{
  "abstract": "We investigate the hypercyclicity of Toeplitz operators on the Bergman space $L_{A}^{2}(\\mathbb{D})$ with symbols of the form $Ψ(z) = γ\\bar{z}+ψ(z)$, where $γ\\in \\mathbb{C} \\setminus \\{0\\}$ and $ψ$ is analytic on an open neighborhood of the closed unit disc $\\overline{\\mathbb{D}}$. Our approach bypasses the classical Hardy space techniques (reproducing kernel linear combinations, Nevanlinna factorization) by directly solving the integro-differential resolvent equation arising from the Bergman projection. A key winding number argument shows that for sense-reversing symbols ($|γ| > \\sup_{z \\in \\overline{\\mathbb{D}}} |ψ'(z)|$), the symbol's image $Ψ(\\mathbb{D})$ is contained in the point spectrum of $T_Ψ$. In the tridiagonal case $Ψ(z) = a\\bar{z}+b+cz$, we fully resolve the longstanding eigenvector completeness problem by linking the recurrence coefficients to a rotated Favard spectral measure on the major axis of the symbol's ellipse. Combined with self-commutator positivity, this establishes an unconditional, exact necessary and sufficient characterization of hypercyclicity: $T_Ψ$ is hypercyclic if and only if $|a| > |c|$ and $Ψ(\\mathbb{D})$ intersects both the unit disc and its exterior, completely eliminating the $(3+\\sqrt{2})$ restriction of previous literature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22017",
  "author_total": 1,
  "authors": [
    {
      "name": "Othman Abad"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0gh809m89kfbn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22017",
  "fulltext_url": null,
  "id": "0gh809m89kfbn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22017",
  "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.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7213903354",
  "pdf_url": "https://arxiv.org/pdf/2609.22017",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0gh809m89kfbn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.22017",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22017"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the dynamics of Toeplitz operators over Bergman spaces",
  "title_text": "On the dynamics of Toeplitz operators over Bergman spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/0gh809m89kfbn",
  "venue": "arXiv",
  "versions": []
}