Generalized de Branges-Rovnyak spaces

Given the reproducing kernel k of the Hilbert space Hₖ we study spaces Hₖ(b) whose reproducing kernel has the form k(1-bb^*), where b is a row-contraction on Hₖ. In terms of reproducing kernels this it the most far-reaching generalization of the classical de Branges-Rovnyaks spaces, as well as their very recent generalization to several variables. This includes the so called sub-Bergman spaces in one or several variables. We study some general properties of Hₖ(b) e.g. when the inclusion map into H is compact. Our main result provides a model for Hₖ(b) reminiscent of the Sz.-Nagy-Foia{s} model for contractions. As an application we obtain sufficient conditions for the containment and density of the linear span of :w\ in Hₖ(b). In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture regarding polynomial approximation in spaces with kernel (1-b(z)b(w)^*)ᵐ/(1-z w)^β, 1≤ m<β, m.