{
  "abstract": "We examine the generalized Hilbert (matrix) operators $$ H_{g,γ} : (a_n) \\mapsto \\sum_{n=1}^{\\infty} \\bigg(\\frac{k}{n}\\bigg)^γ \\frac{g_k a_n}{n+k} $$ on the $\\ell^p$ spaces, $1<p<\\infty$, where $g=(g_n)$ is a sequence and $-1/p<γ<1-1/p$. Given a partition of the natural numbers $\\bigcup_j I_j = \\mathbb N$, we encode $g$ to our investigation via its $\\ell^p$-means $G_j$ over the sets $I_j$. We prove that if $I_j$ increases exponentially, the data $(G_j)$ characterizes boundedness, but is insufficient to determine the exact value of the essential norm. Nonetheless, if the growth rate of $(I_j)$ is subexponential, the corresponding data $(G_j)$, given that the tail converge, is sufficient to determine the exact value, in which case we also calculate it. Moreover, it is shown that the boundedness of $(G_j)$ is sufficient, but not necessary (and necessary, but not sufficient) to ensure $H_{g,γ}$ is bounded if the underlying partition $(I_j)$ increases subexponentially (and superexponentially, respectively). Using the dual operator, we also prove that the essential norm of $H_{g,γ}$ is comparable with $\\limsup_j G(j)$ when $(I_j)$ grows exponentially.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13403",
  "author_total": 1,
  "authors": [
    {
      "name": "David Norrbo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/691kc248xyj2a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13403",
  "fulltext_url": null,
  "id": "691kc248xyj2a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13403",
  "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": "W7213328731",
  "pdf_url": "https://arxiv.org/pdf/2609.13403",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/691kc248xyj2a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13403",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The essential norm, block sizes, and some Generalized Hilbert operators on lp",
  "title_text": "The essential norm, block sizes, and some Generalized Hilbert operators on lp",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/691kc248xyj2a",
  "venue": "arXiv",
  "versions": []
}