{
  "id": "2506.20143",
  "version": "v1",
  "published": "2025-06-25T05:37:29.000Z",
  "updated": "2025-06-25T05:37:29.000Z",
  "title": "Dirichlet-type spaces of the unit bidisc and toral completely hyperexpansive operators",
  "authors": [
    "Santu Bera"
  ],
  "comment": "16 pages, comments are welcome",
  "categories": [
    "math.FA"
  ],
  "abstract": "We discuss a notion, originally introduced by Aleman in one variable, of Dirichlet-type space $\\mathcal D(\\mu_1,\\mu_2)$ on the unit bidisc $\\mathbb D^2,$ with superharmonic weights related to finite positive Borel measures $\\mu_1,\\mu_2$ on $\\overline{\\mathbb D}.$ The multiplication operators $\\mathscr M_{z_1}$ and $\\mathscr M_{z_2}$ by the coordinate functions $z_1$ and $z_2,$ respectively, are bounded on $\\mathcal D(\\mu_1,\\mu_2)$ and the set of polynomials is dense in $\\mathcal D(\\mu_1,\\mu_2).$ We show that the commuting pair $\\mathscr M_{z}=(\\mathscr M_{z_1},\\mathscr M_{z_2})$ is a cyclic analytic toral completely hyperexpansive $2$-tuple on $\\mathcal D(\\mu_1,\\mu_2).$ Unlike the one variable case, not all cyclic analytic toral completely hyperexpansive pairs arise as multiplication $2$-tuple $\\mathscr M_z$ on these spaces. In particular, we establish that a cyclic analytic toral completely hyperexpansive operator $2$-tuple $T=(T_1,T_2)$ satisfying $I-T^*_1 T_1-T^*_2T_2+T^*_1T^*_2T_1T_2=0$ and having a cyclic vector $f_0$ is unitarily equivalent to $\\mathscr{M}_z$ on $\\mathcal{D}(\\mu_1, \\mu_2)$ for some finite positive Borel measures $\\mu_1$ and $\\mu_2$ on $\\overline{\\mathbb{D}}$ if and only if $\\ker T^*$, spanned by $f_0$, is a wandering subspace for $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-25T05:37:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A13",
      "32A10",
      "47B38",
      "31C05",
      "46E20"
    ],
    "keywords": [
      "dirichlet-type space",
      "unit bidisc",
      "hyperexpansive operator",
      "cyclic analytic toral",
      "finite positive borel measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}