{
  "abstract": "Let $ω$ be a radial $\\widehat{\\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\\mathbb D$. We prove that the Toeplitz operator \\(T_{ω,u}\\) is compact on \\(A_ω^2\\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_ω^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{ω,u}$. Even in the unweighted Bergman space \\(A^2\\), our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \\(\\widehat{\\mathcal D}\\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \\(\\widehat{\\mathcal D}\\)-weight \\(ω\\) and bounded symbols \\(u,v\\) such that the product \\(T_{ω,v}T_{ω,u}\\) is noncompact, whereas its Berezin transform vanishes at the boundary.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23733",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuerang Li"
    },
    {
      "name": "Zipeng Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/enbeaa7pvvgtk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23733",
  "fulltext_url": null,
  "id": "enbeaa7pvvgtk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23733",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7204220458",
  "pdf_url": "https://arxiv.org/pdf/2608.23733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/enbeaa7pvvgtk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.23733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces",
  "title_text": "Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/enbeaa7pvvgtk",
  "venue": "arXiv",
  "versions": []
}