2024/05/11 by Alexandru Aleman, Aleman, Alexandru, Frej Dahlin +1 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2405.07016
openalex publication_date 2024/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-Foiaş model for contractions. As an application we obtain sufficient conditions for the containment and density of the linear span of \kw:w\inX\ 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∈ℕ.