Vladimir Bolotnikov

On boundary interpolation for matrix valued Schur functions

Introduction Preliminaries Fundamental matrix inequalities On

On Degenerate Interpolation, Entropy and Extremal Problems for Matrix Schur Functions

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On an operator approach to interpolation problems for Stieltjes functions

spaces Parametrizations of all solutions The equality case Nontangential limits The Nevanlinna-Pick boundary problem A multiple analogue of the Caratheodory-Julia theorem On the solvability of a Stein equation Positive definite solutions of the Stein equation A Caratheodory-Fejer boundary problem The full matrix Caratheodory-Fejer boundary problem The lossless inverse scattering problem Bibliography.

The Higher Order Carathéodory—Julia Theorem and Related Boundary Interpolation Problems

We consider a general bitangential interpolation problem for matrix Schur functions and focus mainly on the case when the associated Pick matrix is singular (and positive semidefinite). Descriptions of the set of all solutions are given in terms of special linear fractional transformations which are obtained using two quite different approaches. As applications of the obtained results we consider the maximum entropy and the maximum determinant extension problems suitably adapted to the degenerate situation.

Interpolation for multipliers on reproducing kernel Hilbert spaces

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapovs method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.

On a bitangential interpolation problem for contractive-valued functions on the unit ball☆

The higher order analogue of the classical Caratheodory-Julia theorem on boundary angular derivatives has been obtained in [7]. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result.

Multivariable backward-shift-invariant subspaces and observability operators

All solutions of a tangential interpolation problem for contractive multipliers between two reproducing kernel Hilbert spaces of analytic vector-valued functions are characterized in terms of certain positive kernels. In a special important case when the spaces consist of analytic functions on the unit ball of C d and the reproducing kernels are of the form (1 - (z, w) -1 )I p and (1 - (z, w) -1 Iq, the characterization leads to a pararnetrization of the set of all solutions in terms of a linear fractional transformation.

The Schur algorithm and reproducing kernel Hilbert spaces in the ball

We solve the bitangential interpolation problem with a finite number of interpolation nodes for a multivariable analogue of the Schur class consisting of matrix-valued analytic functions on the ball. The interpolation conditions are formulated via a generalized functional calculus with operator argument, thereby generalizing in a compact way the simple, first-order interpolation conditions considered for this class of functions in earlier work. The description of all solutions is given via a Redheffer transform whose entries are given explicitly in terms of the interpolation data.

Two-Sided Interpolation for Matrix Functions With Entries in the Hardy Space

It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator. We discuss two multivariable extensions of this structure, where the classical Hardy space is replaced by (1) the Fock space of formal power series in a collection of d noncommuting indeterminates with norm-square-summable vector coefficients, and (2) the reproducing kernel Hilbert space (often now called the Arveson space) over the unit ball in