On an Interpolation Problem for J-Potapov Functions

TL;DR

This paper addresses a complex interpolation problem for J-Potapov functions, providing a comprehensive characterization of solutions, especially in the non-degenerate case, and analyzing the behavior of associated Weyl matrix balls.

Contribution

It offers the most general solution to the interpolation problem for J-Potapov functions and describes Weyl matrix balls in the non-degenerate case.

Findings

Characterization of the solution set for the interpolation problem.

Description of Weyl matrix balls in the non-degenerate case.

Analysis of the limit behavior of Weyl matrix balls.

Abstract

Let, J, be an m-by-m-signature matrix and let D be the open unit disk in the complex plane. Denote by P{J,0}(D) the class of all meromorphic m-by-m-matrix-valued functions, f, in D which are holomorphic at 0 and take J-contractive values at all points of D at which f is holomorphic. The central theme of this paper is the study of the following interpolation problem: Let n be a nonnegative integer, and let A_0, A_1, ..., A_n be a sequence of complex m-by-m-matrices. Describe the set of all matrix-valued functions, f, belonging to the class P{J,0}(D), such that the first n+1 Taylor coefficients of f coincide with A_0, A_1, ..., A_n. In particular, we characterize the case that this set is non-empty. In this paper, we will solve this problem in the most general case. Moreover, in the non-degenerate case we will give a description of the corresponding Weyl matrix balls. Furthermore, we will investigate the limit behaviour of the Weyl matrix balls associated with the functions belonging to some particular subclass of P{J,0}(D).