{
  "id": "1610.01978",
  "version": "v1",
  "published": "2016-10-06T18:12:59.000Z",
  "updated": "2016-10-06T18:12:59.000Z",
  "title": "Properties of operators related to a $Γ_3$-contraction and unitary invariants",
  "authors": [
    "Sourav Pal"
  ],
  "comment": "14 pages, First version, Few results yet to be added. arXiv admin note: text overlap with arXiv:1312.0322 by other authors",
  "categories": [
    "math.FA"
  ],
  "abstract": "The closed 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. For a $\\Gamma_3$-contraction $(S_1,S_2,P)$ there are two unique operators $A_1,A_2$ such that \\[ S_1-S_2^*P=D_PA_1D_P\\;,\\; S_2-S_1^*P=D_PA_2D_P. \\] The operator pair $(A_1,A_2)$ plays central role in determining the structure of a $\\Gamma_3$-contraction. We shall discuss various properties of the fundamental operator pairs of $\\Gamma_3$-contractions. For two operator pairs $(A_1,A_2)$ and $(B_1,B_2)$ we provide conditions under which there exists a $\\Gamma_3$-contraction $(S_1,S_2,P)$ such that $(A_1,A_2)$ is the fundamental operator pair of $(S_1,S_2,P)$ and $(B_1,B_2)$ is the fundamental operator pair of its adjoint $(S_1^*,S_2^*,P^*)$. We shall show that such fundamental operator pair plays pivotal role in determining a set of unitary invariants for the $\\Gamma_3$-contractions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-06T18:12:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "contraction",
      "unitary invariants",
      "fundamental operator pair plays pivotal",
      "operator pair plays pivotal role",
      "properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}