{
  "id": "2002.01298",
  "version": "v1",
  "published": "2020-02-04T14:26:13.000Z",
  "updated": "2020-02-04T14:26:13.000Z",
  "title": "$\\mathbb K$-homogeneous tuple of operators on bounded symmetric domains",
  "authors": [
    "Soumitra Ghara",
    "Surjit Kumar",
    "Paramita Pramanick"
  ],
  "comment": "17 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Omega$ be an irreducible bounded symmetric domain of rank $r$ in $\\mathbb C^d.$ Let $\\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\\Omega$. The group $\\mathbb K$ consisting of linear transformations acts naturally on any $d$-tuple $\\boldsymbol T=(T_1,\\ldots, T_d)$ of commuting bounded linear operators. If the orbit of this action modulo unitary equivalence is a singleton, then we say that $\\boldsymbol T$ is $\\mathbb{K}$-homogeneous. In this paper, we obtain a model for all $\\mathbb{K}$-homogeneous $d$-tuple $\\boldsymbol{T}$ as the operators of multiplication by the coordinate functions $z_1,\\ldots ,z_d$ on a reproducing kernel Hilbert space of holomorphic functions defined on $\\Omega$. Using this model we obtain a criterion for (i) boundedness, (ii) membership in the Cowen-Douglas class (iii) unitary equivalence and similarity of these $d$-tuples. In particular, we show that the adjoint of the $d$-tuple of multiplication by the coordinate functions on the weighted Bergman spaces are in the Cowen-Douglas class $B_1(\\Omega)$. For a bounded symmetric domain $\\Omega$ of rank $2$, an explicit description of the operator $\\sum_{i=1}^d T_i^*T_i$ is given. In general, based on this formula, we make a conjecture giving the form of this operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-04T14:26:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bounded symmetric domain",
      "homogeneous tuple",
      "cowen-douglas class",
      "coordinate functions",
      "action modulo unitary equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}