{
  "abstract": "For $α>1$, we study the weighted multiplicative Hilbert operator \\[ (T_αf)(s) = \\int_{1/2}^{\\infty} f(t)\\bigl(ζ(s+t)^α-1\\bigr) \\left(t-\\frac12\\right)^{α-1}\\,dt \\] on the Hilbertian Bergman space $A_{α,0}^2$ of Dirichlet series with zero constant term. In the natural orthonormal basis, $T_α$ is represented by the weighted multiplicative Hilbert matrix \\[ \\left( \\frac{Γ(α)\\sqrt{d_α(m)d_α(n)}} {\\sqrt{mn}\\,(\\log(mn))^α} \\right)_{m,n\\ge2}, \\] where $d_α$ is the generalized divisor function defined by $ζ(s)^α=\\sum_{n\\ge1}d_α(n)n^{-s}$ for $\\Re s>1$. Set $Λ_α=Γ(α/2)^2/Γ(α)$. We prove that the essential spectrum of \\(T_α\\) is \\([0,Λ_α]\\) and that the spectrum in this interval is purely absolutely continuous with multiplicity one. Above \\(Λ_α\\), there are at most finitely many eigenvalues, all of which are simple. Moreover, there is a unique \\(α_*\\in(1,2)\\) such that \\(T_α\\) has eigenvalues if and only if \\(α>α_*\\). Consequently, the associated sharp embedding inequality admits a non-zero extremizer precisely when $α>α_*$, and this extremizer is unique up to scalar multiplication. We also establish the asymptotic formula $\\|T_α\\|\\simΓ(α+1)/(2(2\\log 2)^α)$ as $α\\to\\infty$. A key step in the proof combines Mellin-transform asymptotics, strict total positivity, and eigenvalue counting to exclude embedded eigenvalues. The proof includes a general result on the essential and singular continuous spectra of weighted integral Hankel operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22882",
  "author_total": 3,
  "authors": [
    {
      "name": "Tengfei Ma"
    },
    {
      "name": "Yufeng Lu"
    },
    {
      "name": "Chao Zu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/62zexf3xghd4b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22882",
  "fulltext_url": null,
  "id": "62zexf3xghd4b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22882",
  "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": "W7204166778",
  "pdf_url": "https://arxiv.org/pdf/2608.22882",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/62zexf3xghd4b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2608.22882",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22882"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series",
  "title_text": "On the Spectra of Multiplicative Hilbert Operator on Bergman Spaces of Dirichlet Series",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/62zexf3xghd4b",
  "venue": "arXiv",
  "versions": []
}