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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.


This is the originally-submitted version of an article that later underwent peer review and was accepted for publication in Linear Algebra and Its Applications, volume 633, in 2021. The definitive publisher-authenticated version may differ and is available online at



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