{
  "id": "1405.2980",
  "version": "v1",
  "published": "2014-05-12T22:01:53.000Z",
  "updated": "2014-05-12T22:01:53.000Z",
  "title": "de Branges-Rovnyak spaces: basics and theory",
  "authors": [
    "Joseph A. Ball",
    "Vladimir Bolotnikov"
  ],
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "For $S$ a contractive analytic operator-valued function on the unit disk ${\\mathbb D}$, de Branges and Rovnyak associate a Hilbert space of analytic functions ${\\mathcal H}(S)$ and related extension space ${\\mathcal D(S)}$ consisting of pairs of analytic functions on the unit disk ${\\mathbb D}$. This survey describes three equivalent formulations (the original geometric de Branges-Rovnyak definition, the Toeplitz operator characterization, and the characterization as a reproducing kernel Hilbert space) of the de Branges-Rovnyak space ${\\mathcal H}(S)$, as well as its role as the underlying Hilbert space for the modeling of completely non-isometric Hilbert-space contraction operators. Also examined is the extension of these ideas to handle the modeling of the more general class of completely nonunitary contraction operators, where the more general two-component de Branges-Rovnyak model space ${\\mathcal D}(S)$ and associated overlapping spaces play key roles. Connections with other function theory problems and applications are also discussed. More recent applications to a variety of subsequent applications are given in a companion survey article.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-12T22:01:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "branges-rovnyak space",
      "unit disk",
      "analytic functions",
      "non-isometric hilbert-space contraction operators",
      "reproducing kernel hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2980B"
    }
  }
}