{
  "id": "1202.3841",
  "version": "v4",
  "published": "2012-02-17T05:12:04.000Z",
  "updated": "2015-07-28T00:54:09.000Z",
  "title": "A functional model for pure $Γ$-contractions",
  "authors": [
    "Tirthankar Bhattacharyya",
    "Sourav Pal"
  ],
  "comment": "Journal of Operator Theory, 71 (2014), 327-339",
  "categories": [
    "math.FA"
  ],
  "abstract": "A pair of commuting operators $(S,P)$ defined on a Hilbert space $\\mathcal H$ for which the closed symmetrized bidisc $$ \\Gamma= \\{(z_1+z_2,z_1z_2):: |z_1|\\leq 1,\\, |z_2|\\leq 1 \\}\\subseteq \\mathbb C^2, $$ is a spectral set is called a $\\Gamma$-contraction in the literature. A $\\Gamma$-contraction $(S,P)$ is said to be pure if $P$ is a pure contraction, i.e, ${P^*}^n \\rightarrow 0$ strongly as $n \\rightarrow \\infty $. Here we construct a functional model and produce a complete unitary invariant for a pure $\\Gamma$-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation $$ S-S^*P=D_PXD_P, \\textup{where} X\\in \\mathcal B(\\mathcal D_P), $$and is called the fundamental operator of the $\\Gamma$-contraction $(S,P)$. We also discuss some important properties of the fundamental operator.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-06-07T07:26:27.000Z",
      "comment": "11 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-07-28T00:54:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "functional model",
      "fundamental operator",
      "complete unitary invariant",
      "spectral set",
      "important properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.3841B"
    }
  }
}