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.
Citation for published, peer-reviewed article:
D. Alpay, F. Colombo, I. Sabadini, and B. Schneider. Beurling-Lax type theorems and Cuntz relations. Linear Algebra and Its Applications, 633:152-212, 2021. https://doi.org/10.1016/j.laa.2021.10.008