The Spectral Carathéodory-Fejér Problem

The problem of the title is to construct an analytic k × k matrix-valued function in the unit disc with a number of prescribed derivatives at 0 and with spectral radius bounded by 1. We show that the problem can be reduced to an interpolation problem for the symmetrized polydisc , and thereby show that, in the case of derivatives of orders 0 and 1 being prescribed, the problem is equivalent to the infinitesimal Kobayashi extremal problem for , which is solved completely in the case k = 2.     

Mixed Löwner and Nevanlinna-Pick Interpolation

A mixed type, L?wner and Nevanlinna-Pick directional two-sided interpolation problem is considered. A necessary and sufficient condition for the problem to have a solution is established, in terms of properties of the Pick kernel to the problem. As well, a parametrization of the set of all real rational solutions of minimal degree is given. The corresponding Nevanlinna-Pick boundary-interior interpolation problem is also considered and a solvability condition for it is obtained. The approach to the problem is via functional Hilbert spaces.     

A Bitangential Interpolation Problem on the Closed Unit Ball for Multipliers of the Arveson Space

We solve a bitangential interpolation problem for contractive multipliers on the Arveson space with an arbitrary interpolating set in the closed unit ball . The solvability criterion is established in terms of positive kernels. The set of all solutions is parametrized by a Redheffer transform. Submitted: February 2, 2002.     

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

Spectral properties of a compactly perturbed linear span of projections

Spectral properties of the sum of a linear span of projections and a compact nonnegative operator are considered. In particular, we prove partial completeness results for certain parts of the system of eigenfunctions. The main tool is a transformation the original spectral problem to that of a monic weakly hyperbolic pencil.     

A Theorem of Beurling-sLax Type for Hilbert Spaces of Functions Analytic in the Unit Ball

Schur multipliers on the unit ball are operator-valued functions for which the N-variable Schwarz-Pick kernel is nonnegative. In this paper, the coefficient spaces are assumed to be Pontryagin spaces having the same negative index. The associated reproducing kernel Hilbert spaces are characterized in terms of generalized difference-quotient transformations. The connection between invariant subspaces and factorization is established.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

Onl p solutions of discrete Volterra equations

Summary In the paper a discrete analog to the Volterra nonlinear integral equation is discussed. Weighted norms are used to find sufficient conditions that all solutions of such equations are elements of anl ^p space.Some generalizations ofl ^p spaces are also considered and the corresponding sufficient conditions are established.     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

Commutative <Emphasis Type="Italic">C</Emphasis>*-Algebras of Toeplitz Operators on the Unit Ball,I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators

Extending known results for the unit disk, we prove that for the unit ball there exist n+2 different cases of commutative C*-algebras generated by Toeplitz operators, acting on weighted Bergman spaces. In all cases the bounded measurable symbols of Toeplitz operators are invariant under the action of certain commutative subgroups of biholomorphisms of the unit ball. This work was partially supported by CONACYT Projects 46936 and 44620, México.     

The Bitangential Inverse Input Impedance Problem for Canonical Systems, II: Formulas and Examples

This paper continues the study of the bitangential inverse input impedance problem for canonical integral systems that was initiated in [ArD6]. The problem is to recover the system, given an input impedance matrix valued function c() (that belongs to the Carathéodory class of p × p matrix valued functions that are holomorphic and have positive real part in the open upper half plane) and a chain of pairs of entire inner p × p matrix valued functions (that are identified with the associated pairs of the second kind of the matrizant of the system). Formulas for recovering the underlying canonical integral systems are derived by reproducing kernel Hilbert space methods. A number of examples are presented. Special attention is paid to the case when c() is of Wiener class and also when it is both of Wiener class and rational.     

An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL ₂ (, ), where is a curve with cusps of arbitrary order and is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995). Dedicated to Professor Israel Gohberg on the occasion of his 70-th birthday Supported by EPSRC grant GR/K01001     

Extensions with positive real part. A new version of the abstract band method with applications

This paper presents a new version of the abstract band method. The new scheme applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions of the abstract band method.     

A normalization problem for a class of singular integral operators with carleman shift and unbounded coefficients

The solution to a normalization problem for singular integral operators with Carleman shift and degenerate and unbounded coefficients inL _p () is obtained, where is either the unit circle or the real line. The approach followed consists mainly in two steps: the reduction to a singular integral operator with bounded coefficients and the use of the solution to an abstract normalization problem.This research was supported by JNICT under the grant PBIC/C/CEN/1040/92.     

On an Inverse Spectral Problem for a Convolution Integro-differential Operator

The operator of double differentiation perturbed by the composition of a Volterra convolution operator and the differentiation one on a finite interval with Dirichlet boundary conditions is considered. It is proved that the standard asymptotics is necessary and sufficient for an arbitrary sequence of complex numbers to be the spectrum of such an operator, which is determined uniquely. A constructive procedure for solving the inverse problem is given. Received: March 5, 2007.     

Spectral radius properties for layer potentials associated with the elastostatics and hydrostatics equations in nonsmooth domains

By producing a L² convergent Neumann series, we prove the invertibility of the elastostatics and hydrostatics boundary layer potentials on arbitrary Lipschitz domains with small Lipschitz character and 3D polyhedra with large dihedral angles.