An Abstract Interpolation Problem and the Extension Theory of Hermitian Operators

TL;DR

This paper explores a general interpolation problem using extension theory of Hermitian operators, connecting classical and recent problems through a unified algebraic and functional model framework.

Contribution

It formulates a comprehensive Abstract Interpolation Problem and describes its solutions via the Arov-Grossman formula, advancing the extension theory approach.

Findings

Unified framework for classical and recent interpolation problems

Solution characterization using Arov-Grossman formula

Extension theory of isometric operators as a key tool

Abstract

The algebraic structure of V.P. Potapov's Fundamental Matrix Inequality (FMI) is discussed and its interpolation meaning is analyzed. Functional model spaces are involved. A general Abstract Interpolation Problem is formulated which seems to cover all the classical and recent problems in the field and the solution set of this problem is described using the Arov--Grossman formula. The extension theory of isometric operators is the proper language for treating interpolation problems of this type.