Orthogonal polynomials in de Branges--Rovnyak spaces

TL;DR

This paper investigates the structure of de Branges--Rovnyak spaces using orthogonal polynomials, especially for rational functions, and examines invariance properties under certain composition operators.

Contribution

It introduces new representations of de Branges--Rovnyak spaces through orthogonal polynomials and analyzes invariance under composition operators for rational cases.

Findings

Identified relevant structures in orthogonal polynomials for rational $b$.

Established invariance properties under composition operators.

Provided new descriptions of $H(b)$ spaces via orthogonal polynomials.

Abstract

Given a function $b$, holomorphic on the disc and bounded by 1, one can construct an associated reproducing kernel Hilbert space called the de Branges--Rovnyak space $H(b)$. We explore representations of such spaces via descriptions of the corresponding families of orthogonal polynomials. We find relevant structures in the linear systems involved in a diversity of cases when $b$ is rational. We also establish a form of invariance under some composition operators on $H(b)$ spaces.