{
  "id": "1804.08492",
  "version": "v1",
  "published": "2018-04-23T15:12:14.000Z",
  "updated": "2018-04-23T15:12:14.000Z",
  "title": "Abstract interpolation in vector-valued de Branges-Rovnyak spaces",
  "authors": [
    "Joseph A. Ball",
    "Vladimir Bolotnikov",
    "Sanne ter Horst"
  ],
  "comment": "32 pages",
  "journal": "Integral Equation and Operator Theory 70 (2011), 227-263",
  "doi": "10.1007/s00020-010-1844-1",
  "categories": [
    "math.FA"
  ],
  "abstract": "Following ideas from the Abstract Interpolation Problem of Katsnelson et al. (Operators in spaces of functions and problems in function theory, vol 146, pp 83-69, Naukova Dumka, Keiv, 1987) for Schur class functions, we study a general metric constrained interpolation problem for functions from a vector-valued de Branges-Rovnyak space $\\mathcal{H}(K_S)$ associated with an operator-valued Schur class function $S$. A description of all solutions is obtained in terms of functions from an associated de Branges-Rovnyak space satisfying only a bound on the de Branges-Rovnyak-space norm. Attention is also paid to the case that the map which provides this description is injective. The interpolation problem studied here contains as particular cases (1) the vector-valued version of the interpolation problem with operator argument considered recently in Ball et al. (Proc Am Math Soc 139(2), 609-618, 2011) (for the nondegenerate and scalar-valued case) and (2) a boundary interpolation problem in $\\mathcal{H}(K_S)$. In addition, we discuss connections with results on kernels of Toeplitz operators and nearly invariant subspaces of the backward shift operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-23T15:12:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E22",
      "47A57",
      "30E05"
    ],
    "keywords": [
      "branges-rovnyak space",
      "general metric constrained interpolation problem",
      "boundary interpolation problem",
      "operator-valued schur class function",
      "abstract interpolation problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}