{
  "abstract": "We determine the smallest square size at which the operator norm and the completely bounded norm of a Schur multiplier on $S^4$ can differ. More precisely, for $A=\\begin{bmatrix}1&2&0\\\\1&-2&0\\\\2&2\\mathrm{i}&0\\end{bmatrix}$ we prove \\[ \\|M_A\\|_{S^4_3\\to S^4_3}^2\\leq\\frac{1119}{250} <\\left(\\frac{18983532}{947485}\\right)^{1/2} \\leq \\|\\mathrm{Id}_{S^4_2}\\otimes M_A\\|_{S^4_6\\to S^4_6}^2. \\] We also prove that no $2\\times2$ Schur multiplier can have unequal operator and completely bounded norms on $S^4$, so square size three is optimal. In addition, we determine the exact ordinary norm of the previous multiplier analytically and give an independent computer-assisted certification of a rational upper bound.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29643",
  "author_total": 1,
  "authors": [
    {
      "name": "Cédric Arhancet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35w1zpng6awpm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29643",
  "fulltext_url": null,
  "id": "35w1zpng6awpm",
  "issns": [],
  "landing_url": "https://hal.science/hal-05739888v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "0.phys"
    ],
    [
      "hal",
      "1.math.math-fa"
    ],
    [
      "hal",
      "1.math.math-gr"
    ],
    [
      "hal",
      "1.math.math-kt"
    ],
    [
      "hal",
      "1.math.math-mp"
    ],
    [
      "hal",
      "1.math.math-oa"
    ],
    [
      "hal",
      "1.math.math-pr"
    ],
    [
      "hal",
      "1.math.math-qa"
    ],
    [
      "hal",
      "1.math.math-sp"
    ],
    [
      "hal",
      "1.phys.qphy"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05739888/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35w1zpng6awpm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:29Z",
      "native_id": "hal-05739888",
      "source": "hal",
      "url": "https://hal.science/hal-05739888v1"
    },
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29643",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29643"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-gr",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-mp",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-pr",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "hal:1.phys.qphy",
      "subfield": "quantum"
    }
  ],
  "title": "Optimal square size for separating the operator and completely bounded norms of Schur multipliers on S^4",
  "title_text": "Optimal square size for separating the operator and completely bounded norms of Schur multipliers on S^4",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/35w1zpng6awpm",
  "venue": "HAL",
  "versions": []
}