{
  "id": "2408.03711",
  "version": "v1",
  "published": "2024-08-07T11:55:04.000Z",
  "updated": "2024-08-07T11:55:04.000Z",
  "title": "Representations of the Möbius group and pairs of homogeneous operators in the Cowen-Douglas class",
  "authors": [
    "Jyotirmay Das",
    "Somnath Hazra"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let M\\\"ob be the biholomorphic automorphism group of the unit disc of the complex plane, $\\mathcal{H}$ be a complex separable Hilbert space and $\\mathcal{U}(\\mathcal{H})$ be the group of all unitary operators. Suppose $\\mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc $\\mathbb D^n$ and contains all the polynomials. If $\\pi : \\mbox{M\\\"ob} \\to \\mathcal{U}(\\mathcal{H})$ is a multiplier representation, then we prove that there exist $\\lambda_1, \\lambda_2, \\ldots, \\lambda_n > 0$ such that $\\pi$ is unitarily equivalent to $(\\otimes_{i=1}^{n} D_{\\lambda_i}^+)|_{\\mbox{M\\\"ob}}$, where each $D_{\\lambda_i}^+$ is a holomorphic discrete series representation of M\\\"ob. As an application, we prove that if $(T_1, T_2)$ is a M\\\"ob - homogeneous pair in the Cowen - Douglas class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each $T_i$, the diagonal operators are identified. We also prove that if $\\mathcal{H}$ consists of symmetric (resp. anti-symmetric) holomorphic functions over $\\mathbb D^2$ and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists $\\lambda > 0$ such that $\\pi \\cong \\oplus_{m = 0}^\\infty D^+_{\\lambda + 4m}$ (resp. $\\pi \\cong \\oplus_{m=0}^\\infty D^+_{\\lambda + 4m + 2}$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T11:55:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cowen-douglas class",
      "möbius group",
      "homogeneous operators",
      "upper triangular form",
      "kernel hilbert space consisting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}