{
  "id": "2205.08962",
  "version": "v1",
  "published": "2022-05-18T14:40:01.000Z",
  "updated": "2022-05-18T14:40:01.000Z",
  "title": "A Family of Homogeneous Operators In The Cowen-Douglas Class Over The Poly-disc",
  "authors": [
    "Prahllad Deb",
    "Somnath Hazra"
  ],
  "comment": "Pages: 14",
  "categories": [
    "math.FA"
  ],
  "abstract": "We construct a large family of positive-definite kernels $K: \\mathbb{D}^n\\times \\mathbb{D}^n \\to \\mbox{M} (r, \\mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $\\mbox{M\\\"ob} \\times\\cdots\\times \\mbox{M\\\"ob}$ ($n$ times) of the bi-holomorphic automorphism group of $\\mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $\\mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreducible, are in the Cowen-Douglas class $\\mathrm B_r(\\mathbb D^n)$ and that they are mutually pairwise unitarily inequivalent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-18T14:40:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B13",
      "47B32",
      "20C25",
      "53C07"
    ],
    "keywords": [
      "cowen-douglas class",
      "homogeneous operators",
      "bi-holomorphic automorphism group",
      "hilbert spaces",
      "positive-definite kernels"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}