Bernd Kirstein

On the theory of matrix-valued functions belonging to the Smirnov class

A theory of matrix-valued functions from the matricial Smirnov class is systematically developed. In particular, the maximum principle of V.I. Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman’s theorem are presented for matrix-valued functions from the Smirnov class. We also consider a family of functions belonging to the matricial Smirnov class which is indexed by a complex parameter λ. We show that with the exception of a “very small” set of such λ the corresponding inner factor in the inner-outer factorization of the function Fλ is a Blaschke-Potapov product.

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System

Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form

On a moment problem for rational matrix-valued functions

{\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}

Topics in interpolation theory

, where

Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

{\phi}

Recovery of the Dirac system from the rectangular Weyl matrix function

is a strictly proper rational matrix function and D = D* ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case.