Interpolation in de Branges-Rovnyak spaces

TL;DR

This paper investigates a general interpolation problem within de Branges-Rovnyak spaces, focusing on operator-argument conditions and their implications for function theory and operator models.

Contribution

It introduces a new interpolation framework for functions in de Branges-Rovnyak spaces involving operator arguments and provides a novel characterization of solutions.

Findings

Established a generalized interpolation condition in de Branges-Rovnyak spaces.

Connected the interpolation problem to operator theory and function models.

Provided conditions for the existence of solutions to the interpolation problem.

Abstract

A general interpolation problem with operator argument is studied for functions f from the de Branges-Rovnyak space H(s) associated with an analytic function s mapping the open unit disk D into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form f(A)c = b (with a suitable interpretation of f(A)c) for given Hilbert space operator A and two vectors b; c from the same space.