Generalized de Branges-Rovnyak spaces

Abstract

Given the reproducing kernel $k$ of the Hilbert space ${\mathcal{H}}_k$ we study spaces ${\mathcal{H}}_k(b)$ whose reproducing kernel has the form $k(1-b{b}^{\ast })$, where $b$ is a row-contraction on ${\mathcal{H}}_k$. 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 ${\mathcal{H}}_k(b)$ e.g. when the inclusion map into $\mathcal{H}$ is compact. Our main result provides a model for ${\mathcal{H}}_k(b)$ reminiscent of the Sz.-Nagy-Foiaş model for contractions. As an application we obtain sufficient conditions for the containment and density of the linear span of $\{k_w:w\in \mathcal{X}\}$ in ${\mathcal{H}}_k(b)$. 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 $\frac{(1-b(z)b(w{)}^{\ast }{)}^m}{(1-z\overline{w}{)}^2},1\le m<{\hat{I}}^2,m\in \mathbb{N}$.