniennes positives et continues, Dokl. Akad. Nauk. SSSR, 26(1940), 17–22.

Ball, Joseph

Virginia, USA

Multivariable linear systems, scattering, unitary dilations, and operator model theory for row contractions

It is well known that unitary linear system theory, Lax-Phillips scattering theory, and Sz.-Nagy-Foiaş model theory are closely related. A complete unitary invariant for each of these three theories is given y a Schur-class operator-valued function on the unit disk, the transfer function for the unitary system, the scattering function for the scattering framework, and the characteristic operator function for the contraction operator T in the operator model theory case. We discuss extensions of these ideas to a multivariable setting, where the unitary system becomes a certain type of multivariable unitary system with time axis ℤ replaced by a certain collection of words in d letters (d > 1) (a proper subset of the free group on d letters containing as a proper subset the free semigroup generated by the d letters), where the Lax-Phillips scattering framework involves a row unitary operator system (U_{1}, …, U_{d}) (a d-tuple of operators on a Hilbert space K for which the row operator matrix [U_{1} … U_{d}] is unitary) rather than a single unitary operator U, and the operator model is for a row contraction (a d-tuple of operators (T_{1}, …, T_{d}) on the Hilbert space H for which the row operator matrix [T_{1} … T_{d}] is a contraction). We then recover the same correspondence between system trajectories, a scattering framework and an operator model as in the one-variable case. There is associated a formal power series in d noncommuting variables which plays the role of the transfer function, scattering function and characteristic operator function; however, in this setting, this function is a complete unitary invariant only in a special case (the completely noncoisometric case). We describe what invariants must be added to this function to get a complete set of unitary invariants for the three cases. The added invariants involve a choice of row unitary extension of a given row isometry; we give explicit parameters for describing such extensions. There are also versions of the Sz.-Nagy-Foiaş, de Branges-Rovnyak and Pavlov models, as well as a coordinate-free formulation of the model in the sense of Nikolskii-Vasyunin for this setting. These results extend and complement earlier dilation results of G. Popescu, K. Davidson and D. Pitts by incorporating the system and scattering point of view. This talk reports on joint work with Victor Vinnikov.

Belishev, Mikhail

St. Petersburg, Russia

Triangular models in boundary value dynamical inverse problems

The subject of the talk is an attempt to generalize ’experimental material’ accumulated in the BC-method which is an approach to the inverse problems (IP) based upon their relations to the Boundary Control Theory [1, 2]. The generalization is given in the frameworks of the System Theory: we consider an abstract dynamical system with boundary control and analyze an operator scheme lying in the background of the approach. The main steps of the procedure solving the IP by the BC-method are the following:

1. a model (isometrical copy) of the dynamical system is constructed via inverse data;

2. the model is used for triangular factorization of some operator determined by inverse data;

3. factorization gives another (triangular) model which solves the IP.

The main tool of factorization is the operator integral generalizing classical construction of M.S.Livshits and M.G.Krein.

References

1. M. Belishev, Boundary control in reconstruction of manifolds and metrics, Inverse Problems, 13 (1997), no. 5, R1–R45.

2. M. Belishev, V. Isakov, L. Pestov, V. Sharafutdinov On reconstruction of gravity field via external electromagnetic measurements, POMI preprint - 10/1999.

Belitskii, Genrich

Beer Sheva, Israel

Multidimensional functional equations generated by affine transformations Solvability conditions for equation

(T φ)(x) ≡ ∑_{j=1}^{t} A_{j}(x)(φ(λ_{j}x + e_{j}) = γ(x), λ_{j} ∈ ℝ^{1}, e_{j} ∈ ℝ^{n}

in classes of continuous or smooth functions φ(x) in ℝ^{n} are investigated. We establish that this equation is normally solvable in some class of smooth functions. Moreover, we prove that the operator T with constant coefficients is semi-Fredholm with dim KerT^{∗} < ∞, and it is Fredholm with ind T = 0 in the case λ_{i} ≠ λ_{j}.

Berezansky, Yurij

Kiev, Ukraine

The applications of spectral theory of Jacobi fields to integration of nonlinear equations

A Jacobi field is understood to be a family J of commuting selfadjoint real operators acting in a symmetric Fock space F (H), having a Jacobi structure, and depending linearly on the vectors from real Hilbert space H. This notion is some generalization of a single classical Jacobi matrix J.

There exists some application (belonging to author) of spectral theory of classical Jacobi matrices to integration of Cauchy problem for differen- ce-differential nonlinear equations (for equations of Toda type). In this talk we give an analogous application of spectral theory of Jacobi fields to integration of some type of non-Abelian difference-differential nonlinear equations.Now their solutions are operator-valued functions of time t.

Birman, Mikhail

St. Petersburg, Russia

Perturbation theory and the spectral properties of the periodic operators near the edge of the spectrum

The spectrum of periodic operators of mathematical physics has band structure. The bands are the ranges of the band functions; the latter describe the dependence of the spectral papameter on the quasimomentum. Often it is important to know the behavior of the band functions that realize the lower edge of the spectrum. The notions of effective mass, effective medium, and effective Hamiltonian arising in various applications are related with that. A unified approach to these notions can be given on the basis of the analytic perturbation theory. It is useful to take into account the fact that the operators in question admit a convenient factorization. For analytic operator families, the notion of the spectral jet is introduced. The equivalence relation for the families having one and the same spectral jet is discussed. The basic results presented in the lecture were obtained jointly with T. Suslina.

Böttcher, Albrecht

Chemnitz, Germany

The Cauchy singular integral operator and its beautiful spectrum

The Cauchy singular integral operator S is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations. While the boundedness of S has been studied for many decades, final results on the spectrum of S were obtained only in recent times. During the last few years it was discovered that, in dependence on the curves the operator acts on and the weights involved in the spaces, there is a surprising and undreamt-of metamorphosis of the (local) spectra of S from circular arcs via horns and logarithmic double-spirals to so-called logarithmic leaves with a halo. The talk, based on joint work with Yu.I. Karlovich, is a survey of this fascinating development. Technical details will be omitted, but many beautiful pictures of local spectra will be shown.

de Branges, Louis

Lafayette, USA

Nevanlinna factorization and invariant subspaces

A construction is made of a canonical unitary linear system whose state space is a Krein space and whose transfer function is a given power series W (z) with operator coefficients. A factorization U(z) = W (z)V (z) is an application of the spectral theorem for self-adjoint operators when a Toeplitz operator is defined by the function. Canonical unitary linear systems exist whose state spaces are Hilbert spaces and whose transfer functions are the produced power series U(z) and V (z) with operator coefficients. A construction results of continuous and contractive transformations of a Krein space into itself which admit maximal nonnegative invariant subspaces. The existence of a nontrivial proper closed invariant subspace is also obtained for certain continuous transformations of a Hilbert space into itself. The transformation is required to be contained in an operator algebra which is closed in the weak topology induced by the trace class and which does not contain the identity transformation.

Brudnyi, Alexander

Beer Sheva, Israel

Polynomial type inequalities for analytic functions

We prove Bernstein and Bourgain type inequalities for real analytic functions defined on a convex domain in ℜ^{n}. As an application we obtain Cartwright’s type inequalities for entire functions of an exponential growth and Hormander type estimates for difference-differential operators.

Bruzual, Ramon

Caracas, Venezuela

On the local operator commutation relations

This is joint work with M. Domínguez.

We consider a group of unitary operators and a local semigroup of isometries which satisfies the commutation relation and the following result is proved.

Let H be a Hilbert space. Let (U_{r})_{r∈(−∞,+∞)} ⊂ L(H) be a strongly continuous group of unitary operators and let (S_{r}, H_{r})_{r∈[0,a)} be a local semigroup of isometries on H such that

(a) U_{x}H_{y} ⊂ H_{y} for all x ∈ ℝ, y ∈ [0, a).

(b) There exists α ∈ ℝ, α ≠ 0 such that U_{x}S_{y}f = e^{iαxy}S_{y}U_{x}f , for all f ∈ H_{y}, x ∈ ℝ, y ∈ [0, a).

(c) One of the deficiency indices of the infinitesimal generator of (S_{r}, H_{r})_{r∈[0,a)} is finite.

Then

1. There exists a strongly continuous semigroup of contractions (V_{r})_{r∈[0,+∞)} ⊂ L(H) such that

(i) S_{r} = V_{r}|_{H_{r}} for all r ∈ [0, a).

(ii) U_{x}V_{y}f = e^{iαxy}V_{y}U_{x} for all x ∈ ℝ, y ∈ [0, +∞).

2. There exists a Hilbert space G containing H as a closed subspace and two strongly continuous groups of unitary operators (Ũ_{r})_{r∈(−∞,+∞)} ⊂ L(G) and (Ṽ_{r})_{r∈(−∞,+∞)} ⊂ L(G) such that

(i) U_{r} = Ũ_{r}|_{H} for all r ∈ ℝ.

(ii) S_{r} = Ṽ_{r}|_{H_{r}} for all r ∈ [0, a).

(iii) Ũ_{x}Ṽ_{y}f = e^{iαxy}Ṽ_{y}Ũ_{x} for all x, y ∈ ℝ.

As an application a result about extension of a class of locally defined positive definite functions on the Heisenberg group is obtained.

References

1. Bruzual, R., Local semigroups of contractions and some applications to Fourier representation theorems, Integral Equations Operator Theory, 10 (1987), 780-801 .

2. Bruzual, R. and Domínguez, M., On the local operator commutation relations and extensions of locally defined positive definite functions on the Heisenberg group, preprint.

3. Jorgensen, P. Positive definite functions on the Heisenberg group, Mathematische Zeitschrift, 201 (1989), 455-476.

4. Jorgensen, P. and Muhly, P., Selfadjoint extensions satisfying the Weyl operator commutation relations, Journal d’Analyse Mathématique, 37 (1980), 46-99.

Castro, Glaysar

Caracas, Venezuela

Two dimensional covariance extensions

An important problem in statistical estimation is to obtain the unknown spectral density of a stationary random field from the given covariances over a limited region of Z^{2} (a partial covariance). This is the classical truncated two-dimensional trigonometrical moment problem: Given a finite sequence {c_{k}}_{k∈I} of complex numbers in I ⊂ Z^{2}, this problem asks whether it exists a finite positive Borel measure ν on the two-dimensional torus T^{2} such that the Fourier coefficients of ν verify ν̂_{k} = c_{k} for k ∈ I. We will call the measure ν a positive extension of the given partial covariance (p.c.).

Given a p.c. c_{k} = k+0, 1, ..., n of a time series (t.s.) {X_{k}; k ∈ Z}, a necessary and sufficient condition for the existence of a solution is that the Toeplitz matrix C = (c_{i−j})_{i,j=0}^{n} is positive definite (p.d.). This result is also true for t.s. taking vector values, in this case the matrix C is Toeplitz by blocks [2]. Between all the extensions we distinguish one that maximizes the Burg entropy, this one is an auto regressive (AR) t.s. X_{k} − ∑_{j=1}^{n} a_{k}X_{k−j} = ϵ_{k} ϵ_{k} a white noise with variance σ^{2}, the spectral density of the AR t.s. is

h(λ) = \frac{σ^{2}}{|p_{0,n}(λ)|^{2}} with ĥ(k) = c_{k} k = 0, 1, ..., n

At present, there is not a simple test for extensibility of a given p.c. in Z^{2}. Given a p.c. (c_{k})_{k∈I}, I ⊂ Z^{2}, a necessary condition for the existence of a positive extension is that the doubly indexed Toeplitz matrix T = (c_{i−j})_{(i,j)∈I×I} must be positive definite, but this conditions is not more sufficient [6]. Observe that unlike the one-dimensional case, some of its entries are unknown. In fact, by symmetric property, the covariance values are only known at the indexes in the set I ∪ (−I) which is strictly contained in the set of indexes I − I = {a − b : (a, b) ∈ I × I}. For instance if I = {(0, 0), (0, 1), (1, 0)} ⊂ Z^{2}, the matrix T related to the given p.c. (c_{k})_{k∈I} is:

[ \begin{matrix}c_{(0,0)} & c_{(0,1)} & c_{(1,0)} \\ c_{(0,−1)} & c_{(0,0)} & ? \\ c_{(−1,0)} & ? & c_{(0,0)} \end{matrix} ]

where the ?’s are the entries of the unknown covariances c_{(1,−1)} and c_{(−1,1)}. If there exists a positive extension, we give a necessary condition for the existence of an autoregressive (AR) one in terms of a completion of the matrix T = (c_{i−j})_{(i,j)∈(I−I)×(I−I)}. By a fixed point method we obtain the explicit AR solution.

Under very general conditions, we construct a family of extensions of a given two-dimensional p.c. Let Λ_{+} be a finite subset of a Helson-Lowdenslager type halfspace S containing (0, 0) and denote Λ = Λ_{+} − Λ_{+}. Suppose that {c_{k}}_{k∈Λ} are given complex numbers so that the |Λ_{+}| × |Λ_{+}| matrix (c_{k−l})_{k,l∈Λ_{+}} is p.d. Let P_{0} = ∑_{k∈Λ_{+}} \frac{p_{k}}{p_{00}} χ^{k} where

col(p_{k})_{k∈Λ_{+}} = [(c_{k−l})_{k,l∈Λ_{+}}]^{−1}col( delta_{k,(0,0)})_{k∈Λ_{+}}

with δ_{k,l} = 1 when k = l and 0 otherwise. Let ĥ(k) denote the kth Fourier coefficient of the function h on T^{2}.

Proposition 1 Assume that P_{0}(z) ≠ 0 for z ∈ T^{2}. There exists a unique Q = ∑_{k∈Λ} q_{k}χ^{k} so that h = \frac{Q}{|P_{0}|^{2}} satisfies ĥ(k) = c_{k}, k ∈ Λ. Moreover, we have that q_{00} = \frac{1}{p_{00}} and Q(z) ∈ R for z ∈ T^{2}.

Our main result is a necessary condition for obtaining an AR fields (q_{k} = 0 if k ≠ (0, 0)) that realizes the given p.c. in terms of the reflection coefficients of the related covariance matrix. Some numerical examples are presented.

References 1. Castro G., Coefficients de Réflexion Généralisés. Extension de Covariances Multidi-

mensionnelles et autres Applications, Thesis, Université de Paris Sud. Orsay, 1997.

2. Constantinescu T., Schur parameters, factorization and dilation problems. Operator Theory: Adv. Appl., 82, Birkhaüser Verlag, Basel-Boston-Berlin, 1996.

3. Guyon Xavier, Random Fields on a Network Modeling, Statistics and Applications Springer Berlin, 1995.

4. Helson H. and Lowdenslager D., Prediction Theory and Fourier Series in Several Variables, Acta Math., 99(1958), pp. 165-212.

5. Lev-Ari H., Parker S. P. & Kailath T., Multidimensional maximum-entropy covariance extension, IEEE Trans. Inf. Theory, 35(1989), pp. 497-508.

6. Rudin W., The Extension Problem for Positive Definite Functions, Ill. J. Math., 7(1963), pp 532-539.

7. Seghier A., Extension de fonctions de type positif et entropie associée. Cas multidimensionnel, Ann de l’I.H.P analyse non linéaire, 8, 6(1991) pp. 651-675.

8. Woods J. W., Two-dimensional Markov Spectral Estimation. IEEE Trans. Inform. Theory, 22(1976), pp. 552-559.

Cheremnikh, E.

L’vov, Ukraine

Spectral singularities amd some analogue of B. S.-Nagy-C.Foias’ model

Nonselfadjoint Friedrichs’ model in the space of squared integrables functions on half line with a weight is considered. Suppose that the kernel of the perturbation is smooth and the set of spectrals singularities is finite. Using the inverse of corresponding Forier transformation some estimation from below for the norm of function of the operator is given. In this sense the construction proposed is some analogue of Nagy-Foias’ model. The model considered contains certains Sturm-Liouville’ operators.

Coburn, Lewis

Buffalo, USA

On the Berezin-Toeplitz calculus

I consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex n-dimensional Euclidean space, ℂ^{n}. For several interesting algebras of functions on ℂ^{n}, we have T [f]T [g] = T [f ∗ g] for all f, g in the algebra, where T [f] is the Berezin-Toeplitz operator associated with f and f ∗ g is a "twisted" associative product on the algebra of functions. On the other hand, there is an infinitely differentiable function f for which T [f] is a bounded operator but T [f]T [f] is not T [g] for any g.

Curtain, Ruth

Groningen, The Netherlands

Stabilization of disspative systems by collocated feedback

Many papers on the control of flexible structures have the mathematical structure (A, B, B^{∗}), where A is skew adjoint, B models boundary control action and B^{∗} corresponds to an observation at the same location. In engineering circles, this is called collocated actuating and sensing and it is very popular because it is well known that the static output feedback law u = −y stabilizes in the input-output sense and, more importantly, it is robust with respect to a large class of uncertainties. On the other hand, p.d.e. experts recognize that asymptotic (and often exponential) stability of the closed-loop system generator A − BB^{∗} can be established using a classical Lyapunov approach or LaSalle’s principle with the energy of the system as a Lyapunov functional. Many flexible beams, plates and hybrid structures have been analyzed using this approach. In the seventies, the study of the system theoretic properties of the class (A, B, B^{∗}) with bounded B was popular and it was shown that if B has finite rank, A − BB^{∗} will never generate an exponentially stable semigroup. However, under mild assumptions, it will generate a strongly stable one. In this talk, we consider the case that B is unbounded, A is skew adjoint and (A, B, B^{∗}) is a regular linear system on a Hilbert space X. We show that A − BB^{∗} can generate a semigroup with very different properties: it may be exponentially stable, exponentially unbounded or weakly or strongly stable. However, there always exists a constant c > 0 such that κI + B_{Λ}^{∗}(sI − A)^{−1}B is positive and for any κ ∈ (0, \frac{1}{c}), u = −κy is a stabilizing feedback in the input-output sense. We give conditions for the closed-loop semigroup to be weakly, strongly or exponentially stable.

Davidson, Kenneth

Waterloo, Canada

The structure of free semigroup algebras (joint work with E.Katsoulis and D.Pitts)

A free semigroup algebra is the WOT-closed algebra generated by an n-tuple of isometries with pairwise orthogonal ranges. The interest in these algebras derives primarily from two interesting connections. The first is that they provide useful information about unitary invariants of representations of the Cuntz–Toeplitz algebras. The second is that they form an interesting class of nonself-adjoint operator algebras with a singular representative 𝔏_{n}, generated by the left regular representation of the free semigroup on n letters, which has been coined the non-commutative analytic Toeplitz algebra. This paper provides a general structure theorem for all free semigroup algebras which extends results for important special cases in the literature. It focuses attention on those representations of type L which are isomorphic to 𝔏L_{n}. Every free semigroup algebra has a 2 × 2 lower triangular form where the first column is a slice of a von Neumann algebra and the 22 entry is a type L algebra. We develop the structure of type L algebras in more detail. As an application of our structure theorem, we are able to obtain a theorem of Russo–Dye type showing that the convex hull of the isometries in any free semigroup algebra contains the whole open unit ball.

Dehghan, Yadollah

Tabriz, Iran

Weighted composition operatoprs

Derfel, Gregory

Beer Sheva, Israel

Divide-and-conquer recurrences, functional equations and their asymptotical analysis

We deal with a class of requrrence equations arising in the algorithms theory, stochastic processes, computer science, etc. We study these recurrences by means of their generating function, which is solution of linear nonhomogenous functional equation with rescaling. Classification theorem on the asymptotics of divide-and-conquer recurrences with binomial coefficients is derived. This work is joint with Fritz Vogl.

Derkach, Volodymyr

Donetsk, Ukraine

On indefinite generalized interpolation

Denote by S_{κ} or H_{κ}^{∞} the class of matrix valued functions s(z) which can be represented by s_{0}(z)b(z)^{−1}, where b(z) is a Blaschke–Potapov product and s_{0}(z) is from the Schur class or s_{0} ∈ H^{∞}, respectively. We consider the following problem IP_{κ}(w, θ): Given are an inner matrix function θ and w ∈ H^{∞}. Find s ∈ S_{κ} such that [s(z) − w(z)]θ(z)^{−1} ∈ H_{κ}^{∞}.

In the case where θ is a Blaschke–Potapov product the problem IP_{κ}(w, θ) is a tangential interpolation problem. We associate with the problem IP_{κ}(w, θ) an isometric operator V in a functional Pontryagin space. Solutions of the problem IP_{κ}(w, θ) and unitary extensions of the operator V are shown to be in a one–to–one correspondence. Making use of Kreĭn – Naimark formula for generalized resolvents we obtain a description of all solutions of the problem.

Domínguez, Marisela

Caracas, Venezuela

Extensions of positive definite functions on the lexicographic discrete plane

This is joint work with R. Bruzual.

M. G. Krein proved that every continuous positive definite function on a bounded interval can be extended to a continuous positive definite function in the whole line.

W. Rudin proved that the result of Krein can not be extended to ℤ^{2} or ℝ^{2} and a rectangle. However A. Devinatz gave some conditions for extensions of Krein’s result to these cases.

Several aditional results about extensions of locally defined positive definite functions have been given: A general result about extensions of scalar valued positive definite functions on the product of two groups was given by Z. Sasvári. Also Z. Sasvári ([4], page 105) proved that a scalar valued positive definite function defined on an interval of an ordered group can be extended to a positive definite function on the whole group.

The problem of the extension of scalar and matrix valued positive definite functions defined on a subset of ℤ^{2} to positive definite functions on the whole ℤ^{2} has been recently studied, see for example [1] and [2].

We will deal with operator valued positive definite functions on ℤ^{2} with the lexicographic order. The following result is proved.

Let H be a Hilbert space, let a⃗ ∈ ℤ^{2}, a⃗ > 0 and let k : (−a⃗, a⃗) → L(H) be a positive definite function. Then there exists a positive definite function K : ℤ^{2} → L(H) such that K|_{(−a⃗,a⃗)} = k.

In order to prove this result it is used a result about commuting unitary extensions of a pair of isometries and a general result about multiplicative families of isometries.

References

1. Bakonyi M., Nævdal G., Characterization of finite subsets of ℤ^{2} having the extension property, C. R. Acad. Sci. Paris, 326 (1998), 417–420.

2. Bakonyi M., Rodman L., Spitkovsky I. M. and Woerdeman H. J., Positive extensions of matrix functions ot two variables with support in an infinite band, C. R. Acad. Sci. Paris, 323 (1996), 859–863.

3. Bruzual, R. and Domínguez, M., Extensions of operator valued functions on an interval of ℤ^{2} with the lexicographic order, Acta Sci. Math., 66 (2000), 373–381.

4. Sasvári, Z., Positive definite and definitizable functions, Akademie Verlag, 1994.

Douglas, Ronald

College Station, USA

Quotient Hilbert modules and multivariate canonical models

One approach to a theory of multivariate canonical models considers quotient Hilbert modules over algebras of holomorphic functions on bounded domains in complex n-space. The role of the Hardy module in the Sz.-Nagy-Foias theory is played by an arbitrary kernel Hilbert space for the domain with the submodule conisting of all functions that vanish on an analytic hypervariety. In earlier work, G. Misra and I provided geometric invariants for the quotient along with a realization using a kernel Hilbert space on the subvariety. Further, these examples were also characterized in terms of the complex geometirc approach to operator theory developed by M. Cowen and me.

If we consier instead the submodule consisting of the functions that vanish to a fixed higher order, then describing the quotient module is more involved. In this talk, I report on some joint work with G. Misra and C. Verughese in which a realization is given for this case using a vector-valued kernel Hilbert space. To provide the realization, one needs to generalize the Cowen-Douglas classes to include pull-backs of bundles based on arbitrary flag manifolds and not just Grassmanians. This study is still in progress and is likely related to earlier work of Martin and Salinas.

Time permitting, I will describe more recent work that considers the case when the subvariety has higher complex codimension but is a complete intersection. In this case, the short exact resolution which exists for quotient bundles is replaced by resolutions of longer length.

Dyakonov, Konstantin

Barcelona, Spain, and St. Petersburg, Russia

Compact Bernstein inequalities in star-invariant subspaces

Let H^{p} be the usual Hardy space of the upper half-plane, and let K_{θ}^{p} stand for the starinvariant subspace (alias, model subspace) in H^{p} generated by an inner function θ. Consider the differentiation operator D : f ↦ f′ acting on f ∈ K_{θ}^{p}. Earlier I showed that D is a bounded map from K_{θ}^{p} to L^{p}(ℝ) iff θ′ ∈ L^{∞}(ℝ). The current result is that D is compact iff θ′ ∈ C_{0}(ℝ).

Dym, Harry

Rehovot, Israel

On the role of the Muckenhoupt condition in the theory of J-inner matrix functions

This talk is based on joint work with Damir Z. Arov.

The class of so-called strongly regular J-inner matrix valued functions is important in: interpolation theory for matrix valued functions, inverse problems for canonical integral and differential systems and the theory of characteristic functions of operators and operator nodes. In this talk we shall present a new characterization of this class in terms of the matricial Muckenhoupt condition that was considered by Treil and Volberg. We shall also review another characterization in terms of reproducing kernel Hilbert spaces that was obtained in our previous work. A number of related results on the parametrization and classification of J-inner matrix valued functions will also be discussed.

References

1. S. Treil and A. Volberg,Wavelets and the angle between past and future, J. Funct. Anal., 143(1997), pp. 269-308.

2. D. Z. Arov and H. Dym, On matricial Nehari problems, J-inner matrix functions and the Muckenhoupt condition, preprint.

Farforovskaya, Yuliya

St. Petersburg, Russia

Frechet derivative for function of selfadjoint operator

Let A be a selfadjoint operator having finite spectrum. Then any bounded operator T can be represented as T = K + AX − XA, where the operator K commutes with A and X is a bounded operator. The Frechet derivative exists and may be calculated by the formula

lim_{ε→0} \frac{f(A + εT) − f(A)}{ε} = f′(K) + f(A)X − Xf(A)

for any function f belonging to C^{1} class.

If bounded selfadjoint operator A has an infinite spectrum, then can be shown a concrete compact direction T and a function fϵC^{1} such that

lim_{ε→0} \frac{∥ f(A + εT) − f(A) ∥}{ε} = ∞.

Some applications are discussed.

Fedorov, Sergei

Auckland, New Zealand

On invertibility of Toeplitz operators in the unit disk and in a multiply connected domain

Let Ω be finitely connected domain with piece-wise analytic nondegenerate boundary in the complex plane. It is well known that any analytic single-valued function in Ω may be thought of as an analytic function in the unit disk D (universal covering of Ω) automorphic with respect to discrete group Σ of covering transformations in D (which is in fact a Fuchsian group of the second kind). In a same way any L^{∞} function f on the boundary of Ω may be considered as a bounded function on the unit circle T automorphic with respect to Σ. Therefore for any such f one may consider together with ordinary Toeplitz operator T_{f}^{D} = P_{+}f|H_{+}^{2}(D) in D the Toeplitz operator with the same symbol f in Ω: T_{f} = P_{H_{+}^{2}(Ω)}f|H_{+}^{2}(Ω_{+}) where H_{+}^{2}(Ω) is the least harmonic majorant Hardy space in Ω. It is known from M.Abrahamse papers on multiply connected regions dated back to seventies that the version of Nehari theorem in Ω involves not only single Toeplitz (or Hankel) operator T_{f} with prescribed symbol f, but the whole scale of such operators parameterized by any H^{∞}-invariant subspace S in H_{+}^{2}(Ω), T_{f}^{S} = P_{S}f|S. In particular Ahrahamse had shown that for unimodular symbol f the inequality dist(f, H^{∞}(Ω)) < 1 is equivalent to the left invertibility not only of T_{f} but of all T_{f}^{S} (for all subspaces S) simultaneously. >From this fact he concluded that if for all S the operators T_{f}^{S} are invertible then the Toeplitz operator T_{f}^{D} in the disk is also invertible.

At the very end of that his paper (Amer. J. Math. 96, no.2, (1974), 261–297) Abrahamse posed the question if the last implication is actually an equivalence, i.e if it is true that the invertibility of T_{f}^{S} for all S in Ω is equivalent to invertibility of T_{f} in the unit disk. He noted that in a case of doubly connected domain this is true.

We show that the answer to this Abrahamse’ question is negative in general (for connectivities greater then 2) by producing the example of a unimodular symbol f for which T_{f} is invertible in D, but T_{f}^{S} is not invertible in Ω. Moreover such an example enables us to give the correct condition in terms of T_{f}^{S} which is equivalent to invertibility of T_{f} in D. In a weaker setting the last result states that if T_{f} is invertible in the unit disk then T_{f}^{S} is invertible in Ω at least for one S.

Ferguson, Sarah

Detroit, USA

Higher-order Hankel forms on the n-ball

Joint work with R. Rochberg, Washington University in St. Louis.

I will discuss a very general realization result for symbols of higher order Hilbert-Schmidt Hankel forms defined on a pair of analytic reproducing kernel Hilbert spaces. The realization involves decomposing the Hilbert space tensor product of the two kernel spaces, identified as analytic functions on the product of the underlying domains, into semi-invariant subspaces determined by higher-order vanishing along the diagonal of the product domain. These higherorder subspaces are functional Hilbert spaces of holomorphic functions in n-variables (as opposed to 2n variables) and one of the problems is to identify the kernel for these subspaces. This would give, among other things, a concrete realization for these symbols.

This was successfully done by Janson/Peetre and Peetre/Rochberg and Rochberg/myself for positive powers of the Szego kernel on the polydisk. In the Janson/Peetre viewpoint, higher-order Hankel forms arise as the irreducible components of the induced action of the automorphism group of the disk and this is also the viewpoint in a more recent paper by Janson, Peetre et al, for Hankel forms on the n-ball. Peetre/Rochberg define higher-order Hankel forms in terms of the module action of polynomials on the underlying functional Hilbert spaces. For the polydisk, the two definitions yield the same classes, however for the nball this is unclear. I will present partial results on the realization problem for the n-ball. The kernels involved are positive powers of the "Arveson/Popescu" kernel. With time permitting, I will discuss extensions to the non-commutative setting of Hankel forms on weighted Fock spaces.

Foias, Ciprian

Bloomington, USA

Unitary invariants for commuting isometries

This is a joint work with R. G. Douglas and H. Bercovici in which we provide unitary models for finite systems of mutually commuting isometries on a Hilbert space. As illustrated, we discuss several cases in which our models provide simple explicit descriptions for those systems

Gasimov, Zaid

Baku, Azerbaijan

The singular Sturm-Liouville operators and their applications in inverse problems

The periodic and antiperiodic boundary value problems, generated on the finite interval by the Sturm-Liouville equation with real potential with nonintegrable singularity at the ends were observed. Fundamental system of solutions of equation were constructed by special way, asymptotic of these solutions were studying. Asymptotic eigenvalues of considered boundary value problems were obtained.Inverse spectral problems for differential operator of Sturm-Liouville on the finite interval with nonintegrable singularity at one of the ends were investigated.The necessary and sufficient conditions, to which should satisfy two sequences of real numbers to be a spectrums of periodic and antiperiodic boundary value problems for some singular classes of real potentials, and construction method of such equations was obtained.

References 1. Gasimov Z., On the solution of the inverse problem of two spectrums for singular Sturm-

Liouville equation, DAN Russia, 365 (1999), 304-305.

Gaşpar, Dumitru, Suciu, Nicolae

Timisoara, Romania

On the generalized von Neumann inequality

The classical von Neumann inequality states that ∥p(T)∥ ≤ sup_{|z|=1} |p(z)|, for any contraction

T on a given Hilbert space H and any scalar polynomial p. Moreover, Arveson and Parrott have proved the above inequality even for any operator valued polynomial p, it’s coefficients being (bounded linear) operators on H, which doubly commute with T (i.e. commute with both T and T^{∗}).

The natural analogue of von Neumann - Arveson - Parrott inequality for n variables (n > 1) is

∥P (T_{1}, T_{2}, …, T_{n})∥ ≤ sup_{{|z_{i}|=1 \atop 1≤i≤n}} ∥P (z_{1}, z_{2}, …, z_{n})∥,  (1)

where T = (T_{1}, T_{2}, …, T_{n}) is a commuting multi-contraction on H and P = P (z_{1}, z_{2}, …, z_{n}) is any polynomial of n complex variables, such that the coefficients of P are opertors on H, which doubly commute with each contraction T_{i}. Notice that the inequality (1) is not true if the multi-contraction T doesn’t admit a commuting isometric dilation, even if T satisfies (1) for any polynomial P having scalar coefficients (see Parrott’s example for n = 3).

Our purpose is to show the inequality (1) for a special class of commuting multi-contractions which contains the multi-contractions having regular (or ∗-regular) isometric dilations. More precisely, if a multi-contraction T = (T_{1}, T_{2}, …, T_{n}) has the property that both T′ := (T_{1}, T_{2}, …, T_{n−2}, T_{n−1}) and T′′ := (T_{1}, T_{2}, …, T_{n−2}, T_{n}) have regular (or ∗-regular) isometric dilations, then the generalized von Neumann inequality (1) holds for T. We remark some particular cases of our theorem. First, we infer that the inequality (1) is true for any pair of commuting contractions T_{1} and T_{2} on H (the case n = 2). We also obtain (1) for any sequence T = {T_{ij}} of contractions on H which has poperty (P) in the sense of Gelu Popescu, and in particular, for any spherical contraction T = (T_{1}, T_{2}, …, T_{n}) on H.

Gaşpar, Păstorel

Timisoara, Romania

Wold decompositions and the Douglas - Foiaş model for bi-isometries

Recently R.G. Douglas and C. Foiaş introduced the so-called unitary model for a c.n.u. multi-isometry (i.e. a n-tuple of commuting isometries, for which the product isometry is completely non-unitary).

For a c.n.u. bi-isometry [H, (V_{1}, V_{2})] the Douglas - Foiaş unitary model gets a simpler form. It consits of a triplet {E, W, P }, with W a unitary operator and P a self-adjoint projection on the same Hilbert space E (E = ker(V_{1}V_{2})^{∗}).

The parametrisation w.r.t. W and P gives a classification of all c.n.u. bi-isometries. Moreover, this unitary model allows the contruction of the functional model [H^{2}(E), (V_{1}(·), V_{2}(·))] for [H, (V_{1}, V_{2})], where V_{1}(·) and V_{2}(·) are the inner operator functions

V_{1}(z) = W (zP + (I − P)) V_{2}(z) = (P + z(I − P))W^{∗}, z ∈ 𝕋.

Our aim is to characterise the particular c.n.u. bi-isometires, which appear in different Woldtype decompositions of a bi-isometry, in terms of the parameters W and P. For instance a c.n.u. bi-isometry is a bi-shift (i.e. it exists a wandering subspace R w.r.t. V_{1}, V_{2}, such that H = ⨁_{(n_{1},n_{2})∈ℤ_{+}^{2}} V_{1}^{n_{1}}V_{2}^{n_{2}}R) iff P W = P W P and W is a bilateral shift with defect space

W P E ⊖ P E. The bi-isometies of shift-type, modified bi-shift type are also characterized.

Finally by proving a one-to-one correspondence between the orthogonal decompositions of the bi-isometry and the decompositions of its unitary model, we obtain new proofs of the Wold decompositions for a bi-isometry by means of the appropriate orthogonal decompositions of the corresponding unitary model.

Gãvruţã, Pasc

Timisoara, Romania

Wavelets and spectral measures

We give a construction of the scaling vectors and (pre)wavelets in a Hilbert space using the spectral measure of an unitary operator.

We make also a discussion concerning the connection coefficients of an operator in a Hilbert space.

Gevorgyan, Levon

Erevan, Armenia

Local spectral theory of equiparanormal operators

Let X be a Banach space, A− a (bounded) linear transformation, referred in what follows as operator. The most known, well studied, important and the simplest class of operators is the class of self-adjoint operators, acting in a Hilbert space. We show that many of pleasant properties of self-adjoint operators may be deduced from a single algebraic condition.

The operator A is said to be equiparanormal, if for any λ ∈ ℂ and x ∈ X the inequality

∥(A − λI) x∥^{2} ≤ ∥(A − λI)^{2} x∥ ∥x∥

is satisfied.

We show that for an equiparanormal operator A

a) for any x ∈ X there exists the limit

lim_{n→∞} ∥A^{n}x∥^{\frac{1}{n}} = r (x, A) ,

b) A satisfies Dunford’s spectrality conditions (A), (C), c) for any λ from the local resolvent set ρ (x) = ℂ \ σ (x) of the element x the inequality

∥x (λ)∥ ≤ \frac{∥x∥}{dist (λ, σ (x))}

holds,

c’) the closure of the numerical range of equiparanormal operator A coincides with the convex hull of the spectrum of A,

d) the element x satisfies the inequality

∥A^{n}x∥ ≤ ε^{n} ∥x∥

if and only if r (x, A) ≤ ε, etc.

Using these properties, we generalize and refine some inequalities due to M. Krein-P. Lax and W. Reid-P. Halmos and prove theorems, concerning the intersection of ranges.

Gohberg, Israel

Tel Aviv, Israel

Regularized determinants for linear operators and applications

A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Hilbert and Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer’s rule and the formulas for the resolvent which are expressed via the extended traces tr(A^{k}) of iterations and regularized determinants. Examples and illustrations are included. A continuous analogue of the Jacobi-Sturm rule of counting the number of negative eigenvalues of a Hermitian matrix is derived.

The paper is based on joint work with Seymour Goldberg and Naum Krupnik.

Golinskii, Leonid

Kharkov, Ukraine

Orthogonal polynomilas on the unit circle from operator-theoretic viewpoint

Let μ be a probability measure on the unit circle 𝕋 such that its support, supp μ (which is, by definition, the smallest closed set whose complement has μ measure 0), is an infinite set.

Orthonormal polynomials {ϕ_{n}(μ)}_{0}^{∞} are defined by

∫_{𝕋} ϕ_{n}(ζ, μ)\overline{ϕ_{m}(ζ, μ)} dμ(ζ) = δ_{m,n} , m, n ∈ ℤ^{+} ,

where

ϕ_{n}(z, μ) = κ_{n}(μ)z^{n} + lower degree terms, κ_{n}(μ) > 0.

The monic orthogonal polynomials are Φ_{n}(z, μ) = κ_{n}^{−1}(μ)ϕ_{n}(z, μ), and the complex numbers {a_{n}(μ) = Φ_{n}(0, μ)}_{n=0}^{∞} are known as the reflection coefficients. They play the similar role in the theory as the entries of Jacobi matrices in the theory of orthogonal polynomials on the real line.

We develop an operator–theoretic approach to orthogonal polynomials on the unit circle based on studying the multiplication operator on L^{2}(μ, 𝕋) and its matrix representation in the orthonormal polynomials basis. The results from the spectral theory of unitary operators in Hilbert space can now be viewed in this new setting. For instance, the perturbation theory results can be paraphrased as the statements about the relation between orthogonality measures and asymptotic behavior of their reflection coefficients.

Gomilko, Alexander

Kiev, Ukraine

Fourier method for biharmonic Dirichlet problem in a semistrip

We study whether the following double expansion is possible and what is the character of its convergence:

( \begin{matrix}f(x) \\ g(x) \end{matrix} ) = ∑_{k=1}^{∞} c_{k} ( \begin{matrix}y_{k}(x) \\ iλ_{k}y_{k}(x) \end{matrix} ) , |x| < 1, Im λ_{k} > 0.  (1)

Here y_{k}(x) are eigenfunctions of the spectral problem

y^{(4)}(x) − 2λ^{2}y^{2}(x) + λ^{4}y(x) = 0, y^{(k)}(±1) = 0, k = 0, 1,

corresponding to the eigenvalues λ_{k} with Im λ_{k} > 0. The problem of expansion (1) is connected with the Fourier method for solving the Dirichlet boundary value problem for a homogeneous biharmonic equation in the semistrip z > 0, |x| < 1. The statement of this problem, as one of the mathematical problems in the theory of plates, comes from P. F. Papkovich (1941). This problem was considered (by absolutely different methods) by R. D. Gregory (1980), A. A. Shkalikov (1980), A. M. Gomilko (1993). In this paper we prove new estimates of the rate of convergence of the series in (1) to a given pair of functions in various function spaces.

Gorbachuk, Myroslav

Kiev, Ukraine

Classes of infinitely differentiable vectors of a closed linear operator and dissipative systems

We consider a differential equation of the form

y′(t) = Ay(t), t ∈ (0, ∞),  (1)

where A is a maximal dissipative linear operator on a Banach space. A description of all weak solutions of equation (1) on (0, ∞) is given, and their behavior near the point 0 and at infinity is studying. We find also necessary and sufficient conditions on the initial value, under which the solution of the Cauchy problem for equation (1) belongs to the class of entire vector-functions of preassigned growth order and type. Presented are the criterions for density of the set of solutions with initial values satisfying to such conditions in the set of all weak solutions of (1).

Gorbachuk, Valentyna

Kiev, Ukraine

Entire functions of eponential type of a closed operator and some approximation problems

Let A be a closed linear operator on a Banach space 𝔅. For a number α > 0 we put

E^{α}(A) = {f ∈ ⋂_{n∈ℕ_{0}={0,1,2,... }}D(A^{n})|∃c = c(f) > 0 : ∥A^{n}f∥ ≤ cα^{n}, ∀n ∈ ℕ_{0}}.

The set E(A) = ∪_{α>0}E^{α}(A) is called the space of vectors of exponential type for the operator A. By the type of f ∈ E(A) we mean σ = σ(f) = inf{α : f ∈ E^{α}(A)}. If we take the spaces C̃([0, 2π]), C([−1, 1]), or L_{p}(ℝ^{n}) (p ≥ 1) for 𝔅, and the operators i\frac{d}{dx}, (\sqrt{1 − x^{2}} \frac{d}{dx})^{2}, i\frac{d}{dx}, respectively, for A, then E(A) coincides with the set of all trigonometric or algebraic polynomials, or entire functions of exponential type, and the type σ is none other than a polynomial degree or the type of an entire function of exponential type.

We find the conditions on the operator A, under which \overline{E(A)} = 𝔅 (an analog of the Weierstrass theorem in the approximation theory ), and establish some direct and inverse theorems that characterize the "smoothness" of a vector f ∈ 𝔅 in terms of the best approximation E_{r}(f) = inf_{g∈E(A):σ(g)≤r} ∥f − g∥ of f by vectors of exponential type of the operator A. The

results are applied for approximate solutions of an operator equation of the form Ax = y in a Hilbert space.

Gubreev, Gennady

Odessa, Ukraine

The structure of Volterra operators and biorthogonal decompositions

Let E be an entire function satisfying the condition |E(\overline{z})| < |E(z)| (Im z > 0), and H(E) be the corresponding de Branges space. Suppose that some function S_{α}(z) := e^{iαz} (α ∈ R) accompanies the space H(E). Consider in H(E) the family B_{α} (−ω ≤ α ≤ ω) of Volterra nondissipative operators whose conjugate ones are given by formula

(B_{α}^{∗}f)(λ) = λ^{−1}(f(λ) − e^{iαz}f(0)), ω := lim sup_{y→+∞} y^{−1} log |E(iy)|.

Denote by I_{γ} the operator of integration in L_{2}(0, γ):

(I_{γ}h)(x) = i ∫_{0}^{x} h(s) ds, h ∈ L_{2}(0, γ).

We shall call the space H(E) regular if the weight |E(x)|^{2} satisfies the Muckenhoupt condition (A_{2}) on the real axis.

Theorem 1 Let some function S_{α} accompany the de Branges space H(E). If H(E) is regular then each operator B_{α} (−ω ≤ α ≤ ω) is similar to the orthogonal sum I_{ω+α} ⊕ (−I_{ω−α}) of the integration operators, and there exists an ϵ > 0 such that for each β ∈ (0, ϵ]

0 < inf_{z∈R+iβ} |E^{∗}(z)/E(z)| ≤ sup_{z∈R+iβ} |E^{∗}(z)/E(z)| < 1.  (1)

Conversely, let the strip 0 ≤ Im z ≤ ϵ exist free of zeros of E^{∗}, and for some β_{0} ∈ (0, ϵ] one has (1). If some operator B_{α} is similar to the orthogonal sum B_{1} ⊕ (−B_{2}), where B_{i} are dissipative operators in separable Hilbert spaces H_{i} (i = 1, 2), then the space H(E) is regular.

This result is used for the description of unconditional bases consisting of values of the de Branges space reproducing kernel.

Hasanov, Mahir

Istanbul, Turkey

On the operator-valued functions of quasihyperbolic type

According to the definition given in [1], a monic selfadjoint operator polynomial of the form

L(λ) = λ^{n}I + ∑_{k=0}^{n−1} λ^{k}A_{k}, A_{k}^{∗} = A_{k}

is said to be quasihyperbolic(Qh) if its spectrum σ(L) is real and any λ ∈ σ(L) is of determinate type in the sense that the conditions ∥f_{n}∥ = 1, L(λ)f_{n} → 0, as n → ∞ imply that either \underline{lim}(L′(λ)f_{n}, f_{n}) > 0 (when λ is said to be of plus type), or \overline{lim}(L′(λ)f_{n}, f_{n}) < 0 (when λ is said to be minus type). There is a connection between definitizable operators and Qh pencils ([1], [2]). A continuously differentiable operator-valued functions L(λ) will be called a function of Qh type if its spectrum σ(L) is real and any λ ∈ σ(L) is of determinate type. We note that operator valued- functions of Qh type does note form a Rayleigh system ([3]). We study variational characterizations of the spectrum. Particularly, we establish some minimax principles for eigenvalues. The forms [x, y] and [x, y|α] are defined and analyzed for operator-valued functions of Quasihyberbolic type. Using the forms we obtain an analog of Riesz’z integral representation theorem for the orthogonal projection, associated with L(λ) at the eigenvalue λ_{0}.

References

1. P. Lancaster, A. S. Markus and V. I. Matsaev, Definitizable operators and quasihyperbolic operator polynomials, J. Func. Anal., 131 (1995), 29-53.

2. H. Langer, Spectral Functions of Definitizable Operators in Krein Spaces, Lecture Notes in Math., 948 (1982), Springer-Verlag, 1-46.

3. M. Hasanov and Y. Cesur, On a nonlinear eigenvalue problem, Integral Equatons Operator Theory, 29 (1997), 491-500.

Hassi, Seppo

Helsinki, Finland

On the class of extremal extensions of a nonnegative operator

Let A be a densely defined nonnegative operator in a Hilbert space ℌ. Its defect numbers are equal and it has selfadjoint extensions in ℌ. A nonnegative selfadjoint extension Ã of A is said to be extremal if inf{(Ã(φ − f), φ − f) : f ∈ dom A} = 0 for all φ ∈ dom Ã. A new construction of all extremal extensions of A is presented. It employs a fixed auxilliary Hilbert space ℌ_{A} associated to A and yields a factorization of each extremal extension of A. This approach is used to give various functional-analytic interpretations of extremal extensions and it results in some new types of characterizations for them. In particular, a purely analytic description of extremal extensions is established, based on a class of functions introduced by M. G. Kreĭn and I. E. Ovčarenko. Applications to some second order differential operators are treated. This talk reports on joint work with Yu. M. Arlinskiĭ (Lugansk), Z. Sebestyén (Budapest), and H. S. V. de Snoo (Groningen).

Helton, William

San Diego, USA

Operator theory and noncommutative computer algebra

The talk describes some recent progress in noncommutative computer algebra. There is an advance in methods and new results on matrix completion, singular perturbations, determination of convexity of a polynomial on matrices, etc.. The talk will describe some of these.

Hryniv, Rostyslav

L’vov, Ukraine

Spectral problems with eigenparameter dependent boundary conditions

We study the spectral problem

Ax = λx  in X,  (1)
Cx = λBx in Y,  (2)

where X, Y are Banach spaces and

A : X → X and B, C : X → Y

are linear (in general unbounded) operators such that A is closable in X and D(A) ⊂ D(B) ∩ D(C). Of this type are, e. g., spectral problems for differential operators with eigenparameter dependent boundary conditions; then A is a differential operator in a domain and B and C are boundary operators. In particular, for A = −Δ in a bounded smooth domain Ω ⊂ ℝ^{n} with B the normal derivative on the boundary ∂Ω and C the trace on ∂Ω, we can take X = L_{p}(Ω), Y = L_{p}(∂Ω), and D(A) = W_{p}^{2}(Ω), p ≥ 1.

We identify (1)-(2) as a spectral problem Ax = λX for some operator A in X × Y and then establish conditions for A to be closable, describe its closure Ā, and study in detail the spectral properties of Ā.

Ivanov, Sergei

St. Petersburg, Russia

Interpolation of subspaces and exponential bases

We give precise conditions under which the real interpolation space [Y_{0}, X_{1}]_{θ,p} coincides with a closed subspace of [X_{0}, X_{1}]_{θ,p} when Y_{0} is a closed subspace of codimension one. For the case of X_{0} = L_{μ}^{1}, X_{1} = L_{μ}^{∞}, R. Wallsten proved that there exist two parameters 0 ≤ s_{0} ≤ s_{1} ≤ 1 such that the metric of Y_{θ} is stronger not at one point but on the segment s_{0} ≤ θ ≤ s_{1}. We show that for general case we have a similar situation where two critical values are needed, even for Hilbert spaces. We then apply this result to nonharmonic Fourier series in Sobolev spaces. In terms of the generating function the critical exponents s_{−}, s_{+} are given such that the exponential family, which is an unconditional basis in L^{2}(0, a), forms an unconditional basis in Sobolev spaces H^{s}(0, a) for s_{+} − 1 < s < s_{−}. For s_{−} ≤ s ≤ s_{+} the family is not an unconditional basis in its span. The examples with arbitrary 0 < s_{−} < s_{+} < 1 are presented.

This is joint work with Nigel Kalton.

Jaming, Philippe

Orleans, France

The discrete radar ambiguity problem

In this talk, I will consider the following problem, known as the radar ambiguity problem: For u ∈ L^{2}(ℝ) fixed, find all v ∈ L^{2}(ℝ) such that, for all x, y ∈ ℝ, |A(u)(x, y)| =

|A(v)(x, y)| where A(u)(x, y) = ∫_{−∞}^{+∞} u(t)\overline{u(t + x)}e^{iyt}dt.

This problem has been partially solved for time limited (compactly supported) signals u. We will here pursue this study via a discretization of the problem. This transforms the initial problem into an orthogonality-like condition on matrices that are Schur products of Hankel and Toeplitz matrices. This new approach allows us to get new examples and counterexamples for the first problem as well as to get an insight on its complexity.

This is joint work with A. Bonami and G. Garrigós.

Janas, Jan

Krakow, Poland

Jacobi matrices with periodically modulated entries (jointly with S. Naboko)

Let J be a Jacobi matrix acting in l^{2} by the formula

(Ju)_{n} = λ_{n−1}u_{n−1} + q_{n}u_{n} + λ_{n}u_{n+1},

where λ_{n} > 0 and q_{n} are real valued. The talk will be devoted to the spectral analysis of J in the case λ_{n} = a_{n}μ_{n} and q_{n} = b_{n}r_{n}, where a_{n} is a fixed N periodic sequence of positive numbers and b_{n} is also a fixed M periodic sequence of real numbers. Spectral phase transition of the first type and the interplay and the interplay between M and N will be shown.

Kaashoek, Marinus

Amsterdam, The Netherlands

The Perelson determinant and limit theorems of Kac-Achiezer type

In this talk we review the main properties of the Perelson determinant, and use these properties to derive generalizations of Kac-Acheizer type limit theorems for Wiener-Hopf integral operators. The entries of the kernel functions of the Wiener-Hopf operators are both summmable and square summable and have an additional continuity property at the origin. Special attention will be paid to the case when the symbol is a rational matrix function. The talk is based on joint work with I. Gohberg.

Kac, Israel

Odessa, Ukraine

The spectral property of dual strings as consequence of the existence of spectral matrix functions for canonical system

The spectral property of dual strings as a consequence of the existence of spectral matrix functions for canonical system.

The term "dual string" for the scalar strings was introduced in [1]. The interrelation between the spectral functions of the fully dual scalar strings was established in [2]. Recently this result was generalized on the regular matrix strings with the continuous everywhere invertible matrix density by H. Dym and L.A. Sakhnovich [3]. The main aim of the present talk is the generalization of the last result on the matrix strings which may be singular, and with the matrix density perhaps everywhere discontinuous and even if everywhere noninvertible.

References

1. I. S. Kac, M. G. Krein, Izv. VysŽ. Ucebn. Zaved. Matematika, 3 (1958), no. 2, 136–153 (Russian).

2. I. S. Kac, M. G. Krein, Amer. Math. Soc. Transl. (2), 103 (1974), 19–102.

3. H. Dym, L. A. Sakhnovich, Integral Equations Operator Theory (to appear).

Kalyuzhniy, Dmitriy

Odessa, Ukraine

Certain state space realization theorems

A multiparametric time-invariant linear system α is determined by Hilbert spaces U (the input space), Y (the output space), X (the state space), and an N-tuple G = (G_{1}, …, G_{N}) of bounded linear operators

G_{k} = ( \begin{matrix}A_{k} & B_{k} \\ C_{k} & D_{k} \end{matrix} ) : X ⊕ U → X ⊕ Y (k = 1, …, N).

If one denotes zT := ∑_{k=1}^{N} z_{k}T_{k} for N-tuples z = (z_{1}, …, z_{N}) of complex numbers and T = (T_{1}, …, T_{N}) of operators then the transfer function of α is defined as

θ_{α}(z) = zD + zC(I_{X} − zA)^{−1}zB.

If J = J^{∗} = J^{−1} is a bounded linear operator on X, and for each ζ ∈ T^{N} (the N-fold unit torus) the operator ζG = ∑_{k=1}^{N} ζ_{k}G_{k} is (J ⊕ I_{U}, J ⊕ I_{Y})-unitary then we call α a Jconservative scattering system. In the special case J = I_{X} such an α is called a conservative scattering system. We prove that any operator-valued function θ(z) holomorphic on some neighbourhood of z = 0 in C^{N} and vanishing at z = 0 can be realized as the transfer function of some N-parametric linear system. Then we prove that any N-parametric linear system α has a dilation which is a J-conservative scattering system. Since the transfer function of a dilation of α is the same as θ_{α}(z), we obtain that any θ(z) as above has a J-conservative scattering system realization.

In the proofs of these results we use our earlier result on the existence of conservative scattering system realizations for operator-valued Agler–Schur-class functions on the polydisk 𝔻^{N} that vanish at z = 0.

Kapustin, Vladimir

St. Petersburg, Russia

One-dimensional perturbations of unitary operators and exposed points in H^{1}

The problem of description of the exposed points of the unit ball in H^{1} is equivalent to a description of log-summable weights w on the unit circle, for which the intersection I_{w} of the closures of analytic and antianalytic polynomials is zero. (It is well known that the positiveness of the angle between these subspaces is characterized by the Helson-Szego and Muckenhoupt conditions). On the space I_{w} we can consider the operator T,

T f = zf − lim(zf(z))_{z→∞}.

We observe that the complement of the spectrum of this operator coincides with the set where all the functions from I_{w} are analytic, and we obtain some applications of this observation to the question about the exposed points in H^{1}. In particular, we show that this property is local, i.e., it depends on the behaviour of the modulus of a function from H^{1} (which is equal to w) at each point of the unit circle.

The other point of our talk is the theorem which says that one-dimensional perturbations of singular unitary operators of a general form (with some very natural restrictions) are unitarily equivalent to an operator of the form as above, but instead of the spaces I_{w} we have to consider a generalization of them, where the fuctions from the space are allowed to have inner factors in the denominator. The latter fact has also some natural extensions to a wider class of operators (which are not necessarily contractions) close to a unitary operator.

Karlovich, Alexei

Lisbon, Portugal and Odessa, Ukraine

Algebras of singular integral operators in rearrangement-invariant spaces with Muckenhoupt weights

In this talk we study Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces with weights X(Γ, w) over arbitrary Carleson curves Γ. Suppose a weight w belongs to the Muckenhoupt classes A_{\frac{1}{α_{X}}} (Γ) and

A_{\frac{1}{β_{X}}} (Γ), where α_{X} and β_{X} are Boyd indices of a rearrangement-invariant space X(Γ). We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator S in the weighted rearrangement-invariant space X(Γ, w). Under some “disintegration condition” we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients. We prove criteria for Fredholmness and get a formula for the index of any operator from this algebra in terms of its symbol. We give nontrivial examples of spaces, for which this “disintegration condition” is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve. Another nontrivial example of space satisfying this “disintegration condition” is a reflexive Orlicz space with distinct Boyd indices over a smooth curve. Note that for weighted Lebesgue spaces corresponding results were obtained by I. Gohberg and N. Krupnik for power weights (the end of sixties); and I. Spitkovsky, A. Böttcher and Yu. I. Karlovich for arbitrary Muckenhoupt weights (the middle of nineties).

Karlovich, Yuri

Mexico, Mexico and Odessa, Ukraine

Toeplitz operators with matrix oscillating symbols on weighted Hardy spaces

The talk is devoted to a present state of the theory of Toeplitz operators with semi-almost periodic and more general oscillating symbols on weighted Hardy spaces on the upper halfplane. Invertibility criteria for Toeplitz operators with matrix almost periodic symbols as well as Fredholm criteria and index formulas for Toeplitz operators with matrix semi-almost periodic symbols are established. These results are extended to Toeplitz operators with matrix symbols belonging to the algebra generated by semi-almost periodic and slowly oscillating matrix functions. Presented results are based on the theory of almost periodic factorization, an isomorphism theorem for C^{∗}-algebras of nonlocal operators, the inverse closedness of some Wiener algebras, limit operators techniques and the theory of convolution type operators.

Close relations between the factorability of bounded measurable matrix-valued functions, the corona problem, and the solvability of an associated Riemann boundary value problem are studied. Criteria for the existence of a generalized canonical factorization and a bounded canonical factorization as well as explicit formulas for their factors are obtained in terms of solutions of two associated corona problems (in the upper and lower half-planes). A separation principle for constructing corona solutions from simpler ones is developed. Applications to Toeplitz operators with oscillating matrix symbols and convolution type operators on finite intervals are considered.

Kerchy, Laszlo

Szeged, Hungary

Invariant subspace lattices of isometries

In the eighties J. B. Conway and T. A. Gillespie characterized those reductive normal operators, which have isomorphic invariant subspace lattices. They have also given several necessary conditions of isomorphism in the class of nonreductive isometries. In this talk we provide a new necessary condition, when the isometry contains a bilateral shift. Furthermore, we give complete characterization of isomorphism if the nonreductive components of the isometries are cyclic. It turns out that this characterization is of different type in the unitary and in the nonunitary case. We describe also when absolutely continuous unitary operators have spatially isomorphic invariant subspace lattices. Our results provide answers for questions posed by Conway and Gillespie, and have been appeared in [J. Funct. Anal. 170 (2000), 475–511].

Kheifets, Alexander

Ariel, Israel

Couplings, scales and wave operators in interpolation problems

Consider the Lax-Philips scattering system: L_{0} be a Hilbert space, U_{0} be a unitary operator on it and L_{0} admits the following orthogonal decomposition

L_{0} = … ⊕ N_{2}^{(1)} ⊕ N_{2}^{(0)} ⊕ H_{0} ⊕ N_{1}^{(−1)} ⊕ N_{1}^{(−2)} ⊕ …,

where N_{1}^{(k)} = U_{0}N_{1}^{(k+1)}, k ≤ −1; N_{2}^{(k)} = U_{0}N_{2}^{(k+1)}, k ≥ 0.

The natural "wave" operator W is considered for arbitrary feedback loading of this scattering system. Generically it is not a bounded operator. The first result is a description of it’s domain in terms of spectral functions and function model spaces.

Let E be an auxiliary Hilbert space and ρ_{0} : E → L_{0} be a scale of unloaded system. If range of ρ_{0} is in the domain of W then a scale of the loaded system can be defined as follows: ρ = W ρ_{0}. In this case the scattering function of the loaded system is computed with respect to the scale ρ . This can be applied to parameterizing solution sets to interpolation problems of Schur type (including general Commutant Lifting Problem and others). A more general setting of Abstract Interpolation problem can be treated on this way.

Kissin, Edward

London, UK

On Johnson-Williams-Fong’s condition for operators

Any S ∈ B(H) defines a derivation δ_{S} on Schatten ideals C_{p} : δ_{S}(X) = [S, X]. Johnson and Williams proved that the range inclusion δ_{T} (B(H)) ⊆ δ_{S}(B(H)), for normal S, implies that T = g(S) where g is a Lipschitz function on σ(S). Fong showed that the condition T = g(S) is equivalent to a weaker range inclusion δ_{T} (C_{1}) ⊆ δ_{S}(B(H)).

For p ∈ [1, ∞] ⋃ b and normal S, we study the conditions:

1) δ_{T} (Cp) ⊆ δ_{S}(Cp),

2) |δ_{T} (X)|_{p} ≤ |δ_{S}(X)|_{p}, for X ∈ C_{p} and some D > 0.

We establish that they are equivalent and, if they hold, then

(i) δ_{T} (Cr) ⊆ δ_{S}(Cr), where min(p, p′) ≤ r ≤ max(p, p′) and p′ = p/(p − 1);

(ii) there is K > 0 such that |δ_{T} (X)|_{p} ≤ K|δ_{S}|_{p}, for X ∈ B(H);

(iii) δ_{g(A)}(C_{p}) ⊆ δ_{A}(C_{p}) for any A with σ(A) ⊆ σ(S).

We show that the range inclusion 1) implies that g is also an Operator C_{p}-Lipschitz function on σ(S), i.e., there is D > 0 such that, for all normal A and B with spectrums in σ(S), |g(A) − g(B)|_{p} ≤ D|A − B|_{p}. Conversely, we prove that, for a selfadjoint S and any Operator C_{p}-Lipschitz function g on σ(S), we have δ_{g(S)}(C_{p}) ⊆ δ_{S}(C_{p}). For non-selfadjoint S, this only holds for p ∈ (1, ∞) and fails if p ∈ 1, ∞, b.

This is joint work with Victor Shulman.

Konstantinov, Alexei

Kiev, Ukraine

A commutation formula for unbounded closed operators

It is well-known that for bounded operators A and B the spectra of the products AB and BA coincide away from zero:

σ(AB) \ {0} = σ(BA) \ {0}.

We prove this equality for unbounded closed operators A and B acting in Bahach spaces under the natural assumption that the resolvent sets of AB and BA are not empty. Moreover, for λ ∈ ρ(AB) \ {0} we prove the commutation formula:

\overline{A(BA − λ)^{−1}B} − λ(AB − λ)^{−1} = 11.

The talk is based on the joint work with V.Hardt and R.Mennicken.

Koshmanenko, Volodymyr

Kiev, Ukraine

On negative eigenvalue problem under singular perturbations of self-adjoint operators

Let A = A^{∗} ≥ 0 be a self-adjoint operator in Hilbert space H. A singular perturbation Ã of A is defined as Ã = A ˜+V, where V acts in the A-scale of Hilbert spaces, V : H_{1}(A) → H_{−1}(A) and ˜+ stands for the generalized operator sum. We investigate the negative eigenvalue problem: Ãψ = Eψ, E < 0. Under suitable assumptions on V we find conditions when −1 is the lowest negative eigenvalue for Ã, show that σ_{disc}(Ã) ∩ (−∞, c), c < 0 is finite, construct examples when σ_{disc}(Ã) contains infinite number of points.

Applications deal with Schrödinger operator −Δ_{β,V,Γ} = −Δ ˜+ βV, β ∈ ℝ with a generalized potential (distribution) V acting in the Sobolev scale, V : W_{2}^{1}(ℝ^{n}) → W_{2}^{−1}(ℝ^{n}) and supported by a self-similar fractal Γ ⊂ ℝ^{n}. For this perturbation we show in particular that −Δ_{β,V,Γ} converges in the strong resolvent sense to the Friedrichs extension −Δ_{∞} of the symmetric restriction −Δ̇ = −Δ | {f ∈ W_{2}^{2}(ℝ^{n}) : f(γ) = 0, γ ∈ Γ}.

References

1. T. V. Karataeva, V. D. Koshmanenko, Generalized sum of operators, Math. Notes 66 (1999), no. 5, 671-681.

2. S. Albeverio, V. Koshmanenko, Form-sum approximation of singular perturbation of self-adjoint operators, J. Funct. Anal. 168 (1999).

Kosiek, Marek

Krakow, Poland

Lebesgue-type decompositions for n-tuples of contractions without von Neumann inequlity

It is well known, that for any contraction T acting on a Hilbert space H we have the following decomposition

H = H_{a} ⊕ H_{s},

where H_{a}, H_{s} are reducing subspaces for T, the restriction of T to H_{a} is absolutely continuous, and the restriction of T to H_{s} is unitary singular.

It appears that the subspaces H_{a}, H_{s} are reducing also for the commutant of T. Consequently for any n-tuple T_{1}, …, T_{n} of commuting contractions on H we get the following decomposition

H = ⨁_{j=1}^{2^{n}} H_{j},

where all H_{j} reduces T_{1}, …, T_{n}, and for every i the operator T_{i} restricted to any of the subspaces H_{j} is either absolutely continuous or unitary singular. We does not require any kind of von Neumann inequality for T_{1}, …, T_{n}.

Kupin, Stanislav

Bordeaux, France and Kharkov, Ukraine

Some criteria for similarity of integral dissipative operator to a normal operator

Let μ be a finite positive measure on [0, 1], α be a measurable real-valued function and c_{k} ∈ L^{2}(μ), k = 1, n, n ≤ ∞. We define an operator A by the formula

(Af)(x) = α(x)f(x) + i ∑_{k=1}^{n} \overline{c_{k}(x)} ∫_{0}^{x+} c_{k}(t)f(t)dμ(t),

where f ∈ D(A), the domain of A. The symbol ∫_{0}^{x+} denotes

∫_{0}^{x+} f(t)dμ(t) = ∫_{[0,x)} f(t)dμ(t) + 1/2 μ({x})f(x).

In this talk we concern with the question on the similarity of A to a normal operator. We obtain necessary as well as sufficient conditions for the similarity for arbitrary n. The conditions turn out to be necessary and sufficient in the case n = 1 as well as in the case of a continuous measure μ and arbitrary n.

Studying the case n = ∞, we suppose that A satisfies the so-called uniform trace boundedness condition

sup_{z∈ℂ_{+}}(Im z) tr [(A^{∗} + \overline{z}I)^{−1}Im A(A + zI)^{−1}] < ∞.

The technique we apply is in essential based on linear resolvent growth tests for dissipative operators. The results generalize certain conclusions of a paper by I. Gohberg and M. Krein.

Kurasov, Pavel

Stokholm, Sweden

Krein’s formula for supersingular perturbations of self-adjoint operators

Krein’s formula for resolvents describes the relations between the resolvents of two selfadjoint operators that are extensions of one symmetric operator. This formula has been applied to study finite rank singular perturbations formally determined by elements from the Hilbert space H_{−2} being the set of bounded linear functionals on the domain of the original operator. We are going to discuss supersingular finite rank perturbations determined by the vectors from a certain Hilbert space H_{−n}, n = 3, 4, …. Such perturbations can be defined in extended Hilbert space. The main tool in the investigation of the spectral properties of these operators is a certain generalization of Krein’s formula. The structure of the corresponding Q-function is analyzed. Some of the results under discussion were obtained in collaboration with S.Albeverio, B.Pavlov and K.Watanabe.

References

1. S. Albeverio, P. Kurasov, Singular perturbations of differential operators and exactly solvable Schrödinger-type operators, Cambridge Univ. Press, 2000 (London Mathematical Society Lecture Notes 271).

2. P.Kurasov, K.Watanabe, On rank one H_{−3}-perturbations of positive self-adjoint operators, Research report 10, Dept. of Mathematics, Stockholm Univ., 1999 (accepted for publication in Proc. S. Albeverio conference in Leipzig, 1999).

Kuzhel, Sergei

Kiev, Ukraine

A new operator approach to determination of analytic continuation of scattering matrix

A simple operator algorithm for the construction of analytic continuation of the Lax-Phillips scattering matrix for an operator-differential equation u_{tt} = −Lu, where L is a positive self-adjoint operator acting in an abstract separable Hilbert space H is presented.

The proposed algorithm is based on the following observation: the existence of Lax-Phillips incoming D_{−} and outgoing D_{+} subspaces for the group W (t) of solutions of the Cauchy problem for equation u_{tt} = −Lu is equivalent to the existence of a simple maximal symmetric operator B acting in the space H_{1} ⊆ H and such that the operator L is an extension (with the exit in H) of the symmetric operator B^{2}. Furthermore, the subspaces D_{±} are expressed by terms of the operator B. By using this fact, we construct explicitly incoming and outgoing spectral representations for the group W (t) and obtain an analytic continuation of the Lax-Phillips scattering matrix in terms of Positive Boundary Value Space of the operator B^{2}.

These results enable us to establish the direct relationship between singularities of analytic continuation of the Lax-Phillips scattering matrix and properties of the operator L.

By using this abstract approach, one can significantly extend the applicability of the Lax-Phillips scattering method and consider various new scattering problems for the classical wave equation.

Lakaev, Saidakhmat

Samarkand, Uzbekiston

The Faddev’s equation and Efimov’s effect for the three-particle difference Schrodinger operator

In models of solid physics [1] a Schrodinger operators are considered on a three dimensional lattice.

It is considered the Hamiltonians of some systems of three particles (three identical and three arbitrary) interacting by means of zero-range pairwise and three-particle potentials moving on a three-dimensional lattice.

These Hamiltonians it have been discribed as a bounded self-adjoint operators (discrete Schrodinger operators - DSO) in corresponding Hilbert spaces of the square-summable functions on three-dimensional lattice [2].

It have been obtained a Faddeev’s and Weinberg’s type compact integral equations for eigenfunctions of the DSO and using these equations it have been established a location and structure of essential spectrum.

The absence of eigenvalues lying to the right of the essential spectrum and the fineteness or the infineteness (Efimov’s effect) of discrete spectrum lying to the left of the essential spectrum have been proved.

The influence of the number of virtual levels on the fineteness or the infineteness of discrete spectrum of the DSO have been manifested.

This is joint work with A. Kholkhujaev and S. Samatov.

References

1. D. C. Mattis, Rev. Modern Phys., 58 (1986), no. 2, 361–379.

2. S. N. Lakaev, Funktsional. Anal. i Prilozhen, 27 (1993), no. 3, 15–28.

Langer, Heinz

Veinna, Austria

The Schur algorithm for generalized Schur functions.

Langer, Matthias

Vienna, Austria

Embedded eigenvalues and eigenvalues in a gap of the essential spectrum of some block operator matrix

We consider eigenvalues of the block operator matrix

Ã = ( \begin{matrix}−\frac{d^{2}}{dx^{2}} + p & v \\ v & u \end{matrix} ) ,

in the space L^{2}(0, 1) ⊕ L^{2}(0, 1) with Dirichlet boundary conditions for the first component, where p, u, and v are real functions. Assume that u is piece-wise continuously differentiable with a positive derivative. Then the essential spectrum of Ã, which is the range of u, consists of a finite number of intervals. The number of eigenvalues which are embedded in the essential spectrum can be estimated by the number of eigenvalues of the operator −\frac{d^{2}}{dx^{2}} + p which are below the maximum of the essential spectrum. If v is large, there are no embedded eigenvalues, but the number of eigenvalues in the gaps of the essential spectrum can be arbitrarily large. Moreover, the eigenvalues in the gaps can be characterized by a min-max principle using the Schur complement of the matrix.

Larson, David

College Station, USA

Some connections between wavelets, frames and operator theory

Operator-theoretic considerations lead naturally to certain results for orthonormal wavelets, Riesz wavelets, frame wavelets and frame sequences that might not seem so natural from a purely function-theoretic point of view, yet can be stated independently from operatortheoretic terminology. We present several instances of this in our work, and we discuss some relevant open problems.

Latushkin, Yuri

Columbia, USA

Spectral mapping theorems in control theory and for nonlinear Schrödinger and Euler equations

There are many examples of strongly continuous semigroups on Banach spaces for which the spectral mapping theorem σ(e^{tA}) \ {0} = exp tσ(A) does not hold. In particular, the position of the spectrum σ(A) does not, generally, determine uniform stability of the semigroup:

It is well-known that on Hilbert spaces the semigroup is uniformly stable if and only if the resolvent of the generator is bounded in the right-hand half-plane. However, even this result does not, generally, hold on Banach spaces.

In the talk we will use operator-valued L^{p}–Fourier multipliers to give a generalization of this result for Banach spaces, and to present its applications for stability in control theory. We also prove the spectral mapping theorem for the group generated by the linearization around a bound state of the nonlinear Schrödinger equation with space-dependent nonlinearity. As a consequence, we derive the existence of locally invariant manifolds around the bound state. Also, we prove the spectral mapping theorem for linearizations of two dimensional Euler equations. Moreover, using Friedlander–Vishik bicharacteristic amplitude equations, we give explicit construction for approximate eigenfunctions of the linearized Euler equations in dimensions two and three. As a consequence, we give a rigorous poof of the fact that a steady state of the Euler equations is hydrodynamically stable if and only if the linearization has pure imaginary spectrum.

Leblond, Juliette

Sophia-Antipolis, France

Diagonalization of Toeplitz operators, convergence properties and asymptotic estimates for H^{2} approximation on part of the boundary

This is a joint work with L. Baratchart, J. Grimm, and J.R. Partington.

The present work stands as a natural sequel to [1, 2, 3] where a family of bounded extremal problems has been studied that generalizes classical dual extremal problems in Hardy spaces H^{p} of the unit disk to a subset of the unit circle 𝕋 (of the complex plane). Being given a function f on some subset I of 𝕋, we are looking for an H^{p} function satisfying some norm constraint on 𝕋 \ I that is as close as possible to f in L^{p} norm on I. Existence and uniqueness results are already available, together with a constructive characterization of the solutions for p = 2, ∞, leading to convergent algorithms that allow one to compute these solutions numerically. In the Hilbertian case p = 2, this procedure relies on iterative inversions of a family of Toeplitz operators indexed by some Lagrange-type parameter. We are concerned here with a precise numerical analysis of this scheme for which we establish sharp L^{2} convergence rates. To this aim, we explicitly diagonalize the involved Toeplitz operators using a concrete spectral theory from [4]. Pointwise convergence results are also derived, as consequences.

References

1. L. Baratchart and J. Leblond, Hardy approximation to L^{p} functions on subsets of the circle with 1 ≤ p < ∞, Constructive Approximation 14 (1998) 41–56.

2. L. Baratchart, J. Leblond and J.R. Partington, Hardy approximation to L^{∞} functions on subsets of the circle, Constructive Approximation 12 (1996), 423–436.

3. M.G. Krein and P.Y. Nudel’man, Approximation of L^{2}(ω_{1}, ω_{2}) functions by minimum– energy transfer functions of linear systems, Problemy Peredachi Informatsii, 11 (1975), 37–60; English translation.

4. M. Rosenblum and J. Rovnyak, Hardy classes and operator theory, Oxford, 1985.

Le Merdy, Christian

Besancon, France

Sums of commuting operators with L^{p}-maximal regularity

Let X be a complex Banach space and let B be an operator on X such that −B generates a bounded analytic semigroup on X, with domain D(B). Given p ∈ (1, ∞), we say that B satisfies L^{p}-maximal regularity (MR in short) if there is a constant C > 0 such that for any u ∈ W^{1,p}(R; X) ∩ L^{p}(R; D(B)), we have ∥u′∥_{p} ≤ C∥u′ + Bu∥_{p}. This property, which does not depend on p ∈ (1, ∞), implies that for any T ∈ (0, ∞) and any f ∈ L^{p}(0, T; X), the strong solution u of the Cauchy problem {u′(t) + B(u(t)) = f(t); u(0) = 0} satisfies u′ ∈ L^{p}(0, T; X). For several years, abstract operator theoretic techniques have been developed to try to determine classes of operators with or without MR, with applications to evolution equations and PDEs. This includes McIntosh’s H^{∞} functional calculus and its connections with bounded imaginary powers. It is well-known that any X isomorphic to a Hilbert space has the property that MR holds for any B such that −B generates a bounded analytic semigroup on X. Very recently, N. Kalton and G. Lancien proved that this property actually characterizes Hilbert spaces among separable Banach lattices. In this talk, I shall present results on MR obtained in the last three years with various coauthors (F. Lancien, G. Lancien, A. Simard). Extending McIntosh’s H^{∞} functional calculus in a suitable way, we exhibit new classes of operators on non Hilbertian spaces satisfying MR. We are mainly concerned by the following issue. Take two commuting operators C and D on X and assume that C and D satisfy MR. Does their sum C + D, or its closure (which is the negative generator of (e^{−tC}e^{−tD})_{t≥0}) satisfy MR as well?

Li, Wing Suet

Atlanta, USA

Inequalities of eigenvalues of sums of self-adjoint operators

Consider two self-adjoint operators A, B : H → H on a finite-dimensional Hilbert space. Let {λ_{j}(A)}, {λ_{j}(B)}, and {λ_{j}(A + B)} be sequences of eigenvalues of A, B, and A + B counting multiplicity, arranged in decreasing order. In 1912, H. Weyl raised the following question: what are all the inequalities that {λ_{j}(A)}, {λ_{j}(B)}, and {λ_{j}(A + B)} must satisfy? This problem was only solved completely by A. Klyachko in 1996. The proof uses the moduli space of stable bundles on the projective plane. In this talk I will give a very brief sketch of its history and connections to algebraic geomety and representation theory. Also I will talk about analogus of Weyl’s question for self-adjoint elements in von Neumann algebras with finite traces.

Litvinchuk, Georgiy

Madeira, Portugal

On a Hilbert value problem with shift

We consider the following problem. Find a function ϕ^{+}(z) analytic in |z| < 1 and satisfying the condition

Re{A(t)ϕ^{+}(t) + B(t)ϕ^{+}(α(t))} = h(t) on |t| = 1,  (1)

where A, B, h are continuous functions on |t| = 1, α(t) is a diffeomorphism of |t| = 1 onto itself preserving or changing orientation on |t| = 1. The Fredholm theory of problem (1) is constructed.

If α(t) is a linear-fractional diifeomorphism on |t| = 1 satisfying the Carleman condition α(α(t)) = t then the numbers of linear independent solutions and solvability conditions of problem (1) are calculated, and the solutions are constructed in the terms of factorization of (2 × 2)-matrix function determined by coefficients A, B. The particular cases are analysed when this factorization is realized effectively.

Lomonosov, Victor

Kent, USA

Positive functionals on general operator algebras

We show that any weakly closed algebra of bounded operators acting on a Banach space and different from the algebra of all bounded operators admits positive vector-functionals continuous in the essential operator norm.

Luger, Annemarie

Vienna, Austria

Factorization results for regular generalized Nevanlinna functions

Recall that a symmetric operator valued function Q:D ⊂ℂ→ L(H) which is meromorphic in the upper and the lower half plane belongs by definition to the generalized Nevanlinna class N_{κ}(H), if the so–called Nevanlinna kernel

N_{Q}(z, ζ) := \frac{Q(z) − Q(ζ)^{∗}}{z − \overline{ζ}}

has κ negative squares. The function Q is called regular if there exists at least one point w_{0} ∈ ℂ^{+} such that the operator Q(w_{0}) is boundedly invertible. We prove that for each regular function Q ∈ N_{κ}(H) there exists a rational factor B(z) that reduces the negative index of Q, that is the function

B(\overline{z})^{∗}Q(z)B(z)

belongs to the class N_{κ−1}(H).

Lyubarskii, Yurii

Trondheim, Norway

Approximation of subharmonic functions

We consider the following problem. Given a subharmonic function u(z) in ℂ find an entire function f(z) such that u(z) − log |f(z)| be the smallest possible in some natural sense. Some related problems are also considered.

McCarthy, John

St. Louis, USA

Qualitative properties of Pick interpolants

It is well known that an extremal Nevanlinna-Pick interpolation problem on the disk has a unique solution which is a Blaschke product. We shall discuss qualitative properties of solutions to other extremal interpolation problems. We shall consider both other Pick kernels, and also interpolation on the bidisk.

This is joint work with Jim Agler of U.C. San Diego.

Maestripieri, Alejandra

Buenos Aires, Argentina

Differential geometry of positive operators

Consider the algebra L(H) of bounded linear operators on a Hilbert space H, and L(H)^{+} the set of positive elements of L(H). For each A ∈ L(H)^{+} we study the differential geometry of the Thompson component of A,C_{A} = {B ∈ L(H)^{+} : A ≤ rB and B ≤ sA for some s, r > 0}. The set of components is parametrized by means of all operator ranges of H Each C_{A} is a differential manifold modelled in an appropriate Banach space and a homogeneous space with a natural connection. Morover, given arbitrary B, C ∈ C_{A} there exists a unique geodesic with endpoints B and C. Finally, we introduce a Finsler metric on C_{A} for which the geodesics are short and show that it coincides with the so-called Thompson metric.

This is joint work with G. Corach.

Malamud, Mark

Donetsk, Ukraine

Spectral analysis of some tensor products and applications

Let K be a unicellular Volterra operator on L_{p}[0, 1] and B be an arbitrary nonsingular n × n matrix, B ∈ ℂ^{n×n}. We consider the tensor product A := K ⊗ B in L_{p}^{n} = L_{p} ⊗ ℂ^{n} and present descriptions of the lattice LatA of invariant subspaces and the set CycA of cyclis vectors. We investigate the problem of similarity of two operators A_{i} = K ⊗ B_{i} (i = 1, 2) and prove (under additional assumptions) the "cancellation" result: the operators A_{i} are similar in L_{p}^{n} iff so are the matrices B_{i} (i = 1, 2)

Descriptions of the operator algebras {A}′, {A}′′ and AlgA, Alg_{u}A are prsented too. Here as usual {A}′ and {A}′′ are the commutant and the double commutant of A resepctively, and AlgA (Alg_{u}A) stands for the weakly (uniformly) closed subalgebra of L(L_{p}) generated by A and the identity I.

These results on operator algebras allow us to describe the lattice HyplatA of hyperinvariant subspaces as well as to construct counterexamples to some hypotheses which are valid for C_{0}-contractions (such as results on the splitting of Alg(T_{1} ⊕ T_{2}), Alg_{u}(T_{1} ⊕ T_{2}), the double commutant theorem and others).

We present some applications to inverse problems for first order systems B ⊗ dy/dx + Q(x)y = λy as well as to investigation of completeness property of eigenfunctions of boundary value problems for such systems on a finite interval.

Marcantognini, Stefania

Caracas, Venezuela

Weighted Nehari-Dym-Gohberg problem in the Wiener class

The talk concerns joint work with M.D. Morán (Universidad Central de Venezuela, Caracas, Venezuela) and A. Octavio (Instituto Venezolano de Investigaciones Científicas, Caracas, Venezuela).

We are given:

• {c_{n}}_{n=1}^{∞} ∈ l^{1}, and

• w ∈ L^{1}(𝕋), w ≥ 0,

such that there exists K < 1 such that

|∑_{n=0}^{∞} ∑_{m=−∞}^{−1}c_{n−m}α_{n}β_{m}|^{2} ≤ K∫_{0}^{2π}|∑_{n=0}^{∞} α_{n}e^{int}|^{2}w(e^{it})dt ∫_{0}^{2π}| ∑_{m=−∞}^{−1}β_{m}e^{imt}|^{2}w(e^{it})dt,

for all {α_{n}}_{n=0}^{∞}, {β_{m}}_{m=−∞}^{−1} ⊆ ℂ with finite support.

The problem we deal with can be split into two parts:

1. Find ϕ in the Wiener class W (𝕋) such that

* ∥w^{−1}ϕ∥_{∞} < 1, and

* ϕ̂(−n) = c_{n}, for all n ≥ 1,

where ϕ̂(p) denotes the p Fourier coefficient of ϕ, and, in case such a solution ϕ exists,

2. Describe all the ϕ’s.

Markus, Alexander

Beer Sheva, Israel

Quadratic numerical range of block operator matrix. Properties and applications

This is a joint work with Heinz Langer, Vladimir Matsaev and Christiane Tretter.

Let H_{1}, H_{2} be Hilbert spaces and H = H_{1} ⨁ H_{2}. We consider a linear bounded operator A in H given by the block operator matrix

A = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .

The quadratic numerical range W_{A}^{2} of the operator A is defined as

W_{A}^{2} = \{λ ∈ ℂ : det ( \begin{matrix}(Af, f) − λ(f, f) & (Bg, f) \\ (Cf, g) & (Dg, g) − λ(g, g) \end{matrix} ) = 0 for some f, g ≠ 0\} .

This definition was introduced in [1] where it was proved, in particular, that the spectrum σ(A) is contained in \overline{W_{A}^{2}}.

We study various properties of the quadratic numerical range and their connection with spectral characteristics of the operator A. Some applications to the factorization of Schur complements corresponding to the block operator matrix A are obtained.

References 1. H. Langer, C. Tretter, Spectral decomposition of some nonselfadjoint block operator

matrices, J. Operator Theory 39 (1998), 339–359.

Matsaev, Vladimir

Tel Aviv, Israel

Division and estimates of resolvents

Megretskii, Alexander

Massachusetts, USA

Relaxations of quadratic programs in Operator Theory and System Analysis

The talk will cover an area of interesting interplay between the operator theory, system analysis, and combinatorics.

"Quadratic programs" are the problems of maximizing a quadratic functional on a Hilbert space (usually real and finite dimensional) subject to quadratic constraints. While almost any combinatorial problem can be reduced to such optimization, it is conjectured that there exist no efficient way of solving general non-convex quadratic programs exactly.

A technique of "relaxation", which essentially means replacing vectors by random variables, is known experimentally to yield high quality approximate solutions in many non-convex quadratic programs arising in engineering applications. However, proving good error bounds for a particular relaxation scheme is usually a challenging mathematical problem.

In this talk, relaxation techniques of dynamical system analysis will be described. It will be shown how operator theoretic methods can be used to give error bounds for these techniques or to provide counterexamples. On the other hand, it will be demonstrated that some basic problems of operator theory have equivalent formulations in terms of relaxation bounds in quadratic programming, and can be approached using the insights from combinatorics and system theory.

Meyer, Yves

Paris, France

Real analysis, complex analysis, partial differential equations and operator theory

I plan to speak about the mysterious cross-fertilization between real analysis, complex analysis, PDE’s and operator theory.

This theme will be developped on the specific example of a famous problem raised by Tosio Kato. It concerns the domain of the square root of some accretive operators L. This conjecture was soon recognized to be too ambitious in its theoretical setting but is hoped to be true whenever L is a second order differential operators written in a divergence form.

Alan McIntosh failed in his many assaults against Kato’s conjecture but these failures helped solving some famous problems in complex analysis raised by Alberto Calderón.

A decade later and at our greatest surprise a better proof of Calderón’s conjecture was discovered.This new proof was using some simply minded tricks in complex analysis. Then Guy David went one step forward and proved Vitushkin conjecture on analytic capacity.

Returning to Kato’s conjecture, Pascal Auscher and Philippe Tchamitchian developped a long term program which culminated in the proof by Steve Hofmann and John Lewis of the 2-dimensional case of Kato’s conjecture. The main ingredient of this beautiful proof is an extremely clever use of Carleson’s measures.

Carleson measures were created for solving the “corona" problem. The “corona" problem concerns the topology of the spectrum of the Banach algebra H^{∞} of holomorphic and bounded functions in the unit disk...

I will then try to conclude and comment on the role played by a broad mathematical culture.

Minkin, Arkadi

Haifa, Israel

Unconditional basicity of dissipative differential operators

Consider a dissipative differential operator L in L^{2}(0, 1) generated by a two-point boundary value problem (D = −i d/dx)

Ly ≡ D^{n}y + ∑_{k=0}^{n−2} p_{k}(x)D^{k}y = λy, p_{k} ∈ L(0, 1)  (1)

with n linearly independent boundary conditions

∑_{k=0}^{n−1}[α_{jk}D^{k}y(0) + β_{jk}D^{k}y(1)] = 0, j = 0, …, n − 1.  (2)

S. Treil has found a strong criterion of unconditional basicity of eigenfunctions of a large class of dissipative operators. However this criterion requires uniform minimality of the eigenfunctions which seems to be unattainable to check directly for the boundary value problem (1)-(2).

Using another approach based on almost orthogonality of a fundamental system of solutions of (1) we proved

Theorem 1 If n is even then eigenfunctions of L constitute an unconditional basis with brackets in L^{2}(0, 1).

Remark This is an even case part of S.G.Krein’s conjecture stated at the mid 80s.

Mityagin, Boris

Columbus, USA

Smoothness of solutions of nonlinear ODEs

Smoothness of infinitely differentiable functions is measured by Carleman sequence M = M_{n}. We analyze smootheness of solutions of a nonlinear differential equation du/dt = F (u, t) where F (z, t) is an entire function of order 1, finite type, of z and of a nonanalytic Carleman class on t [for example, of Gevrey class]. Any solution u has to be in the same Carleman class; this and other results on smoothness of solutions of ODEs will discussed. Technical difficulties are absorbed by special recurrent quadratic inequalities. [This is a joint work with Plamen Djakov, University of Sofia, Bulgaria.]

Muminov, Mukhiddin

Samarkand, Uzbekiston

On the essential spectrum of four body Schrödinger’s operator for the system of two bosons with two fermions

It is considered the discrete Schrödinger operator of a system of four particles, two of which are bosons and two fermions on a lattice.

The essential spectrum of Schrödinger’s operator at fixing total quasi-momentum is expressed in terms of the spectrum of the canal operators and this operator has no eigenvalue lying to the right of the essential spectrum.

The Faddeev-Yakubovsky type integral equation for the resolvent of Schrödingers operator on the Euclid space is obtained.

Muravnik, Andrey

Moscow, Russia

Fourier-Bessel transformation of measures and singular pseudo-differential operators

An equation Au = f is considered where A is the following singular pseudo-differential operator:

Au(x, y) = \frac{1}{y^{ν}} ∫_{0}^{∞}\begin{matrix}∫ \\ ℝ & ^{n} \end{matrix} η^{ν+1}p(x, y, ξ, η)û(ξ, η)e^{ix·ξ}J_{ν}(yη)dξdη,

ˆ denotes Fourier-Bessel transformation (see [1]), parameter ν belongs to (−1/2, +∞).

Operators of this kind appear in models of mathematical physics with degenerative spatial heterogeneities.

Besides the standard requirements to the symbol (see [2]) we also suppose that it is elliptic

and factorable: p(x, y, ξ, η) = a(x, y)b(ξ, η). Then under the assumption 0 ≤ \frac{η^{2ν+1}}{b} ˆ(\frac{f}{a} ) ∈

L_{1}(ℝ^{n} × ℝ_{+}^{1}) we estimate weighted L_{∞}-norm of weighted hemispherical mean of the solution (in the scalar case of n = 0 - of the solution itself) via its weighted L_{p}-norm. We also prove that the solution achieves its supremum at the origin.

The proof is based on the properties of Fourier-Bessel transformation of measures proved in [3].

References

1 I. A. Kiprijanov, Fourier-Bessel transforms and imbedding theorems for weight classes, Proc. Steklov Inst. Math., 89(1967), pp. 149–246.

2 I. A. Kiprijanov and L. N. Lyakhov, A certain class of pseudodifferential operators, Sov. Math. Dokl., 15(1974), no. 5, pp. 1319–1322.

3 A.B. Muravnik, On weighted norm estimates for mixed Fourier-Bessel transforms of non-negative functions, Integral methods in science and engineering 1996: analytic methods, Pitman Res. Notes Math. Ser. 374, Addison Wesley Longman, Harlow, 1997, pp. 119–123.

Naboko, Sergey

St. Petersburg, Russia

Asymptotics of generalized eigenvectors and the spectral analysis of unbounded Jacobi matrices

The spectral analysis of unbounded selfadjoint Jacobi (tridiagonal) matrices J with powerlike entries is considered.It is based on the detailed investigation of the asymptotics at infinify of generalized eigenvectors of J, which do not belong to the Hilbert space in general. This allows to apply the Pearson-Gilbert-Khan subordinacy theory for the investigation of the spectral components of J. We present various methods for studying the asymptotics: grouping in block approach, Stolz smooth method, discrete version of the semiclassical method, etc. Applications to the spectral phase transition for special models of inhomogenious media are given. Here we obtain a jump of the spectral structure from the pure absolutely continuous spectrum case to discrete spectrum with changing some parameters in the matrix J. In particular spectral phase transition has been observed with depending on an interplay between diagonal and nondiagonal entries at infinity. The talk is based on the common works with J. Janas (Krakow).

Nakazi, Takahiko

Hokkaido, Japan

Rudin’s orthogonality problem and isometric operators between two weighted Hardy spaces

In this lecture, we study an isometric operator between two weighted Hardy spaces H^{p}(ν) and H^{p}(μ) on the closed unit disc D̄ when 0 < p < ∞ and p ≠ 2. Then the operator draws an into-isometric composition operator between H^{2}(ν) and H^{2}(μ). When H^{2}(μ) is the classical Hardy space H^{2} and ν is a radial measure on D̄, the composition operator relates to Rudin’s orthogonality problem. That is, if ϕ is a function in H^{∞} with ∥ϕ∥_{∞} = 1 and the different nonnegative powers of ϕ are orthogonal in H^{2}, is ϕ an inner function ?

References

1. P. S. Bourdon, Rudin’s orthogonality problem and the Nevanlinna counting function, Proc. Amer. Math. Soc., 125 (1997), 1187–1192.

2. J. A. Cima, B. Korenblum and M. Stessin, Composition isometries and Rudin’s problem, preprint.

Nevanlinna, Olavi

Helsinki, Finland

Resolvent as a meromorphic function

In the talk I shall outline an approach to treat operator valued meromorphic functions. In the value distribution theory of (scalar) meromophic functions the Nevanlinna characteristic function T := m + N has a central role. Here

m(r) = \frac{1}{2π} ∫ log^{+} |f(re^{iϕ})]dϕ

and

N(r) = ∑ log^{+} \frac{r}{b}

where b denotes the poles (with multiplicities). Replacing the absolute values by operator norms a similar function, I call it T_{∞}, can be defined. This, however, does not satisfy any analogue of the first main theorem, namely that (if normalized f(0) = 1, then)

T (r, f) = T (r, \frac{1}{f}).

We give another generalization, for functions of the form I − F where F is trace class valued function with F (0) = 0. We call this T_{1}, and it utilizes all the singular values σ_{j} of I − F. This generalization of T satisfies the exact inversion identity

T_{1}(r, I − F) = T_{1}(r, (I − F)^{−1})

and combined with T_{∞} it can be used to give perturbation results for resolvents e.g. of the following form: even so the "outlook" of a resolvent may change dramatically under small rank perturbations of the operator, the "size" as meromorphic function is essentially unchanged.

References

1. O. Nevanlinna, Growth of operator valued meromorphic functions, Ann. Acad. Sci. Fenn. Math. 25 (2000), 3–30.

Nikolskaia, Ludmila

Bordeaux, France

Estimates of inverses and finite section inverses of Toeplitz operators on the Wiener algebra

Let W be the Wiener algebra of absolutely convergent Fourier series on the unit circle 𝕋, and let T_{φ} = P_{+}φ : W_{+} ⟶ W_{+} (φ ∈ W) be an invertible Toeplitz operator, that is, δ(T_{φ}) = inf_{𝕋}|φ| = inf{|λ| : λ ∈ σ(T_{φ})} > 0 and Ind(φ) = 0. The problems of estimates of inverses ∥T_{φ}^{−1}∥ and of finite section inverses ∥(P_{n}T_{φ}|P_{n})^{−1}∥, in terms of the lower spectral bound δ(T_{φ}), are considered. It is shown that for δ(T_{φ}) ≥ δ∥T_{φ}∥ and δ > 1/\sqrt{2} there exists a constant c(δ) such that ∥T_{φ}^{−1}∥, ∥(P_{n}T_{φ}|P_{n})^{−1}∥ ≤ ∥T_{φ}∥^{−1}c(δ) for every n ≥ 0, and that for 0 < δ ≤ 1/2 such an estimate is impossible: sup{∥T_{φ}^{−1}∥ : δ(T_{φ}) ≥ δ, Ind(φ) = 0, ∥T_{φ}∥ ≤ 1\} = ∞ and sup\{\overline{lim}_{n}∥(P_{n}T_{φ}|P_{n})^{−1}∥ : δ(T_{φ}) ≥ δ, Ind(φ) = 0, ∥T_{φ}∥ ≤ 1} = ∞. For analytic symbols φ ∈ W_{+}, the similar constant c_{+}(δ) is finite iff 1/2 < δ ≤ 1.

Similar results are obtained for Fredholm regularizers of operators T from the Toeplitz algebra Alg(T_{W}).

Nudelman, Adolf

Odessa, Ukraine

Interpolation problems of Nevanlinna-Pick type with spectral restrictions

Olshevsky, Vadim

Atlanta, USA

Fast and numerically accurate algorithms for structured matrices: a survey

Many important problems in pure and applied mathematics and engineering can be reduced to matrix computations. Unfortunately, practical circumstances often impose limitations on the use of available standard matrix methods. For example, in many applications the size of the associated matrices is prohibitively large, so the available methods often require an extremely large amount of time. Secondly, the standard methods may behave poorly in a floating-point arithmetic leading to what is called catastrophic round-off error accumulation.

This is one reason why one seeks in various applications to identify special/characteristic structures that can be exploited in order to speed-up computations. Such additional assumptions are often provided by particular physical properties leading to various structured matrices, such as Toeplitz, Hankel, Vandermonde, Cauchy, Pick matrices, Bezoutians and others.

In this talk we survey several recent results where exploiting the structure was a unified method that allowed us to design efficient algorithms for several seemingly unrelated applications. The list of algorithms includes: a superfast algorithm for tangential Nevanlinna-Pick problem with confluencies, a first provably backward stable (numerically accurate) algorithm for factorization of positive definite Hankel-like matrices, and a new algorithm for list decoding of algebraic-geometric codes.

Paneah, Boris

Haifa, Israel

Support dependent estimates for Fourier transforms related to Uncertainty principles

The topic of this talk is some relations of new type between functions f ∈ L_{p,Ω} and their Fourier transforms F. Here L_{p,Ω}, 1 ≤ p ≤ ∞, is the space of functions f such that the L_{p}- norm ∥f∥_{p} is finite and F is supported in a convex bounded domain Ω ⊂ ℝ^{n}. The relations we deal with have a form

∥wf∥_{p} ≥ K(Ω)∥f∥_{p}, f ∈ L_{p,Ω},

where w is an unbounded function, and K(Ω) is a constant independent of f. We mainly concentrate on a connection between the behavior at infinity of the weight w and the growth of the corresponding constant K(Ω) as the domain Ω shrinks into a lower dimensional convex set Ω_{∞}. In the one-dimensional case one of the typical results is the following estimate

∥W f∥_{p} ≥ cw(σ^{−1})∥f∥_{p}, f ∈ L_{p,(−σ,σ)},

where W = wφ and φ is an arbitrary almost periodic function while w(t) is an increasing function for |t| > A. This estimate is sharp when σ → 0. (If W = t^{k}, this is nothing but a Hardy inequality; if W = t^{k} sin^{l} t this is a new result).

In the multidimensional case for a weight w = Q ∘ |P | with P as an arbitrary complex valued polynomial in ℝ^{n} and Q as an arbitrary nondecreasing function in ℝ^{1} the constant K(Ω) is explicitely expressed in terms of the support function of the domain Ω, the polynomial P and the function Q. When p = ∞ the result obtained is closely connected with a dual form of the famous S.Bernstein inequality for entire functions, which has not been investigated yet. Some new results are related to bounded weight functions w, too. For example, our method allows to prove the following estimate

k^{−1} sup |w_{1}(t)f(t)| ≤ sup |w_{2}(t)f(t)| ≤ k sup |w_{1}(t)f(t)|,

where w_{1} and w_{2} are arbitrary nontrivial almost periodic functions and f ∈ L_{∞,(−σ,σ)}.

Partington, Jonathan

Leeds, UK

Approximation and interpolation in Hilbert spaces of analytic functions

Several problems involving constrained approximation and interpolation in Hilbert spaces of analytic functions, including Hardy spaces and Paley–Wiener spaces, can be put into a general framework and solved using the theory of certain Toeplitz-like operators naturally expressible as the product of two projections.

We shall touch lightly on some of the applications [4], which include Systems Identification [2], Signal Processing [5], and Inverse Problems for PDEs [1].

This talk is based on some joint work with J. Leblond [3], together with recent extensions being developed in the thesis of M.P. Smith.

References

1. L. Baratchart, J. Leblond, F. Mandrea and E.B. Saff, How can meromorphic approximation help to solve some 2D inverse problems for the Laplacian?, Inverse Problems, 15:79–90, 1999.

2. M.G. Krein and P.Y. Nudel’man, Approximation of L^{2}(ω_{1}, ω_{2}) functions by minimumenergy transfer functions of linear systems, Problemy Peredachi Informatsii, 11(2):37– 60, 1975. English translation.

3. J. Leblond and J.R. Partington, Constrained approximation and interpolation in Hilbert function spaces, J. Math. Anal. Appl., 234:500–513, 1999.

4. J.R. Partington, Interpolation, identification and sampling, Oxford University Press, 1997.

5. D. Slepian and H.O Pollack. Prolate spheroidal wave functions, Fourier analysis and uncertainty I, Bell Systems Tech. J., 40:43–64, 1961.

Pavlov, Boris

Auckland, New Zealand

The Poincaré–Hardy inequality on the complement of a Cantor set

This is joint work with Cristian S. Calude.

In Classical Analysis the Poincaré–Hardy inequality (see, for example, [6] or [7] is one of most popular tools for comparing the generalized smoothness of a given function and its square integrability with a singular weight–function, see, for instance [8], [2], [10]. A version of Poincaré–Hardy inequality on the complement of a uniformly δ-regular set was derived by Ancona in [1] in connection with the question on the uniqueness of the solution of the Dirichlet problem for second order elliptic equations in a domain with a uniformly δ–regular boundary, see also [11], [12], [14], [9]

In the present note we derive the simplest version of the Poincaré–Hardy inequality on the complement of a Cantor set in R_{3} paying a special attantion to the structure of the corresponding constant. We reduce the proof of Poincaré–Hardy inequality to the estimation of a discretized integral of an analog of the strong barrier function, see Theorem 1 below. This estimation is based on analysis of trajectories of some finite automaton generated by Mauldin–Williams graph of the Cantor set together with a proper measure constructed on all cylinder sets of trajectories, see [4], represented by all paths on a binary Bruhat–Tits tree, [3]. The above measure leads to a metric space homeomorphic with the space of tiles endowed with the Euclidean distance. This is the simplest case of a general connection between analysis of smooth functions on the complement of a uniformly δ-regular set (or just a zero–measure perfect set), on one hand, and Symbolic Dynamics (see, for instance, Lind and Marcus [13]), on the other hand. The characteristic features of the corresponding general construction are

• a special self-similar tiling ω_{a} of a neighborhood of the set, parameterized by all possible strings - trajectories generated by some Mauldin–Williams graph which defines the authomorphisms of the set.

• a homeomorphism between an Euclidean metric structure on the tiling and the metric space of trajectories.

The following general statement serves as a base for our calculations.

Theorem 1 For every function u ∈ W_{2}^{1}(R_{3}) and every bounded fractal E ∈ R_{3} with dim_{E} < 1 fulfilling the condition

K_{E} = sup_{x∈E} \frac{d_{x}}{4π} ∫ \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(x)} < ∞,  (1)

the Poincaré–Hardy inequality holds with the constant K_{E}^{2}:

∫ \frac{u^{2}}{d_{E}^{2}(x)} dm ≤ K_{E}^{2} · ∫ | ▽ u|^{2} dm.  (2)

The inequality (2) can be used to derive several useful facts. In what follows we will present two such applications.

A. Consider a real measurable function q, locally bounded on each compact in the complement E′ of the fractal E. Then the Dirichlet form ∫_{R_{3}} (|∇u|^{2} + q(x)|u|^{2}) dm is closed in W_{2}^{1}(R_{3}) if q satisfies the following additional condition:

lim_{d_{E}(x)→0} |q(x)| d_{E}^{2}(x) = 0.  (3)

B. The constant K plays the role of the dimensionless Plank constant in the uncertainty principle. To see this, let us consider the self-adjoint operator (unbounded in L_{2}(R_{3})) of multiplication by the function ε(x)d_{E}(x), where the factor ε(x) = ±1 is chosen such that for every given smooth function u with a compact support in E′ the mean value of the “balanced distance" with respect to some unitary-valued sign-factor ε(x), ε(x)d_{E}(x) to the singular set E is equal to zero:

∫_{R_{3}} ε(x)d_{E}(x)|u|^{2}dm = 0.

We assume that the mean value of momentum is also zero:

∫_{R_{3}} ∇u ū dm = 0.

Under the above hypotheses we may estimate from below the product of the mean quadratic errors of the balanced distance and the mean quadratic error of the momentum, the Dirichlet integral,

\frac{1}{K} · ∫_{R_{3}} |u|^{2} dm ≤ [∫_{R_{3}} |d_{E}(x)|^{2} |u|^{2} dm]^{1/2} · [∫_{R_{3}} |∇u|^{2} dm]^{1/2} ,  (4)

The constant K^{−1} (≈ 0.021819 for Cantor set ) plays the role of estimate for analog of the classical “dimensionless Planck constant" 1/2. It defines the admissible precision of simultaneous measurements of deviation of the the coordinate of the quantum particle from the singular set and the deviation of its total momentum from zero.

Lemma 1 For middle-third Cantor set E situated in R_{3} on z there exist a self- similar tiling {ω_{a}}_{a} of a neighbourhood of the set E enumerated by all possible binary strings a length |a| ≥ 0. In this case the integral coefficient in general Hardy inequality

K = \frac{d_{E} (x)}{4π} ∫ \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)}

can be discretized as a sum over the tiling

∑_{a} \frac{d_{E} (x)}{4π} ∫_{ω_{a}} \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)}

and estimated for x ∈ ω_{a} as follows:

\frac{d_{E}(x)}{4π} ∫_{R_{3}} \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)} = \frac{d_{E}(x)}{4π} ∫_{R_{3}\∪_{a}ω_{a}} \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)}

+\frac{d_{E}(x)}{π} ∫_{ω_{b}, |b|≥1,∂ω_{b}∩∂ω_{a}≠∅} \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)}+

∑_{b} \frac{d_{E}(x)}{π} ∫_{ω_{b}, |b|≥1,∂ω_{b}∩∂ω_{a}=∅} \frac{1}{|x − s|^{2}} \frac{dm(s)}{d_{E}^{2}(s)}

≤ 300 + 900 · ∑_{ω_{b}, |b|≥1,∂ω_{b}∩∂ω_{a}≠∅} 3^{2·|a∩b|−|a|−|b|}.

The following inequality holds true :

∑_{ω_{b}, |b|≥1,∂ω_{b}∩∂ω_{a}≠∅} 3^{2·|a∩b|−|a|−|b|} ≤ 2.  (5)

Lemma 2 The Poincaré–Hardy inequality

∫ \frac{|u|^{2}}{dist^{2}(x, E)} dm ≤ 2, 100 · ∫ | ▽ u |^{2}dm

holds true in R_{3} on the complement of the Cantor set E.

Our estimate for the constant in Hardy inequality is obviously not the best, but we guess, that the structure of this constant clearly reveals the contribution of the corresponding dynamical system to it in form of the inequality (5).

References

1. A. Ancona. On strong barriers and an inequality of Hardy for domains in R_{n}, J. London Math. Soc. (2) 34 (1986), 274-290.

2. M. S. Birman. On spectrum of boundary value problems, Mat. Sb. (N.S.) 55, (1961), 125-174. (in Russian)

3. F. Bruhat and J.Tits. Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251. (French)

4. C. Calude. Information and Randomness–An Algorithmic Perspective, Springer-Verlag, Berlin, 1994.

5. Gerald A. Edgar. Measure, Topology, and Fractal Geometry, Springer-Verlag, New York (1990).

6. W. N. Everitt (ed.). Inequalities: Fifty Years on from Hardy, Littlewood and Polya, Proceedings of the International Conference held at the University of Birmingham, Birmingham, July 13–17, 1987, Lecture Notes in Pure and Applied Mathematics, 129, Marcel Dekker, Inc., New York, (1991).

7. G. H. Hardy, J. E. Littlewood, J. E. Polya. Inequalities, Cambridge Univ. Press, Cambridge, (1934).

8. O. A. Ladyzhenskaja. Problems in the Dynamics of Viscous Incompressible Flow, Gordon & Breach, New-York, (1963).

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Pedersen, Thomas

Bordeaux, France

Translation invariant subspaces of weighted Hilbert spaces on the real line

We discuss the extent to which certain recent results by Esterle and Volberg on biinvariant subspaces of weighted Hilbert spaces on the integers are valid for similar spaces on the real line. Consider the weighted space

L_{ω}^{2}(ℝ) = {f ∈ L^{2}(ℝ) : fω ∈ L^{2}(ℝ)},

where ω : ℝ → [1, ∞) is a continuous, decreasing function with ω(t) = 1 for t ≥ 0, ω(−t) → ∞ as t → ∞, and for which the translation operator T (s) (s ∈ ℝ) defined by (T (s)f)(t) = f(t − s) (t ∈ ℝ, f ∈ L_{ω}^{2}(ℝ)) is bounded on L_{ω}^{2}(ℝ) with σ(T (s)) = 𝕋 (s ∈ ℝ \ {0}). We say that a closed subspace M of L_{ω}^{2}(ℝ) is translation invariant if T (s)M ⊆ M for every s ∈ ℝ, and that M is analytic if M^{+} = M ∩ L^{2}(ℝ^{+}) ≠ \{0\}. When ω satisfies log ω(−t)/\sqrt{t} → ∞ as t → ∞, we show that every singular, inner function U on the right-hand halfplane ℙ^{+} generates a translation invariant subspace { \atop E(U)} of L_{ω}^{2}(ℝ) with

{ \atop E(U)}^{+} = \{f ∈ L^{2}(ℝ^{+}) : L(f) ∈ UH^{2}(ℙ^{+})\},

where L is the Laplace transform. In particular, { \atop E(U)} is non-trivial and analytic. Moreover, every analytic, translation invariant subspace M of L_{ω}^{2}(ℝ) satisfying an extra condition is of the form { \atop E(U)} for some singular, inner function U on ℙ^{+}. Finally, without the assumption that log ω(−t)/\sqrt{t} → ∞ as t → ∞, we prove that there exists a singular, inner function U on

ℙ^{+} for which { \atop E(U)}^{+} = \{f ∈ L^{2}(ℝ^{+}) : L(f) ∈ UH^{2}(ℙ^{+})\}.

Peller, Vladimir

Kansas, USA

Balanced unitary-valued functions and new factorization theorems

A new approach will be given to construct a superoptilmal approximation of a matrix function by analytic matrix functions. A special role will be played with a class of so-called balanced unitary-valued functions which is a generalization of the clqss of thematic matrix functions. They possess many remarkable properties. Related so-called canonical factorizations will be obtained. It turns out that in contrast with thematic factorizations, canonical factorizations possess important invariance properties. All their factors are uniquely determined by the initial matrix function modulo constant unitary factors. The results are obtained jointly with R. Alexeev.

Peterburgsky, Irina

Boston, USA

Description of solutions to a wide range of extremal problems over classes of abstract - valued analytic functions.

A number of classical extremal problems (including E. Landau coefficient problem) has been generalized for a case of analytic functions taking on values in complex Hilbert or Banach space. In particular, a question about existence and form of solutions has been studied.

Petkov, Vesselin

Bordeaux, France

Breit-Wigner approximations for the scattering phase

It is well known that for operators with discrete spectrum, {λ_{j}^{2}}, the counting function N(λ) of eigenvalues satisfies the equality

N(λ + δ) − N(λ − δ) = ∑_{j} δ_{λ_{j}^{2}}((λ − δ, λ + δ]).

In the scattering theory the natural analogue of the discrete spectrum is given by the resonances, λ_{j} ∈ ℂ_{+}, and the counting function N(λ) is replaced by the scattering phase s(λ) = i log det S(λ), S(λ) being the scattering operator. The relation between the resonances and s(λ) is not simple and the Breit-Wigner approximation and scattering experiments predict that the existence of resonances μ_{j} exponentially close to the real axis produce "blips" and, consequently, the derivative s′(λ) has a exponential blow-up for λ close to ℜμ_{j}. In this direction, the analysis of the variation s(λ + δ) − s(λ_{δ}) is of particular interest. In a joint work with M. Zworski we prove for a very large class of trapping perturbations that

s(λ + δ) − s(λ − δ) = ∑_{|λ_{j}−λ|<ϵ} ω_{ℂ_{+}}(λ_{j}, [λ − δ, λ + δ]) + O(δ)λ^{n−1},

where ω_{ℂ_{+}} is the harmonic measure of the upper half plane given by

ω_{ℂ_{+}}(z, E) = \frac{1}{π} ∫_{E} \frac{ℑz}{|λ − z|^{2}} dλ, E ⊂ ℝ = ∂ℂ_{+}.

This result presents a precise version of the Breit-Wigner approximation and, moreover, we treat the scattering by compact perturbations without any assumption on the geometry of trapped trajectories.

Pisier, Gilles

Paris, France

On the space H^{1} with values in the trace class, and related non-commutative analysis

We discuss the analogue of the classical Burkholder-Gundy inequalites for martingales in non-commutative L_{p}-spaces (1 < p < ∞), as recently proved in a joint paper with Q. Xu. We describe an alternate proof of one of these inequalities. This new approach is restricted to p an even integer, but it yields a constant which is O(p) when p → ∞ and it applies to a much more general kind of sums (than just sums of martingale differences) which we call p-orthogonal.

We also discuss the non-commutative martingale versions of H^{1} and BMO. In particular, we will concentrate on the case of either martingales or functions with values in the trace class operators S_{1} (or S_{1}^{d} on d-dimensional Hilbert space). Several different non equivalent versions of “S_{1}^{d}-valued H^{1}" will be compared, and very precise bounds will be given for the constants in the corresponding inequalities as a function of the dimension d. This is closely related to a recent joint work with F. Nazarov, S. Treil and A. Volberg, and to a note by S. Petermichl.

Poltoratski, Alexei

College Station, USA

Maximal properties of the normalized Cauchy transform

The classical theorem by M. Riesz states that the Cauchy (Hilbert) transform is bounded as an operator in L^{p} with respect to the Lebesgue measure for 1 < p < ∞. The result of R. Hunt, B. Muckenhoupt and R. Wheeden gives the condition for such boundedness in the case of weights absolutely continuous with respect to the Lebesgue measure. The associated maximal operator is also shown to have similar properties. In this talk our goal is to find an analog of this theory for the case of arbitrary (possibly singular) weights. To make the corresponding problem correct one has to consider the so-called normalized Cauchy transform.

Let μ be a positive finite measure on the unit circle of the complex plane and f ∈ L^{1}(μ). Denote by Kμ and Kfμ the Cauchy integrals of the measures μ and fμ correspondingly. The normalized Cauchy transform is defined as C_{μ} : f ↦ \frac{Kfμ}{Kμ}.

We prove that C_{μ} is bounded as an operator in L^{p}(μ) for 1 < p ≤ 2 but is unbounded (in general) for p > 2. The associated maximal non-tangential operator is bounded for 1 < p < 2 and has weak type (2, 2) but is unbounded for p > 2.

Popovici, Dan

Timisoara, Romania

On the structure of regular dilations

Introduced by S. Brehmer in the early sixties, regular dilations for multi-contractions seem to be the most amenable among all kinds of dilations. Douglas and Foiaş have recently obtained a remarkable model for any completely non-unitary bi-isometry in terms of a unitary operator and an orthogonal projection. It is our aim in this paper to obtain structure results relative to this model for the minimal regular (respectively ∗-regular) isometric dilation [K, V_{T} = (V_{1}, V_{2})] (respectively [K_{∗}, V_{T^{∗}}]) of a given bi-contraction [H, T = (T_{1}, T_{2})] (respectively T^{∗}). More precisely we characterize the membership of V_{T} and V_{T^{∗}} to some special classes of bi-isometries in terms of T. In this context we can mention that V_{i} is unitary iff T_{i}, S_{T}^{(i)} are co-isometries and V_{i} is pure iff T_{i}, S_{T}^{(i)} ∈ C_{·0}, i = 1, 2 (we have considered contractions S_{T}^{(i)} ∈ L(D_{T_{3−i}}), S_{T}^{(i)}D_{T_{3−i}}h := D_{T_{3−i}}T_{i}h, h ∈ H, i = 1, 2). Moreover V_{T^{∗}} is bi-shift and V_{T} is unitary iff T_{i} are backward shifts and S_{T}^{(i)} co-isometries, i = 1, 2, V_{T^{∗}} is bi-shift, V_{1} is unitary and V_{2} shift iff T_{1} is backward shift, T_{2} ∈ C_{00}, S_{T}^{(1)} is co-isometric and S_{T}^{(2)} ∈ C_{·0}. Similar results are obtained for other remarkable parts of V_{T} and V_{T^{∗}}.

Pott, Sandra

York, UK

Hilbert transform and martingale transforms on operator-weighted L^{2} spaces

Let H be a finite or infinite dimensional Hilbert space, let W : 𝕋 → B(H) be an operator weight, and let L_{W}^{2} be the (operator)-weighted L^{2} space

L_{W}^{2} = {f : 𝕋 → H : ∫_{𝕋}⟨W (t)f(t), f(t)⟩ dt < ∞}

We discuss connections between boundedness results for the Hilbert transform and boundedness for Martingale transforms on L_{W}^{2} , in particular concerning dimensional growth.

Putinar, Mihai

Santa Barbara, USA

The Friedrichs operator of a planar domain (joint work with Harold S. Shapiro)

Let U be a planar domain. The gap operator between the Bergman space of U and its complex conjugate was implicitly introduced by K. Friedrichs in 1938 in connection with two classical problems of planar elasticity.

A series of spectral properties of this operator will be related to the geometry of the domain. In particular, the inverse spectral problem will be shown to fail for simple counting parameters reasons.

The connection with generalized quadrature formulas and some harmonic analysis on translation invariant sets will provide a few interesting examples. The same Friedrichs operator will be naturally linked to boundary value problems for the Laplace and the square of the Cauchy-Riemann equations.

The compactness of the Friedrichs operator with bounded symbol will put in parallel to the known similar results for Hankel operators.

Rodman, Leiba

Williamsburg, USA

Jordan forms of nilpotent matrices having prescribed generic triangular part (with Michael I. Gekhtman)

Sets of possible Jordan forms of nilpotent matrices with a given upper triangular part, including the diagonal, are studied. It is proved that, for generic matrices within a set of triangular band matrices, the set of Jordan forms contains a unique Jordan form which is minimal in the sense of majorization. Moreover, in the generic case, there exists a nilpotent matrix in the orbit of a given upper triangular matrix having the minimal Jordan form, and also having the minimal rank among all matrices with the given upper triangular part. The orbit is understood in the sense of the triangular group action. Normal, or canonical, forms of the triangular group action (again, in the generic case) serve as the main technical tool in proving these results.

Sadosky, Cora

Washington, USA

Almost orthogonality and block diagonal operators in perturbation theory

The classical perturbation theorem of v. Neumann allows the decomposition of each bounded self-adjoint operator in a Hibert space into a sum of two operators, one diagonal, and the other Hilbert-Schmidt with arbitrarily small norm. This theorem provides, in particular, norm estimates for the operators, and led to the method of almost orthogonal expansions. Here we enlarge significantly the range of applications of almost orthogonal expansions, by introducing the notion of block diagonal operators, and we obtain approximations by block diagonals through an explicit simple algorithm, which allows a functional calculus.

Analogs in this functional calculus of the trace formulas of Krein-Koplienko are computable explicitly in some particular cases. While the theory of the spectral shift is not valid in general for Hilbert-Schmidt perturbations, the trace formulas can be extended in very special cases; the explicit computation of the trace formulas for block diagonal operators may lead to new developments in scattering theory.

This is joint work with Mischa Cotlar.

Saff, Edward

Tampa, USA

Best meromorphic approximation of Markov functions

We investigate the convergence rate and asymptotic distribution of poles of best meromorphic approximants in L_{p} on the unit circle to Markov functions (Cauchy transforms) generated by measures that are supported on a compact subset E of (−1, 1). The meromorphic approximants are of the form P/Q, with P in the Hardy class H_{p} of the unit disk and Q a polynomial of degree n. Connections with n-widths and Hankel operators will be discussed.

van Schagen, Frederik

Amstredam, The Netherlands

Finite section method for linear ordinary differential equations

In this paper the finite section method, well-known for integral equations, is developed for differential equations. The aim is to obtain solutions of a linear ordinary differential equation on the full line as the limit of solutions of corresponding equations on finite intervals or half lines. Both the time-variant and the time-invariant case are treated.

To describe in more detail the problems that we have in mind, consider the equations

ẋ(t) − A(t)x(t) = f(t),  −∞ < t < ∞,  (1)

\{ \begin{matrix}ẋ(t) − A(t)x(t) = f(t), & σ ≤ t ≤ τ, \\ M(σ)x(σ) + N(τ)x(τ) = 0, \end{matrix}  (2)

where σ and τ are finite, the n × n state matrix A(t) is locally integrable on (−∞, ∞), and M(σ) and N(τ) are n × n matrices such that rank M(σ) + rank N(τ) = n. The problem is to derive, if possible, the solution of (1) as the limit for −σ, τ → ∞ of the solution of (2).

The first main result is that under certain natural additional conditions for each f in L_{p}^{n}(−∞, ∞) the solution x of (1) is obtained as the limit in L_{p} of the solution x_{σ,τ} of equation (2) provided that

sup_{τ≥τ_{∘},σ≤σ_{∘}} ∥ ( \begin{matrix}U(σ) \\ U(τ) \end{matrix} ) (M(σ)U(σ) + N(τ)U(τ))^{−1} ( M(σ) N(τ) ) ∥ < ∞. (3)

Here U(t) is the fundamental matrix of the equation ẋ(t) − A(t)x(t) = 0, −∞ < t < ∞. Under natural conditions on (1) one can choose M(σ) and N(τ) such that (3) is satisfied. We show that the sufficient condition (3) is also necessary for the time invariant case when A(t), M(t) and N(t) do not depend on t. Included are applications to integral equations with semi-separable kernels.

The talk reports on joint work with I. Gohberg and M.A. Kaashoek.

Seip, Kristian

Trondheim, Norway

Frames of complex exponentials and problems of norm equivalence for Hilbert spaces of entire functions

We solve the problem of Duffin and Schaeffer (1952) of characterizing the frames of complex exponentials for L^{2}(−π, π). This is done by means of L. de Branges’ theory of Hilbert spaces of entire functions, after recasting the problem as a problem about norm equivalence for such spaces. We show that our solution permits us to obtain an inequality of H. Landau as a consequence of the John-Nirenberg theorem, and we discuss how it leads to an approximation problem for subharmonic functions.

The talk is based on a forthcoming joint paper with Joaquim Ortega-Cerdà.

Shimorin, Serguei

Lund, Sweden

Wold-type decompositions and wandering subspaces of operators close to isometries

For operators T satisfying certain inequalities we obtain a weak analog of the classical Wold-Kolmogorov decomposition theorem, representing T as a direct sum of an unitary operator and a shift operator acting in some Hilbert space of analytic functions. A conception of a dual operator is introduced, which reflects relationships between shift operators acting in two Hilbert spaces of analytic functions dual to each other with respect to the Cauchy pairing. Among different aspects of this duality we consider relationships between hereditary inequalities and properties of reproducing kernels.

As an application, we consider wandering subspace properties of z-invariant subspaces in weighted Bergman spaces.

Shkalikov, Andrei

Moscow, Russia

Schrödinger operators with singular potentials from the space of multipliers

The Schrödinger operator with δ(x)-potential was first discussed in the physical literature. Rigourous approuches to the understanding of S-operators with δ-type potensials were proposed in papers of F. Berezin, R. Minlos and L. Faddev in 60-ies. Their investigations were developed by S. Albaverio, F. Gesztezy, R. Hoegh-Krohn, H. Holden, A. Kiselev, V. Koshmanenko, S. Kuroda, B. Simon and many others.

Singular potentials supported by sets of non-zero measure have got less attention. We aware of studies on the \frac{1}{x} (Coulomb) potential in the one dimentional case (F. Atkinson, B. Bodenstorfer, A. Dijiksma, W. Everitt, J. Gunson, P. Kurasov, H. Langer, A. Zettle et. al.) Two approaches will be proposed in the talk to define the operators of the form Δ + q(x) for q(x) belonging to Sobolev spaces with negative indices. The first approach is based on the theory of multipliers on Sobolev spaces. In the one dimentional case it allows to define the S-operator for any q(x) ∈ H_{2}^{−1}(ℝ),

in particular, for q(x) = x^{α}, α > −3/2. The correctness of the definition will be discussed from different points of view. The case q(x) ∉ H_{2}^{−1}(ℝ) is more delicate. However, for some concrete potentials (in particular, for x^{α}, −2 < α < −3/2, a "reasonable" definition of the S-operator can be also given.

The talk is based on the joint papers with M. Neiman-zade and A. Savchuk.

de Snoo, Henk

Groningen, The Netherlands

Rank one perturbations in a Pontryagin space with one negative square

Let N_{1} denote the class of generalized Nevanlinna functions with one negative square and let N_{1,0} denote the subclass of functions Q(z) ∈ N_{1} with the additional properties that lim_{y→∞} Q(iy)/y = 0 and lim sup_{y→∞} y|ℑQ(iy)| < ∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ) = A + τ[·, ω]ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions, such as −z log(1 − z) or iz^{3/2}, either belong to the subclass N_{1,0} or can be associated with the corresponding generalized Friedrichs extension. It is shown that a complete spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is possible. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions. This talk reports on joint work with Seppo Hassi (Helsinki) and Vladimir Derkach (Donetsk).

Solomyak, Michael

Rehovot, Israel

On a class of elliptic operators with non-Weylian spectral asymptotics

Let Ω ⊂ ℝ^{d}, d ≥ 3, be an unbounded domain, such that the spectrum of the Dirichlet Laplacian Δ_{D} in L_{2}(Ω) is not discrete. Denote λ_{0} = min Spec(−Δ_{D}). Let V ≥ 0 be a real-valued potential in Ω and α > 0 be a large parameter (the coupling constant). It is well known that under the single assumption V ∈ L_{\frac{d}{2}} (Ω) the semi-classical asymptotic formula holds true for any c < λ_{0}:

N_{−}(−Δ_{D} − cI − αV) ∼ C_{0}α^{\frac{d}{2}} ∫_{Ω} V+^{\frac{d}{2}} dx, α → ∞.

Here N_{−}(·) stands for the number of negative eigenvalues of a given self-adjoint operator; C_{0} is the classical Weyl constant.

The asymptotics (*) is generally not uniform with respect to c < λ_{0} and may fail for c = λ_{0}. The only case when (*) knowingly survives for c = λ_{0}, is λ_{0} = 0. This follows from the Rozenblum – Lieb – Cwikel inequality.

It is natural to state the following question: What is the asymptotic behaviour of N_{−}(−Δ_{D}−

cI−αV) for c = λ_{0}? There is no universal answer to this question, since this behaviour heavily

depends on the geometry of Ω at infinity. However, the answer is known for many important situations. We are going to discuss some of these cases. The simplest one is the case of cylindrical domain, when the result becomes quite transparent. We also give the answer for the case when Ω is periodic in one direction (a waveguide-type domain). The results for this problem (and for a wide class of more general periodic problems in such domains) were recently obtained by M. Sh. Birman and the author.

Spitkovsky, Ilya

Williamsburg, USA

Those evil Toeplitz operators with semi almost periodic symbols

Explicit invertibility and Fredholmness criteria are well known for several classes of Toeplitz operators, including those with continuous, piecewise continuous, almost periodic, and semialmost periodic (SAP) symbols. A simple reasoning allows to restate these criteria as nice geometric descriptions of the spectra and essential spectra for all the above mentioned classes of operators except for the last one. Surprisingly, the situation is much more complicated in the case of general SAP symbols. We will discuss the possible shapes of the spectra for the respective Toeplitz operators. In particular, we will show that the spectrum is discontinuous as a function defined on the manifold of Toeplitz operators with SAP symbols.

This work is joint with A. Boettcher and S. Grudsky.

Staffans, Olof

Åbo, Finland

Lax-Phillips scattering and well-posed linear systems

We discuss the connection between Lax–Phillips scattering theory and the theory of wellposed linear systems, and show that the latter theory is a natural extension of the former. As a consequence of this, there is a close connection between the Lax–Phillips generator and the generators of the corresponding well-posed linear system. All the essential information

about these two systems is contained in the system operator S = [ \begin{matrix}A B \\ C&D \end{matrix} ], where A is the

generator of the (interior) semigroup, B is the control operator, and C&D is the combined observation/feedthrough operator. If the system is compatible in the sense of Helton or regular in the sense of Weiss, then this system operator can be written in the more familiar form

S = [ \begin{matrix}A & B \\ C & D \end{matrix} ], where C is the observation operator and D is the (generalized) feedthrough

operator. We show that S is closed and densly defined. In the reflexive case the adjoint of S is the system operator of the dual system. We give formulas for the Lax–Phillips generator and resolvent in terms of the system operator. By applying the Hille–Yoshida theorem to the Lax–Phillips semigroup we get necessary and sufficient conditions for the L^{p}-admissibility or joint L^{p}-admissibility of a control operator B and an observation operator C.

Sukhocheva, Lioudmila

Voronezh, Russia

Spectral properties of selfadjoint operator functions

This talk is based on a joint work with T. Azizov and A. Dijksma. Let L : [a, b] → L(H) be a continuous function which values are selfadjoint bounded operators on a Hilbert space H. We set

E(L) = c.l.s. {ker L(λ) | λ ∈ [a, b]},

κ_{+}(L(a)) (κ_{−}(L(b)), resp.) the number of positive (negative) eigenvalues of L(a) (L(b), resp.) counting multiplicity.

Theorem 1 Let dim H = ∞ and let L : [a, b] → L(H) be a continuous selfadjoint operator function which satisfies the conditions:

(i) f ≠ 0, (L(x)f, f) = 0 ⇒ (y − x)(L(y)f, f) > 0, y ≠ x, (ii) κ_{+}(L(a)) < ∞, κ_{−}(L(b)) < ∞, (iii) there is a c ∈ (a, b) such that L(x) is Fredholm for x ≠ c. Then (a) κ_{+}(L(x)) < ∞ for x ∈ [a, c) and κ_{−}(L(x)) < ∞ for x ∈ (c, b], (b) L(c) is compact,

(c) dim E(L) = dim H, (d) codim E(L) ≥ κ_{+}(L(a)) + κ_{−}(L(b), and (e) if σ(L) ∩ [a, c) (σ(L) ∩ (c, b]) is a finite set consisting of κ_{a,c} (κ_{c,b}),say, eigenvalues counting multiplicity, it holds

κ_{+}(L(c)) = κ_{+}(L(a)) + κ_{a,c} (κ_{−}(L(c)) = κ_{−}(L(b)) + κ_{c,b}).

This research is supported by the grant RFBR 99-01-00391.

Tikhonov, Alexey

Simferopol, Ukraine

Spectral theory for operators with spectrum on a curve

We introduce weak extreme subspaces (spectral components) for operators whose continuous spectrum lies on a closed smooth curve. Axiomatic approach is used for definition of such subspaces. As particular cases of general 𝔛-maximal subspace one can consider absolutely continuous

N(S) = clos {f ∈ H : e_{S}^{±}(f, g, w) ∈ E_{±}^{2} ∀g ∈ H} ,

singular

M(S) = {f ∈ H : e_{S}^{+}(f, g, w) = e_{S}^{−}(f, g, w) ∀g ∈ H} ,

singular^{+}, singular^{−}, hypersingular, and other subspaces. Here e_{S}^{±}(f, g, w) = lim_{z→w}((S −

z)^{−1}f, g) are interior and exterior angular boundary values, E_{±}^{2} are Smirnov spaces. Some properties and relationships for these subspaces are established. We apply the developed theory to scattering theory and to extreme factorizations of J-contractive-valued functions.

Timotin, Dan

Lyon, France and Bucharest, Romania

The intertwining lifting theorem for a class of reproducing kernel Hilbert spaces

The aim of this paper is to obtain an intertwining lifting theorem for a certain class of reproducing kernel Hilbert spaces that seems to be the most suitable for interpolation. The framework is the following: we consider an abstract set of indices Λ and a Hilbert space H of complex functions defined on Λ, such that the point evaluations f ↦ f(λ) are continous. A standard procedure is then to introduce the reproducing vectors for these evaluations, which we will denote with k_{λ}; that is, f(λ) = ⟨f, k_{λ}⟩ for any f ∈ H. The reproducing kernel for the space is the function K : Λ × Λ → ℂ, defined by K(λ, μ) = k_{μ}(λ), which is a function of positive type. A function ϕ : Λ → ℂ is called a multiplier if the map f ↦ ϕf is everywhere defined. The algebra of bounded operators T_{ϕ} obtained in this way is denoted by M.

Now, the particular class of reproducing kernels spaces that we consider, apart from some natural conditions that are fulfilled by basically all the spaces that appear in applications, satisfies a special property which is essential to interpolation: the function 1 − 1/K is also of positive type on Λ × Λ. This condition has already been pointed out by Quiggin in 1993, in connection with the Nevanlinna-Pick problem; it originates in the work of Agler on interpolation.

The main result of the paper is the following intertwining lifting theorem.

Theorem 1 Suppose X ⊂ H is a subspace invariant to M^{∗}, and denote M_{X} = P_{X} M|X. If X ∈ L(X) and X ∈ M′_{X} (the commutant of M_{X}), then there exists a multiplier ϕ with ∥ϕ∥_{M} = ∥X∥, such that X = P_{X} T_{ϕ}|X.

Applications of this result to a large variety of particular cases are presented. Also, a central role in the proof of the theorem is played by a fractional transform representation for contractive multipliers, which can be of independent interest.

Tkachenko, Vadim

Beer Sheva, Israel

Non-selfadjoint Sturm-Liouville operators with multiple spectra

We consider Sturm-Liouville operators

H = −\frac{d^{2}}{dx^{2}} + q(x),  q(x) ∈ L^{2}[0, π],  (1)

with one of the following boundary conditions

\begin{matrix}D) y(0) = 0, & N) y(0) = 0, & P) y(0) = y(π), & AP) y(0) = −y(π), \\ y(π) = 0; & y′(π) = 0; & y′(0) = y′(π); & y′(0) = −y′(π). \end{matrix}

Spectrum of each problem is discrete and behaves asymptotically as the spectrum of the corresponding operator with q(x) ≡ 0. Namely,

λ_{n}(D) = (n + \frac{Q}{n} + \frac{f_{n}(D)}{n})^{2} , Q ∈ C;

λ_{n}(N) = (n − \frac{1}{2} + \frac{Q}{n} + \frac{f_{n}(N)}{n})^{2} , \{f_{n}(.)\}_{n=1}^{∞} ∈ ℓ^{2};

λ_{n}^{±}(P) = (2n + \frac{Q}{2n} ± \frac{f_{n}(P)}{n} + \frac{g_{n}(P)}{n^{2}})^{2} ,  \{g_{n}(.)\}_{n=1}^{∞} ∈ ℓ^{2};  (2)

λ_{n}^{±}(AP) = (2n + 1 + \frac{Q}{2n + 1} ± \frac{f_{n}(AP)}{n} + \frac{g_{n}(AP)}{n^{2}})^{2} .

Our aim is to find additional conditions, if any, which guarantee that a given sequence satisfying one of equations (2) is the spectrum of the corresponding boundary problem. In particular, we would like to know whether some points of the spectra may be multiple and whether there are some restrictions on their multiplicities.

In the self-adjoint case, i.-e., ℑq(x) = 0, all above spectra are real and their complete description is well-known. In particular, the spectra of problems D and N are simple (i.e., all eigenspaces are of dimension 1 and there are no associated root functions) and satisfy the interlacing conditions

λ_{n}(N) < λ_{n}(D) < λ_{n+1}(N).

The spectra of problems P and AP are at most of multiplicity 2 (i.e., eigenspaces may be of dimensions either 1 or 2, and there are no root functions) and interlacing conditions

… λ_{2n−2}(D) ≤ λ_{n−1}^{+}(P) < λ_{n}^{−}(AP) ≤ λ_{2n−1}(D) ≤ λ_{n}^{+}(AP)
< λ_{n+1}^{−}(P) < λ_{2n}(D) ≤ λ_{n+1}^{+}(D) ≤ …

are fulfilled.

If a spectral gap [λ_{n}^{−}(P), λ_{n}^{+}(P)] (or [λ_{n}^{−}(AP), λ_{n}^{+}(AP)]) collapses, then the multiplicity of λ_{n}^{±}(P) (or λ_{n}^{±}(AP), respectively) is equal to 2.

The aim of the present communication is to show that there is no restrictions on the multiplicities in non-selfadjoint case, i.e., for potentials q(x) taking on non-real values. Namely, the following statements are true:

Proposition 1 For every finite set D = {λ_{1} …, λ_{n}} of pairwise distinct points in C and every set M(D) = {m_{1}, …, m_{n}} of positive integers there exists a Sturm-Liouville operator (1) for which {λ_{1} …, λ_{n}} are points of the Dirichlet spectrum with multiplicities equal to respective numbers from M(D).

Proposition 2 For every finite set P = {λ_{1}^{+} …, λ_{n}^{+}} of pairwise distinct points in C and every set M(P) = {m_{1}, …, m_{n}} of positive integers there exists a Sturm-Liouville operator (1) for which {λ_{1}^{+} …, λ_{n}^{+}} are points of the periodic spectrum with multiplicities equal to respective numbers from M(P).

The similar statement is valid for anti-periodic spectrum.

Proposition 3 A complex number z may be simultenuously a point of the Dirichlet spectrum of multiplicity m and a point of the periodic spectrum of multiplicity p of some Sturm-Liouville operator if and only if either m ≤ p or m > p and p is an odd number.

We construct Sturm-Liouville operators with above properties basing on the solution of the inverse problem using two spectra, e.g., Dirichlet and Neumann spectra.

Theorem 1 Two sequences {λ_{n}}_{n=1}^{∞} and {μ_{n}}_{n=0}^{∞} of complex numbers are the Dirichlet and Neumann spectra of some operator (1) if and only if the following conditions are fulfilled:

i) λ_{n} ≠ μ_{k} for all n and k, and representations

λ_{n} = (n + \frac{Q}{n} + \frac{f_{n}}{n} )^{2} ;  μ_{n} = (n − \frac{1}{2} + \frac{Q}{n} + \frac{g_{n}}{n} )^{2} ;  n ≥ 1,  (4)

are valid with Q ∈ C, {f_{n}}_{n=1}^{∞} ∈ ℓ^{2}, {g_{n}}_{n=1}^{∞} ∈ ℓ^{2}; ii) if

s(λ) = π ∏_{n=1}^{∞} \frac{λ_{n} − λ^{2}}{n^{2}} , c(λ) = ∏_{n=0}^{∞} \frac{μ_{n} − λ^{2}}{(n − 1/2)^{2}} ,

and

F (x, t) = ∑_{k=−∞}^{∞} \{res_{s(z_{k})=0} (\frac{c(z)}{zs(z)} sin zx sin zt) − \frac{1}{π} sin kx sin kt\} ,

then the homogeneous Gelfand-Levitan equation

g(t) + ∫_{0}^{x} F (t, s)g(s)ds = 0,  0 ≤ t ≤ x,  (5)

has only trivial solution g(s) ∈ L^{2}(0, x) for every x ∈ [0, π].

For real-valued potential q(x) the interlacing conditions and representations (4) imply ii) and Theorem 1 coincides with results of B. Levitan, M. Gasymov and V. Marchenko [1960–1964].

For complex-valued potentials all above parameters may be non-real and properties i) and ii) are independent one from another.

The following proposition describes some class of sequences which may be Dirichlet spectra of Sturm-Liouville operators with complex-valued potentials.

Theorem 2 Let {λ_{n}}_{n=1}^{∞} be a sequence of complex numbers symmetric with respect to the real axis and representable in the form

λ_{n} = (n + \frac{Q}{n} + \frac{f_{n}}{n} )^{2} ;  n ≥ 1,  (6)

with some Q ∈ C and {f_{n}}_{n=1}^{∞} ∈ ℓ^{2}. Then there exists Sturm-Liouville operator H = d^{2}/dx^{2}+ q(x) with q(x) ∈ L^{2}[0, π] for which {λ_{n}}_{n=1}^{∞} coincides with the Dirichlet spectrum.

To prove Proposition 1 it is sufficient to extend a set D = {λ_{1} …, λ_{n}} to a sequence {λ_{n}}_{n=1}^{∞} satisfying conditions of Theorem 2 and to apply it.

Proposition 2 may be proved using the following description of Hill determinants [Tkachenko, 1992].

Theorem 3 For a function Δ(λ) to be Hill’s determinant of some Sturm-Liouville operator (1) it is necessary and sufficient that it be an even entire function of exponential type π which may be represented in the form,

Δ(λ) = cos πλ + \frac{Q}{λ} sin πλ − \frac{Q^{2}}{2λ^{2}} cos πλ + \frac{f(λ)}{λ^{2}}

where Q is a complex number, and f is an entire function of exponential type not exceeding π and satisfying conditions

∫_{−∞}^{+∞} |f(λ)|^{2}dλ < ∞,  ∑_{n=−∞}^{+∞} |f(n)| < ∞.

Tomilov, Yuri

Kiev, Ukraine

On convergence of operator families

Let (U(t, s))_{t≥s≥0} be an evolution family of linear bounded operators in a Banach space X. One of the most important tasks in the study of linear well-posed abstract Cauchy problems is the description of (corresponding to them) evolution families (U(t, s))_{t≥s≥0} such that

lim_{t→∞} ||U(t, s)x|| = 0 (t ≥ s)

for every x ∈ X and every s ∈ ℝ^{+} or, in other words, asymptotically stable evolution families. The main problem here is an absence of natural terms such descrption can be given in. We have made an attempt to circumvent this problem by applying a recent technique of "evolution semigroup". The "evolution semigroup" and also "complete trajectories" methods have been applied for the study of the asymptotic stability of evolution families for the first time. These methods allowed us to obtain necessary and sufficient conditions for the asymptotic stability in terms of bounded complete trajectories of the dual evolution family, ergodicity (and stability) of associated evolution semigroup, convolution properties of evolution families. Such criterias were unknown even in finite-dimensional case. The convolution characterization of the asymptotic stability is "similar" to famous Datko’s characterization of exponential stability. As byproducts, an essential improvement of Curtis-Neumann results on spectral maximal subspaces of isometric semigroups and a multilplier criteria for the asymptotic stability of C_{0}-semigroups were derived.

This is joint work with C.J.K. Batty and R. Chill.

Treil, Sergei

Providence, USA

Recent advances in similarity problem

The talk is devoted to the problem of similarity to a normal operator. Similarity of an operator T to a normal operator means that an operator admits rich functional calculus, namely it is possible to define f(T) for any function f continuous on the spectrum σ(T). The simplest, and easily verified in applications necessary condition for the similarity to a normal operator is the so-called Linear Resolvent Growth (LRG) condition ∥(T − λI)^{−1}∥ ≤ C/dist(λ, σ(T)) . Generally this condition is not sufficient, for example, the shift operator S, Sf(z) = zf(z) on the Hardy space H^{2} satisfies this condition, but clearly is not similar to a normal operator.

We are interested mostly in the problem for a “small” perturbation of a unitary (selfadjoint) operator, namely, when LRG implies that the operator is similar to a normal one.

New results will be presented. The recent progress in the subject is connected, in particular, with the progress in the theory of singular integral operators. For example, Nazarov’s counterexample to Sarason’s Conjecture plays an important role here.

Tretter, Christiane

Regendburg, Germany

Diagonalization of block operator matrices and applications to Dirac operators

In this talk some classes of self-adjoint and non-self-adjoint block operator matrices

A := ( \begin{matrix}A & B \\ C & D \end{matrix} )

in the product H ⊕ Ĥ of Hilbert spaces are considered the entries of which can be unbounded operators. It is assumed that the spectra of the diagonal elements A and D are separated by a strip around the imaginary axis and that the spectrum of A at infinity can be splitted, i.e. the operator A is dichotomous.

Using indefinite inner product methods, it is proved that (under some additional assumptions) there exist invariant subspaces L_{+} and L_{−} of A corresponding to its spectrum in the right and in the left half plane, respectively, which can be represented by means of angular operators K_{+} ∈ L(H, Ĥ) and K_{−} ∈ L(Ĥ, H) satisfying certain Riccati equations. This yields a block-diagonalization of the operator Ã.

Applications of these results to Dirac operators are given. For supersymmetric Dirac operators this block diagonalization coincides with the Foldy–Wouthuysen transformation well-known in quantum mechanics. For Dirac operators with potential no such diagonalization seems to have been known by now.

(Joint work with Heinz Langer, TU Vienna)

Tsekanovskii, Eduard

New York, USA

Interpolation theory in sectorial Stieltjes classes and explicit system solutions

We introduce sectorial classes of matrix–valued Stieltjes functions in which we solve the bitangential interpolation problem of Nudelman and Ball-Gohberg-Rodman. We consider also a new type of solutions of Nevanlinna–Pick interpolation problems, so–called explicit system solutions generated by Brodskii–Livsic colligations, and find conditions on interpolation data of their existence and uniqueness. We point out the connections between sectorial Stieltjes classes and sectorial operators, and find out new properties of the classical Nevanlinna–Pick interpolation matrices (in the scalar case). We present in terms of interpolation data the exact formula for the angle of sectoriality of the main operator in the explicit system solution as well as the criterion for this operator to be extremal.The interpolation model for nonselfadjoint matrices is established.

This is joint work with D. Alpay.

Vasil’ev, Vladimir

Novgorod, Russia

Wave factorization of elliptic symbols

One suggests a new method for studying of pseudodifferential equations in nonsmooth domains which is based on a special factorization ( it was called wave factorization ) of a symbol. It essentially differs from factorization which was applied by M.I. Vishik and G.I. Eskin in halh-space case. In model situations, for example, in a plane angle, the existence of wave factorization permits us to find a general solution of a simplest pseudodifferential equation or solvability conditions in Sobolev - Slobodetskii spaces in dependence on sign of the wave factorization index. For positive wave factorization index one gives correct statements of boundary value problems when boundary conditions are given in form of traces of certain pseudodifferential operators on angle sides. If pseudodifferential operators are homogeneous, then after applying of Fourier transform the boundary value problem is reduced to a system of linear integral equations on the real axis. After some simplifications and using the Mellin transform the system is reduced to either system of difference equations or system of linear algebraic equations, and solvability of original boundary value problem is formulated as solvability of the last system. For negative wave factorization index one considers the equations with additional co-boundary operators. A priori estimates and applications to some problems in mathematical physics are given. For the Laplacian with Dirichlet or Neumann boundary conditions the exact calculations of Mellin transforms for kernels of integral equations obtained are done.

References

1. V. B. Vasil’ev, Wave factorization of elliptic symbols: theory and applications, Dordrecht, Kluwer Academic Publishers, 2000.

Verbitsky, Igor

Columbia, USA

Best constants for some Fourier multiplier operators on L^{p} spaces

We will discuss some recent results concerning the exact values of the norms and essential norms for operators of the form a I + b H on L^{p} spaces, where I is the identity operator, H is the Hilbert transform, and a and b are complex numbers. This includes the norm of the analytic projection on L^{p} spaces with weights, and the best constant for the re-expansion operator, which answers a question of M.S. Birman. Some multidimensional generalizations and applications will be considered.

Verduyn Lunel, S. M.

Leiden, The Netherlands

Periodic differential equations with time lags and their spectral properties

In this lecture we give an overview of the spectral theory for periodic delay equations and discuss some recent results regarding completeness of the system of Floquet solutions for systems of periodic delay equations.

The result extends earlier results for scalar equations. If the time delays are integer multiples of the period of the coefficients, then we have a complete answer. In the scalar case, we shall present necessary and sufficient conditions for completeness of the system of Floquet solutions in case the time delays are rational multiples of the period of the coefficients. Our result also yields insight in what happens if the time delays approach irrational multiples of the period of the coefficients.

Noncompleteness of the Floquet solutions is equivalent to the existence of super-exponenti-ally decaying solutions. We shall explain this connection and present some recent results about bounds for the decay rate of such solutions.

The lecture wil be concluded with some examples and open problems. This work is in part joint work with V. Matsaev (Tel-Aviv University).

Vinnikov, Victor

Beer Sheva, Israel

Commuting isometries and vector bundles on algebraic curves

We show that a pair of commuting isometries with finite defects satisfy an algebraic equation with constant coefficients. Furthermore we associate to such a pair of isometries a certain vector bundle (or more generally, a certain sheaf) on the corresponding algebraic curve and obtain a complete set of unitary invariants for the pair as well as a functional model in a certain H^{2} space of sections of this vector bundle.

These results form a part of the general spectral theory of commuting contractions, and more generally commuting nonunitary operators, with finite defects, which is closely related to the theory of multidimensional input/state/output systems and to the theory of scattering systems of Lax–Phillips type with several evolution operators. The key underlying notion is the notion of an operator vessel which is a certain algebraic structure reflecting the interplay of several nonunitary or nonselfadjoint operators (in both commutative and noncommutative case) and enabling us to establish (in the commutative case) a link with function theory on an algebraic curve leading to results as explicit as in the classical single operator situation.

This is joint work with J. Ball.

Volberg, Alexander

East Lansing, USA

Carleson measures for K_{θ} cannot be checked on reproducing kernels

Carleson measures for Hardy space H^{2} play important role in complex analysis. For a mesure in the disk to be a Carleson measure can be checked by testing it on squares of reproducing kernels of H^{2}. It has been a problem to which W. Cohn and A. Alexandrov devoted a considerable attention whether the same is true for measures which are Carleson for "model spaces" K_{θ} (θ is an inner function). We will show that testing measure on reproducing kernels of K_{θ} is not enough to conclude that it is a Carleson measure for K_{θ}. The technique is based on real analysis methods: on new knowledge about two weight estimates for Calderon-Zygmund operators and on so called Bellman function approach.

Winkler, Henrik

Dresden, Germany

On the generalized Friedrichs and Krein-von Neumann extensions

Let S be a densely defined and closed symmetric relation in a Hilbert space H with defect numbers (1, 1), and let A be some of its canonical selfadjoint extensions. According to M. G. Krein’s formula, to S and A corresponds a so-called Q-function from the Nevanlinna class N. Each function Q ∈ N has a unique representation of the form

Q(z) = bz + a + ∫_{−∞}^{+∞} (\frac{1}{λ − z} − \frac{λ}{1 + λ^{2}} ) dσ(λ),

with a ∈ R, b ≥ 0 and a measure σ with the property that ∫_{−∞}^{+∞} \frac{dσ(λ)}{1+λ^{2}} < ∞. Let N_{γ}, γ ∈ (0, 2), be the class of all functions Q ∈ N with b = 0 and ∫_{−∞}^{+∞} \frac{dσ(λ)}{1+|λ|^{γ}} < ∞. In our talk we show to which subclasses N_{γ} of N the Q-functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q-function of the generalized Friedrichs extension. Further we present similar results for the generalized Krein-von Neumann extension and show some applications for systems of canonical differential equations.

Xia, Jingbo

Buffalo, USA

Diagonalization and unitary equivalence modulo Schatten p-classes

Let 1 < p < ∞. Recall that an operator T is said to be in the Schatten p-class C_{p} if

∥T ∥_{p} = {tr((T^{∗}T)^{p/2})}^{1/p} = {tr(|T |^{p})}^{1/p} < ∞.

A positive, finite, regular Borel measure μ on ℝ^{n} is defined to be p-singular if

∫_{0}^{1} (\frac{μ(B(x, r))}{r^{p}})^{1/(p−1)} \frac{dr}{r} = ∞ for μ-a.e. x ∈ ℝ^{n}.

Let A = (A_{1}, ..., A_{n}) be a commuting tuple of self-adjoint operators on a Hilbert space H and let E be its spectral resolution. For each ξ ∈ H, let μ_{ξ} be the measure on ℝ^{n} defined by the formula μ_{ξ}(Δ) = ⟨E(Δ)ξ, ξ⟩. The tuple A is said to be purely p-singular if μ_{ξ} is p-singular for every nonzero ξ ∈ H. We give the following spectral characterization of the simultaneous diagonalizability of the tuple A modulo C_{p} :

Theorem 1 Let 1 < p < ∞ and let A = (A_{1}, ..., A_{n}) be a commuting tuple of self-adjoint operators. Then there exists a commuting tuple D = (D_{1}, ..., D_{n}) of self-adjoint diagonal operators such that A_{j} − D_{j} ∈ C_{p} for every 1 ≤ j ≤ n if and only if the tuple A is purely p-singular.

Xu, Quanhua

Besancon, France

Non-commutative inequalities

We present non-commutative analogues of some classical martingale inequalities.

Yafaev, Dmitri

Rennes, France

Singular Friedrichs model and Hardy-Rellich inequalities

We consider a model of the perturbation theory where the Hamiltonian H is a sum of a multiplication operator H_{0} by |x|^{2l} in the space L_{2}(ℝ^{d}) and of an integral operator V with oscillating kernel. As typical examples, we note the kernels cos⟨x, y⟩ and sin⟨x, y⟩. We find conditions for the coincidence of the essential spectra of the operators H_{0} and H and for the existence and completeness of the wave operators relating H_{0} and H. Moreover, we find necessary and sufficient conditions for the negative spectrum of the operator H to be finite and calculate expl icitly the total number of its negative eigenvalues.

As a by-product of our considerations, we prove the inequality

∫_{ℝ^{d}} |x|^{−2l}|u(x)|^{2}dx ≤ C_{l} ∫_{ℝ^{d}} |ξ|^{2l}|û(ξ)|^{2}dξ, u ∈ C_{0}^{∞}(ℝ^{d} \ {0}),

with the optimal constant

C_{l} = 2^{−2l} max\{\frac{Γ^{2}((d/2 − l)/2)}{Γ^{2}((d/2 + l)/2)}, \frac{Γ^{2}((d/2 − l + 1)/2)}{Γ^{2}((d/2 + l + 1)/2)}\}.

Here û is the Fourier transform ofu, Γ is the gamma-function and l is not necessarily integer.

Yger, Alain

Bordeaux, France

About the direct realisation of algebraic identities and some relations with operator theory

Let μ_{1}, ..., μ_{m} be m elements in E′(R^{n}) and s an unknown continuous signal in R^{n}, which has been measured through the devices μ_{1}, ..., μ_{m}, which means some “blurred” versions s_{1}, ..., s_{m} of μ_{1} ∗ s, ..., μ_{m} ∗ s are known. Wiener’s filtering process allows to recover s as

ŝ = \frac{∑_{k=1}^{m} \overline{μ̂_{j}}ŝ_{j}}{∑_{k=1}^{m} |\overline{μ̂_{j}}|^{2} + ϵ} ,

where ϵ is interpreted as a “tolerance” ratio noise/signal. We will show how some divisioninterpolation formulas can be realized (in an approximate way) thanks to the Wiener’s filtering procedure. We will see how such a process fits with the realization of interpolation-division formulas in complex analysis or commutative algebra. When the entries are polynomials, the fact that Lagrange’s interpolation formula provides a substitute for Euclide’s division algorithm seems to give some hint towards the realization of algebraic identities as operatorial identities, leading to the transposition of some of the tools involved in geometric or arithmetic intersection theory to an operational context (Bezutian, resultant, Chow forms,...) In this talk, I will summarize (taking into account the operatorial point of view) some of the ideas which appeared in my long-term joint work with C. A. Berenstein about such questions.

Yuditskii, Peter

East Lansing, USA and Kharkov, Ukraine

Character–automorphic scattering function, functional model and asymptotic behavior of polynomials orthonormal on a homogeneous set

Let E be a homogeneous compact set, for instance a Cantor set of positive length. Further let σ be a positive measure with supp(σ) = E. Under the condition that the absolutely continuous part of σ satisfies a Szegö–type condition we give an asymptotic representation, on and off the support, for the polynomials orthonormal with respect to σ. For the special case that E consists of a finite number of intervals and that σ has no singular component this is a nowaday well known result of Widom. If E = [a, b] it becomes a classical result due to Szegö and in case that there appears in addition a singular component, it is due to Kolmogorov– Krein. In fact the results are presented for the more general case that the orthogonality measure may have a denumerable set of mass–points outside of E which are supposed to accumulate on E only and to satisfy (together with the zeros of the associated Stieltjes function) the free–interpolation Carleson–type condition. Up to the case of a finite number of mass points this is even new for the single interval case. Furthermore, as a byproduct of our representations, we obtain that the recurrence coefficients of the orthonormal polynomials behave asymptotically almost periodic. Or in other words the Jacobi matrices associated with the above discussed orthonormal polynomials are compact perturbations of a one–sided restriction of almost periodic Jacobi matrices with homogeneous spectrum. Our main tool is a theory of Hardy spaces of character–automorphic functions and forms on Riemann surfaces of Widom type and ideas of the scattering theory (joint work with Franz Peherstorfer, Austria).

Zemanek, Jaroslav

Warsaw, Poland

Recent advances in operator ergodic theory

We intend to clarify the role of the closedness of the range of some power of I − T in the ergodic behaviour of T, with respect to various operator topologies, and in connection with analytic properties of the resolvent of T.

Zwart, Hans

Enschede, The Netherlands

Conjectures on exact observability

This is a joint work with B. Jacob.

In this talk we discuss exact observability of an infinite-dimensional system on a Hilbert space. Since exact observability is dual to exact controllability, the results as presented here can easily be translated to similar results on exact controllability.

Consider the infinite-dimensional system

ż(t) = Az(t),  y(t) = Cz(t).  (1)

We assume that A generates a (strongly) stable C_{0}-semigroup on the Hilbert space H. Furthermore, we assume that C is a bounded linear operator from the domain of A (D(A)) to the Hilbert space Y and that for all x_{0} ∈ D(A) we have that CT (·)x_{0} is in L^{2}((0, ∞); Y), and ∥CT (·)x_{0}∥_{L^{2}} ≤ ∥x_{0}∥. This last inequality implies that for all x_{0} ∈ H we can regard CT (·)x_{0} as an element of L^{2}((0, ∞); Y).

We say that the system is exactly observable if there exists an m > 0 such that for all x_{0} ∈ H we have that

∫_{0}^{∞} ∥CT (t)x_{0}∥^{2}dt ≥ m∥x_{0}∥^{2}.

Russell and Weiss conjectured in [1] that the system (1) is exactly observable if and only if there exists an m_{2} > 0 such that

∥(sI − A)x_{0}∥^{2} + |Re(s)|∥Cx_{0}∥^{2} ≥ m_{2}Re(s)^{2}∥x_{0}∥^{2}

for all x_{0} ∈ D(A) and for all s with Re(s) < 0.

We shall show that this conjecture holds if the following semigroup conjecture holds. Conjecture Let T (t) be a bounded C_{0}-semigroup on the separable Hilbert space Z. The following assertions are equivalent:

1. There exists an m_{3} > 0 such that ∥T (t)z_{0}∥ ≥ m_{3}∥z_{0}∥ for all z_{0} ∈ Z.

2. There exists an m_{4} > 0 such that

∥(sI − A)z_{0}∥ ≥ m_{4}|Re(s)|∥z_{0}∥

for all z_{0} ∈ D(A) and for all s with Re(s) < 0.

The aim of this talk is to discuss the relation between these conjectures, and to give the latest developments on the proofs of these conjectures.

References

1. D.L. Russell and G. Weiss, A general necessary condition for exact observability, SIAM J. Control Optim., 32(1994), pp. 1-23.

96

List of Participants Liste de participants

Prof. Evgeny Abakumov Equipe d’Analyse et de Mathematiques Appliquees Universite de Marne-la-Vallee Cite Descartes - 5, Bd Descartes Champs-sup-Marne 77454 Marne-la-Vallee CEDEX 2 France e-mail: abakumov@math.univ-mlv.fr

Mr. Hossein Abbaspour 162 Hoyt st. Brooklyn, NY 11217 USA e-mail: habbaspour@gc.cuny.edu

Prof. Vadim Adamyan Department of Theoretical Physics Odessa State University ul. Dvoryanskaya 2 270044 Odessa, Ukraine e-mail: vadamjan@m-vox.odessa.ua

Prof. Daniel Alpay Department of Mathematics Ben-Gurion University of the Negev P.O.B. 653, Beer-Sheva 84105 Israel e-mail: dany@math.bgu.ac.il

Prof. Mohamed Akkar UFR de Mathématiques Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: akkar@math.u-bordeaux.fr

Prof. Eric Amar UFR de Mathématiques Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: eamar@math.u-bordeaux.fr Dr. Kais Ammari Institut Elie Cartan Departement de Mathematiques Universite Nancy I 54506 Vandoeuvre-les-Nancy France e-mail: ammari@iecn.u-nancy.fr

Prof. Grigori G. Amosov

Department of Higher Mathematics Moscow Institute of Physics and Technology Institutski 9, Dolgoprudni 141700 Russia e-mail: gramos@mail.sitek.ru

Prof. Jonathan Arazy

Department of Mathematics

University of Haifa

Mount Carmel, Haifa 31905

Israel

e-mail: jarazy@math.haifa.ac.il

Prof. Yury Arlinskii

East Ukrainian State University Kvartal Molodyozhny 20-A Lugansk, 348034 Ukraine e-mail: yma@vugu.lugansk.ua

Prof. Damir Z. Arov

prosp. Ak. Glushko 15, kv. 18 Odessa, 270104 Ukraine e-mail: aspect@farlep.net vadamjan@m-vox.odessa.ua

Dr. Zoya Arova

Ukraine, 270104 Odessa st. Glushko 15 apt. 18 e-mail: aspect@farlep.net

Prof. Sergei Avdonin Mathematics Department University of Tennessee Knoxville, TN 37996-1300 USA e-mail: avdonin@ist.flinders.edu.au

Prof. Tomas Azizov Dept. of Math. Voronezh State University Universitetskaya pl. 1 Voronezh, 394693 Russia e-mail: azizov@tom.vsu.ru

Dr. Mohamed Bachir UFR de Mathématiques Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: bachir@math.u-bordeaux.fr

Prof. Catalin Badea UFR de Mathematiques Université des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex France e-mail: Catalin.Badea@agat.univlille1.fr

Prof. Mihaly Bakonyi Department of Mathematics Georgia State University Atlanta, GA 30303 USA e-mail: mbakonyi@cs.gsu.edu

Prof. Michel Balazard UFR de Mathématiques Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: balazard@math.u-bordeaux.fr

Prof. Joseph A. Ball Department of Mathematics Virginia Tech Blacksburg VA 24061-0123 USA e-mail: ball@math.vt.edu

Prof. Laurent Baratchart "Projet Miaou", INRIA 2004, route de Lucioles BP93 06902 Sophia Antipolis Cedex France e-mail: Laurent.Baratchart@sophia. inria.fr

Prof. Ismat Beg

Department of Mathematics

Kuwait University P.O.Box 5969 Safat, 13060 Kuwait e-mail: ibeg@math-1.sci.kuniv.edu.kw

Prof. Mikhail I. Belishev

St-Petersburg Dept. of the Steklov Mathematical Institute (POMI) Fontanka 27, St. Petersburg 191011 Russia e-mail: belishev@pdmi.ras.ru

Prof. Genrich Belitskii

Department of Mathematics

Ben-Gurion University of the Negev P.O.B. 653, Beer-Sheva 84105 Israel e-mail: genrich@cs.bgu.ac.il

Prof. Yurij Berezansky Institute of Mathematics Ukrainian Acad. Sci. 3 Tereshchenkivs’ka Str. 01601 Kyiv, Ukraine e-mail: berezan@mathber.carrier. kiev.ua

Prof. Mikhail Birman Department of Physics St. Petersburg State University Ul’yanovskaya 1, Petrodvorets St. Petersburg, 198904 Russia e-mail: tanya@petrov.stoic.spb.su

Mr. Bernhard Bodenstorfer Institute of Analysis and Technical Mathematics E114-1 Vienna University of Technology Wiedner Hauptstrasse 8-10 A-1040 Vienna, Austria e-mail: e9025205@fbma.tuwien.ac.at

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University of West Timişoara

Bul. V. Pârvan No. 4 1900 Timişoara, Romania e-mail: gaspar@math.uvt.ro

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University of West Timişoara

Bul. V. Pârvan No. 4 1900 Timişoara, Romania e-mail: pasto@math.uvt.ro

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"Politehnica"

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School of Mathematical Sciences Tel Aviv University 69978 Tel Aviv, Israel e-mail: gohberg@math.tau.ac.il

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Dept. Math.

Istanbul Tecknical University

80626 Maslak, Istanbul

Turkey

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CMI-LATP Universite de Provence Marseille, France e-mail: youssfi@cmi.univ-mrs.fr

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Department of Mathematics and Statistics P.O. Box 54 00014 University of Helsinki Finland e-mail: hassi@cc.helsinki.fi

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Department of Mathematics

University of California at San Diego La Jolla, USA e-mail: helton@osiris.ucsd.edu

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Institute for Applied Problems of Mechanics and Mathematics 3b Naukova str., 79601 Lviv Ukraine e-mail: hryniv@mebm.lviv.ua

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Russian Center of Laser Physics

St. Petersburg University

Ul’yanovskaya ul. 1

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198904 Russia

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Prof. Alexei Konstantinov Department of Mathematics University of Kiev 64 Volodymyrska Str. 252033 Kiev, Ukraine e-mail: konst@faust.kiev.ua

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UMR CNRS 7534, Universite Paris 9 place du M. de Lattre de Tassigny 75775 Paris Cedex 16, France e-mail: Yves.Meyer@cmla.ens-cachan.fr

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Department of Math.

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Department of Mathematics

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Prof. Quanhua Xu Departement de Mathematiques Universite de Franche-Comte 25030 Besancon Cedex, France e-mail: qx@math.univ-fcomte.fr

Prof. Dmitri Yafaev IRMAR, Universite Rennes 1 Campus Beaulieu, 35042 Rennes France e-mail: Dimitri.Yafaev@univ-rennes1. fr

Prof. Alain Yger Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: yger@math.u-bordeaux.fr

Prof. Mohamed Zarrabi UFR de Mathématiques Université Bordeaux 1 351, cours de la Libération 33405 Talence, France e-mail: zarrabi@math.u-bordeaux.fr Prof. Peter Yuditskii Dept. of Mathematics Michigan State University East Lansing, MI 48824 USA e-mail: yuditski@math.msu.edu

Prof. Jaroslav Zemanek Institute of Mathematics Polish Academy of Sciences P.O.Box 137, 00-950 Warsaw Poland e-mail: zemanek@impan.gov.pl

Prof. Hans Zwart

Po.Box 217, 7500 AE Enschede The Netherlands e-mail: h.j.zwart@math.utwente.nl