Aad Dijksma
In the lecture we relate the operators in the operator representations of a generalized Nevanlinna function N(z) and of the function -N(z)-1 under the assumption that z = ¥ is the only (generalized) pole of non-positive type. The results are applied to the Q-function of S and H and the Q-function for S and H¥, where H is a self-adjoint operator in a Pontryagin space with a cyclic element w, H¥ is the self-adjoint relation obtained from H and w via a rank one perturbation at infinite coupling, and S is the symmetric operator given by S = HÇH¥. The lecture is based on joint work with Heinz Langer (University of Technology, Vienna) and Yuri Shondin (Pedagogical Institute Nihzny Novgorod, Russia).
Backward shift operator and finite dimensional de
Branges-Rovnyak spaces in the ball.
Chen Dubi
Recently V. Bolotnikov and L. Rodman characterized finite dimensional backward shift
invariant subspaces of the Arveson space. In this note we study finite dimensional backward shift invariant
Hilbert spaces contractively included in the Arveson space.
To that purpose we redefine in an appropriate way the backward shift operators in terms of power series expansions.
Some related factorization theorems will be discussed.
The work presented is joint work with Daniel Alpay.
On the Bessmertny class of homogeneous
positive holomorphic functions
on the Cartesian product of matrix half-planes.
Dmitry Kalyuzhny-Verbovetzki
We study the class of holomorphic operator-valued functions on the
Cartesian product of N half-planes which admit the representation
in the form of Schur complement of a block of a linear homogeneous
pencil of operator block matrices with positive semidefinite
coefficients. M. F. Bessmertny showed that in the
finite-dimensional case the class of (rational) n×n
matrix-valued functions which admit such representations (in his
work they were called long resolvent representations) coincides
with the class of characteristic matrix functions of passive
2n-poles considered as functions of impedances z1, ¼,zN of their elements. We give several descriptions of the
general Bessmertny class and show the close relation of the
latter to the Agler-Schur class of holomorphic functions on the
unit polydisk. Then we extend these descriptions, as well as long
resolvent representations, to the case of holomorphic functions on
the Cartesian product of N matrix half-planes. Our motivation
comes also from electrical engineering where sometimes matrix
impedances Z1, ¼, ZN are considered as independent
variables. We show the close relation of this extended
Bessmertny class to the recent works of Ambrozie-Timotin
and Ball-Bolotnikov on the Agler-Schur class of holomorphic
functions on the domain defined by the inequality
||P(z1,¼,zd)|| < 1 where P(z1,¼,zd) is a matrix
polynomial.
Toeplitz operators on Dirichlet-type spaces.
H. Turgay Kaptanoglu
We define and study the basic properties of the Toeplitz operators on certain Dirichlet-type spaces on the unit ball of \mathbb CN. These spaces are a family of reproducing kernel Hilbert spaces with reproducing kernel (1- < z,w > )(-N+1+q) for q > -(N+1) and include Arveson, Hardy, and Bergman spaces. Our results generalize those known especially for Bergman spaces. This is joint work with Daniel Alpay.
Tau functions of rational solutions of the Schlesinger system.
Victor Katsnelson
An explicit expression for rational solutions of the Schlesinger system
is obtained.
This expression is dan in terms of the system representation of
rational matrix functions.
The role of the Lyapunov matrix as the tau function of the rational
solution is highlighted.
Convex invertible cones, matrix sign function and the
Navanlinna-Pick interpolation problem
Izchak Lewkowicz
Let A be a matrix whose spectrum avoids the imaginary axis. It is well known that at least from the computational point of view, there is an interest in obtaining Sign(A). For example, if one wishes to solve the algebraic Lyapunov or Riccati equations, A has a certain Hamiltonian structure. The Matrix Sign Function Algorithm (MSFA) is the better known way to compute Sign(A): For j = 0, 1, ¼ set fo(s) = s and fj+1(s): = qjfj(s)+[(1-qj)/(fj(s))] with qj Î (0, 1). Thus fj(A)
One can view the MSFA as aimed at simultaneously mapping (in C) a (possibly large) neighborhood of each eigenvalue of A, lk, to a vicinity of Sign(lk). Namely, a variant of an interpolation problem. This suggests the enlargement of the class of the mapping functions from the above fj(s) to all Rational Positive Reals, trying to gain the following advantages: (i) to facilitate incorporation of partial data on the spectrum of A, (ii) to reduce the degree of the mapping function and (iii) due to a known parameterization, a reduction in both the computational effort and the error caused by the large number of ``nested" inverses involved in the MFSA.
Infinite Systems of Functional Differential Equations in Frechet
Spaces.
Elena Litsyn
Infinite systems of ordinary differential equations are usually studied in the frames of the theory of differential equations in Banach spaces. An evident advantage of this approach is a possibility of application of functional analysis methods. At the same time, necessity of imposing apriori restrictions on mutual dependence of the solutions' components leads to some losses. Nevertheless, as we will show in our lecture, it is possible to select such a class of infinite system of functional differential equations for which the mentioned restrictions can be overcome.
A Hilbert space approach for strict/wide-sence self-similar
systems
Mamadou Mboup
In a previous work we have investigated some structural properties of continuous time self-similar systems. The self-similar property was interpreted in terms of invariance of the corresponding transfer function space to a given transformation, in a same way as the time invariance property is related to the shift-invariance of the Hardy spaces. A self-similar systems was thus defined as a system whose transfert function belongs to a de Branges homogeneous space. The advantage of this approach is to consider any self-similar process as the ouput of a linear system driven by a white noise. This talk will extend these ideas in the discrete time case. Definition of strict and wide-sense discrete time self-similar systems will be proposed in the framework of Hardy spaces of character-automorphic functions. This talk is based on a joint work with Daniel Alpay.
Matrix completions: open problems.
Leiba Rodman
Various open problems concerning matrix and operator completions
with prescribed properties will be discussed, with emphasis on
multidimensional problems.
Explicit formulas for the defining polynomial of a quadrature domain.
Alexander Shapiro
Let us consider a quadrature domain defined as the image of the unit disc under the action of a conformal function f(z). Using f(z) we construct two rational functions and consider a rational mapping of \mathbb C into \mathbb C2 under the action of these two rational functions. The notion of determinantal representation of an algebraic curve allows us to find the explicit formula for the image of the complex plane. This image is defined by a certain polynomial and this polinomial also defines the quadrature domain.
On some function spaces generated by meta-harmonic functions in the
plane.
Michael Shapiro
In a series of papers by the author and his collaborators including a research book, there
were constructed the fundamentals of the theory of quaternion-valued functions which generalizes
that of usual holomorphic functions and which is related to the Helmholtz, not the Laplace,
equation. In the talk, there will be presented a study of Sobolev-type spaces generated by
such functions which belong additionally to a weighted Lebesgue space together with their
radial and angular derivatives. The resulting right quaternionic Banach spaces are considered
and discussed. In case of a Hilbert space, its reproducing kernel will be given; the latter
proves to be an interesting object in a non-Hilbert case also and its properties will be
explained as well.
This work was partially supported by CONACYT projects and by Instituto Politécnico Nacional via COFAA and CGPi programs.
Evolution equations and geometric
function theory on the operator ball and J*-algebra1
2
David Shoikhet
The talk is based on several joint works with Mark Elin, Lawrence Harris
and Simeon Reich.
In a series of papers [FK-78, 79, 82, 86, 88] Ky Fan (see also
[AT-FK-79]) developped a geometric theory of holomorphic functions of proper
contractions on Hilbert spaces in the sense of the functional calculus. His
theorems and elegant constructions cover some classical results of functional
analysis (for example, a theorem of Von Neumann), as well as many results
of complex function theory (for example, Wolff's theorem
and Harnack's inequality). Ando and Fan also proved several general
operator inequalities in the spirit of Pick and Julia which yield the above
mentioned results. These results are a powerful tool in the functional
calculus as well as in the study of the discrete-time semigroups of
l-analytic functions defined by the Riesz-Dunford integral on the space
L(H) of bounded operators on a Hilbert space H.
In particular, Ky Fan studied some properties of holomorphic functions
of proper contractions produced by univalent star-like complex valued
functions (see [FK-78]).
In a parallel development [STJ-77] (see also [GKR-75] and
[HLF-STJ-79]) T.J. Suffridge defined spiral-lik mappings in general
Banach spaces.
However, such properties as star-likeness with respect to a boundary
point and spiral-likeness were not investigated at all.
Another look at the geometric operator theory is based on the theory of linear
semigroups in locally convex spaces in the spirit of Hille-Yosida (see
[EM-HL-RS-SD] and [EM-SD]). It enables us to study spiral-shaped domains with
respect to a boundary point, as well as to solve the Koenigs embedding
problem for hyperbolic operator functions and the eigenvalue problem for
composition operators in the subcritical case.
All our considerations are carried out in the framework of the so-called
J*-algebras which include, in particular, C*-algebras and
Cartan's factors.
References
[AT-FK-79] T. Ando and K. Fan, Pick-Julia theorem for operators,
Math. Z. 168 (1979), 23-34.
[EM-HL-RS-SD-02] M. Elin, L. Harris, S. Reich and D. Shoikhet,
Evolution equations and geometric function theory in J*-algebras.
Journal of nonlinear and convex analysis, vol. 3 I, (2002), p. 81-121.
[EM-SD-02] M. Elin and D. Shoikhet, Univalent functions of proper contractions
spirallike with respect to a boundary point, Multidimensional Complex
Analysis (2002), p. 28-36.
[FK-78] K. Fan, Analytic functions of a proper contraction, Math. Z.,
160 (1978), 275-290.
[FK-79] K. Fan, Julia's lemma for operators, Math. Ann. 239
(1979), 241-245.
[FK-82] K. Fan, Iterations of analytic functions of operators, Math. Z., 179 (1982)
[FK-86] K. Fan, The angular derivative of an operator-valued analytic
functions, Pacific Jour. Math. 121 (1986), 67-72.
[FK-88] K. Fan, Inequalties for proper contractions and strictly dissipative
operators, Linear Algebra Appl. 105 (1988), 237-248.
[GKR-75] K.R. Gurganus, F-like holomorphic functions in \mathbb Cn
and Banach spaces, Trans. Americ. Math. Soc. 205 (1975),
389-406.
[HLA-71] L.A. Harris, The numerical range of holomorphic functions in
Banach spaces, Amer. J. Math. 93 (1971), 1005-1019.
[HLA-RS-SD-99] L.A. Harris, S. Reich and D. Shoikhet, Dissipative holomorphic
functions, Blocj radii, and the Schwartz Lemma, J. Analyse Math.,
vol. 82, (2001), 221-232.
[STJ-77] T.J. Suffridge, Starlikeness, convexity and other geometric
properties of holomorphic mapps in higher dimensiions, Complex Analysis
(Proc. Conf. Univ. Kentucky, Lexington, KY, 1976), Lecture Notes in math.
599 (1977) 146-159.
[SD-01] D. Shoikhet, Semi-groups in Geometric Function Theory, Kluwer Academic publishers, 2001, Dortdrecht/Boston/London, 222p.
Duality of correspondences and a noncommutative Nevanlinna-Pick
theorem.
Baruch Solel
I shall describe the construction of non selfadjoint operator algebras that are associated with certain bimodules (=correspondences). These algebras generalize the algebra H¥ of the unit circle (and also algebras studied by Davidson and Pitts and by Popescu). I will explain the notion of duality for these bimodules and present an interpolation Nevanlinna-Pick theorem. The theorem generalizes the classical one as well as an interpolaton result of Popescu. This is a joint work with Paul Muhly.
Three term recurrence relation modulo an ideal and orthogonality of polynomials in several
variable.
FH Szafraniec
An effort has been made to include orthogonality of polynomials in several variables with respect to measure supported by an algebraic set into the general framework existing so far (joint work with Dariusz Cicho\'n and Jan Stochel). The talk aims at presenting main ideas of the approach.
Positive functionals in spaces of fractions.
F. Vasilescu
Let W be a compact space, let C(W) be the algebra
of all complex-valued continuous functions on W and let
Q Ì C(W) be a family of nonnull functions such that
1 Î Q, if q Î Q then [`q] Î Q, and q¢,q¢¢ Î Q
implies q¢q¢¢ Î Q. We consider the algebra of fractions
C(W)/Q and study linear functionals defined on some subspaces of
C(W)/Q, which extend to positive linear
functionals, the latter having always integral representations. Applications
to moment problems on unbounded sets are also given.
Point evaluation and Hardy space on an homogeneous tree
Dan Volok
In their works on the multiscale system theory, Basseville, Benveniste, Nikoukhah and Willsky have introduced the notion of stationary transfer functions in the case when the discrete time is replaced by the nodes of an homogeneous tree. Such a function is an operator, acting on the space of sequences, indexed by the nodes of the tree, and commuting with tree isometries. It can be written as a formal power series in one indeterminant (a primitive shift), with coefficients from a C*-algebra of operators. Since the coefficients do not commute, in general, with the indeterminant, this setting is quite different from the classical stationary case. On the other hand, there is a deep analogy with the classical non stationary setting, as treated by Alpay, Dewilde and Dym, since an upper-triangular operator from l2(\mathbb Z) into itself can be formally written as a power series in the primitive shift, with coefficients being diagonal operators. We exploit this analogy to define in the stationary setting on an homogeneous tree a point evaluation with values in the above-mentioned C*-algebra of ``scalars'', the corresponding ``Hardy space'' which is now a non self-dual reproducing kernel Hilbert module, and the counterpart of other classical notions such as Blaschke factors. This is joint work with D. Alpay.
1 1991 Mathematics Subject classification: 46G20, 47H06, 47H20
2 Key words and phrases: functional calculus, holomorphically accretive mappings, linear and nonlinear semigroups, proper contractions, spiral-shaped set.