{
  "id": "2405.07016",
  "version": "v1",
  "published": "2024-05-11T13:48:22.000Z",
  "updated": "2024-05-11T13:48:22.000Z",
  "title": "Generalized de Branges-Rovnyak spaces",
  "authors": [
    "Alexandru Aleman",
    "Frej Dahlin"
  ],
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "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 $\\frac{(1-b(z)b(w)^*)^m}{(1-z\\overline w)^\\beta}, 1\\leq m<\\beta, m\\in\\mathbb{N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-11T13:48:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "branges-rovnyak spaces",
      "reproducing kernel",
      "conjecture regarding polynomial approximation",
      "study spaces",
      "generalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}