{
  "id": "2401.03176",
  "version": "v1",
  "published": "2024-01-06T09:38:42.000Z",
  "updated": "2024-01-06T09:38:42.000Z",
  "title": "On the convexity of the Berezin range of composition operators and related questions",
  "authors": [
    "Athul Augustine",
    "M. Garayev",
    "P. Shankar"
  ],
  "comment": "20 Pages",
  "categories": [
    "math.FA",
    "math.CV"
  ],
  "abstract": "The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\\mathcal{H}$ is the set $B(T)$ := $\\{\\langle T\\hat{k}_{x},\\hat{k}_{x} \\rangle_{\\mathcal{H}} : x \\in X\\}$, where $\\hat{k}_{x}$ is the normalized reproducing kernel for $\\mathcal{H}$ at $x \\in X$. In general, the Berezin range of an operator is not convex. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\\mathbb{C}$ and the Dirichlet space of the unit disc $\\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\\mathcal{H}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-06T09:38:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47B32",
      "52A10"
    ],
    "keywords": [
      "composition operators",
      "related questions",
      "unitarily equivalent berezin range",
      "two-dimensional reproducing kernel hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}