Generalized de Branges-Rovnyak spaces

TL;DR

This paper generalizes de Branges-Rovnyak spaces to a broad class of reproducing kernel Hilbert spaces, providing new models and conditions for polynomial approximation, including solutions to recent conjectures.

Contribution

It introduces a unified framework for generalized de Branges-Rovnyak spaces with a Sz.-Nagy-Foia ext{s}-type model and addresses polynomial approximation conjectures.

Findings

Established a model for generalized spaces akin to Sz.-Nagy-Foia ext{s} model.

Provided conditions for the density of polynomial spans in these spaces.

Resolved a recent conjecture on polynomial approximation in specific kernel 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 $k(1-bb^*)$, 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-Foia\c{s} 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 $\frac{(1-b(z)b(w)^*)^m}{(1-z\overline w)^\beta}, 1\leq m<\beta, m\in\mathbb{N}$.