Beurling-Lax type theorems and Cuntz relations

TL;DR

This paper extends classical Beurling-Lax theorems by replacing the backward-shift operator with a resolvent operator linked to rational functions, exploring connections to Cuntz relations and applications to de Branges-Rovnyak spaces.

Contribution

It introduces a new representation for analytic functions using composition and multiplication operators tied to rational functions, broadening the scope of classical operator theory.

Findings

Generalized Beurling-Lax theorems with resolvent operators

Established connections between resolvent operators and Cuntz relations

Applied results to de Branges-Rovnyak spaces in indefinite metrics

Abstract

We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.