Generalized de Branges-Rovnyak spaces

Given the reproducing kernel k of the Hilbert space H-k we study spaces H-k(b) whose reproducing kernel has the form k (1 - bb & lowast;), where b is a row-contraction on H-k. In terms of reproducing kernels this is 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 [31] in one or several variables. We study some general properties of H-k(b) e.g. when the inclusion map into H-k is compact. Our main result provides a model for H-k(b) reminiscent of the Sz.-Nagy-Foia & scedil; model for contractions (see also [7]). As an application we obtain sufficient conditions for the containment and density of the linear span of {k(w) : w is an element of X} in H-k(b). In the standard cases this reduces to containment and density of polynomials. These methods resolve a very recent conjecture [13] regarding polynomial approximation in spaces with kernel (1-b(z)b(w)& lowast;)(m)/(1-zw)(beta), 1 <= m < beta, m is an element of N. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).