{
  "id": "0907.0026",
  "version": "v4",
  "published": "2009-06-30T21:04:48.000Z",
  "updated": "2010-04-09T09:05:52.000Z",
  "title": "Contractive Hilbert modules and their dilations",
  "authors": [
    "Ronald G. Douglas",
    "Gadadhar Misra",
    "Jaydeb Sarkar"
  ],
  "comment": "17 pages. Title changed, Improved presentation, Typos corrected. To appear in the Israel Journal of Mathematics",
  "categories": [
    "math.FA",
    "math.CV",
    "math.OA"
  ],
  "abstract": "In this note, we show that a quasi-free Hilbert module R defined over the polydisk algebra with kernel function k(z, w) admits a unique minimal dilation (actually an isometric co-extension) to the Hardy module over the polydisk if and only if S^{-1}(z, w) k(z, w) is a positive kernel function, where S(z, w) is the Szeg\\\"{o} kernel for the polydisk. Moreover, we establish the equivalence of such a factorization of the kernel function and a positivity condition, defined using the hereditary functional calculus, which was introduced earlier by Athavale \\cite{Ath} and Ambrozie, Englis and M\\\"{u}ller. An explicit realization of the dilation space is given along with the isometric embedding of the module R in it. The proof works for a wider class of Hilbert modules in which the Hardy module is replaced by more general quasi-free Hilbert modules such as the classical spaces on the polydisk or the unit ball in {C}^m. Some consequences of this more general result are then explored in the case of several natural function algebras.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-04-09T09:05:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "47A20",
      "46E20",
      "46E22",
      "46M20",
      "47B32"
    ],
    "keywords": [
      "contractive hilbert modules",
      "kernel function",
      "general quasi-free hilbert modules",
      "hardy module",
      "natural function algebras"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0026D"
    }
  }
}