{
  "abstract": "Let $A\\in\\mathbb{C}^{n\\times n}$ be positive definite Hermitian, where $n\\ge 2$, and define $\\|X\\|_A^2=\\mathrm{tr}(AX^*X)$ for all $X\\in\\mathbb{C}^{n\\times n}$. We determine the smallest constant $C_A$ for which $$ \\|XY-YX\\|_{\\mathrm{F}}^2 \\le C_A\\|X\\|_A^2\\|Y\\|_A^2 $$ holds for all rank-one $X,Y\\in\\mathbb{C}^{n\\times n}$, where $\\|\\cdot\\|_{\\mathrm{F}}$ denotes the Frobenius norm. The constant depends only on the two smallest eigenvalues of $A$ and is determined by a strictly monotone scalar equation. This answers the sharp-constant question for rank-one matrices formulated by Fang and Cheng. We also characterize all equality pairs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29240",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhekai Pang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/72qdgnbk1hzjp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29240",
  "fulltext_url": null,
  "id": "72qdgnbk1hzjp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29240",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7223230589",
  "pdf_url": "https://arxiv.org/pdf/2610.29240",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/72qdgnbk1hzjp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29240",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29240"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A sharp weighted commutator inequality for rank-one matrices",
  "title_text": "A sharp weighted commutator inequality for rank-one matrices",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/72qdgnbk1hzjp",
  "venue": "arXiv",
  "versions": []
}