{
  "abstract": "We prove that the classical Cesàro operator belongs to the Toeplitz algebra, providing an independent solution to a question raised by Barría and Halmos. Our approach is based on a discrete Mellin calculus for the sampled-ratio matrices \\[ W(κ)_{jk} = \\frac{1}{j+1}\\, κ\\!\\left(\\frac{k+1}{j+1}\\right). \\] For a natural algebra of kernels $A$, we prove that this quantization is multiplicative modulo Hilbert--Schmidt operators, \\[ W(κ)W(η) - W(κ\\star η) \\in S_2, \\qquad κ,η\\in A. \\] We further show that every operator $W(κ)$, $κ\\in A$, belongs to the commutator ideal of the Toeplitz algebra. Since the Cesàro operator corresponds to the kernel $κ= \\mathbf 1_{(0,1]}$, this resolves the Barr\\'ıa--Halmos question as a special case of the general framework.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09453",
  "author_total": 2,
  "authors": [
    {
      "name": "Carlo Bellavita"
    },
    {
      "name": "Georgios Stylogiannis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8t0529zf86mr8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09453",
  "fulltext_url": null,
  "id": "8t0529zf86mr8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09453",
  "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": "W7212165213",
  "pdf_url": "https://arxiv.org/pdf/2609.09453",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8t0529zf86mr8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09453",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09453"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A discrete Mellin calculus in the Toeplitz algebra",
  "title_text": "A discrete Mellin calculus in the Toeplitz algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/8t0529zf86mr8",
  "venue": "arXiv",
  "versions": []
}