{
  "id": "1604.07758",
  "version": "v1",
  "published": "2016-04-26T17:15:50.000Z",
  "updated": "2016-04-26T17:15:50.000Z",
  "title": "Curvature inequalities for operators in the Cowen-Douglas class of a planar domain",
  "authors": [
    "Md. Ramiz Reza"
  ],
  "comment": "18 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Fix a bounded planar domain $\\Omega.$ If an operator $T,$ in the Cowen-Douglas class $B_1(\\Omega),$ admits the compact set $\\bar{\\Omega}$ as a spectral set, then the curvature inequality $\\mathcal K_T(w) \\leq - 4 \\pi^2 S_\\Omega(w,w)^2,$ where $S_\\Omega$ is the S\\\"{z}ego kernel of the domain $\\Omega,$ is evident. Except when $\\Omega$ is simply connected, the existence of an operator for which $\\mathcal K_T(w) = 4 \\pi^2 S_\\Omega(w,w)^2$ for all $w$ in $\\Omega$ is not known. However, one knows that if $w$ is a fixed but arbitrary point in $\\Omega,$ then there exists a bundle shift of rank $1,$ say $S,$ depending on this $w,$ such that $\\mathcal K_{S^*}(w) = 4 \\pi^2 S_\\Omega(w,w)^2.$ We prove that these {\\em extremal} operators are uniquely determined: If $T_1$ and $T_2$ are two operators in $B_1(\\Omega)$ each of which is the adjoint of a rank $1$ bundle shift and $\\mathcal{K}_{T_1}({w}) = -4\\pi ^2 S(w,w)^2 = \\mathcal{K}_{T_2}(w)$ for a fixed $w$ in $\\Omega,$ then $T_1$ and $T_2$ are unitarily equivalent. A surprising consequence is that the adjoint of only some of the bundle shifts of rank $1$ occur as extremal operators in domains of connectivity greater than $1.$ These are described explicitly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-26T17:15:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "curvature inequality",
      "cowen-douglas class",
      "bundle shift",
      "arbitrary point",
      "spectral set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160407758R"
    }
  }
}