Abstract Interpolation Problem and Some Applications. II: Coefficient Matrices

TL;DR

This paper extends classical theorems in the Abstract Interpolation Problem by analyzing residual parts of minimal unitary extensions, revealing boundary properties of coefficient matrices and their relation to problem data structure.

Contribution

It generalizes the Nevanlinna-Adamjan-Arov-Krein theorem and links properties of coefficient matrices to the denseness of sets in function model spaces.

Findings

Residual parts of minimal unitary extensions determine boundary properties of coefficient matrices

Generalization of classical interpolation theorems

Structure of dense sets reflects problem data structure

Abstract

The main content of this paper is Lectures 5 and 6 that continue lecture notes [20]. Content of Lectures 1-4 of [20] is reviewed for the reader's convenience in sections 1-4, respectively. It is shown in Lecture 5 how residual parts of the minimal unitary extensions, that correspond to solutions of the problem, yield some boundary properties of the coefficient matrix-function. These results generalize the classical Nevanlinna - Adamjan - Arov~- Krein theorem. Lecture 6 discusses how further properties of the coefficient matrices follow from denseness of certain sets in the associated function model spaces. The structure of the dense set reflects the structure of the problem data.