{
  "abstract": "In this paper, we consider the Rosenthal -- Halmos problem of almost commuting matrices with respect to the normalized Hilbert -- Schmidt norm $\\|\\cdot\\|_{2,d}=d^{-1/2}\\|\\cdot\\|_2$. We show that if $X$ and $Y$ are self-adjoint matrices with $\\|X\\|\\le 1$, $\\|Y\\|\\le 1$, then there exist commuting self-adjoint matrices $X',Y'$ such that $\\|X-X'\\|_{2,d}+\\|Y-Y'\\|_{2,d}\\le 5\\|[X,Y]\\|_{2,d}^{1/3}$, and $[X,X']=0$. Within these constraints, the exponent $1/3$ cannot be improved.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31000",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohit Bansil"
    },
    {
      "name": "Ilya Kachkovskiy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s5drrd4mkans.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31000",
  "fulltext_url": null,
  "id": "0s5drrd4mkans",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31000",
  "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"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7204893696",
  "pdf_url": "https://arxiv.org/pdf/2608.31000",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s5drrd4mkans.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.31000",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31000"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm",
  "title_text": "On almost commuting matrices with respect to the normalized Hilbert--Schmidt norm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/0s5drrd4mkans",
  "venue": "arXiv",
  "versions": []
}