  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.
