{
  "id": "1610.00936",
  "version": "v1",
  "published": "2016-10-04T11:36:50.000Z",
  "updated": "2016-10-04T11:36:50.000Z",
  "title": "On decomposition of operators having $Γ_3$ as a spectral set",
  "authors": [
    "Sourav Pal"
  ],
  "comment": "Submitted",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "The symmetrized polydisc of dimension three is the set \\[ \\Gamma_3 =\\{ (z_1+z_2+z_3, z_1z_2+z_2z_3+z_3z_1, z_1z_2z_3)\\,:\\, |z_i|\\leq 1 \\,,\\, i=1,2,3 \\} \\subseteq \\mathbb C^3\\,. \\] A triple of commuting operators for which $\\Gamma_3$ is a spectral set is called a $\\Gamma_3$-contraction. We show that every $\\Gamma_3$-contraction admits a decomposition into a $\\Gamma_3$-unitary and a completely non-unitary $\\Gamma_3$-contraction. This decomposition parallels the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $\\Gamma_3$ and $\\Gamma_3$-contractions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-04T11:36:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral set",
      "contraction admits",
      "decomposition parallels",
      "non-unitary contraction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}