Generalized de Branges-Rovnyak spaces

Given the reproducing kernel k of the Hilbert space $H_k$ we study spaces $H_k\left(b\right)$ whose reproducing kernel has the form $k(1-b{b}^{⁎})$, 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 $H_k\left(b\right)$ e.g. when the inclusion map into $H_k$ is compact. Our main result provides a model for $H_k\left(b\right)$ reminiscent of the Sz.-Nagy-Foiaş model for contractions (see also [7]). As an application we obtain sufficient conditions for the containment and density of the linear span of $\{k_w:w\in X\}$ in $H_k\left(b\right)$. 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 $\frac{{(1-b(z\left)b{\left(w\right)}^{⁎}\right)}^m}{{(1-z\bar{w})}^{\beta }},1\le m<\beta ,m\in N$.