{
  "abstract": "We study the numerical radius \\(ω(T)\\) of a bounded linear operator \\(T\\) on a complex Hilbert space. Write \\(T=A+iB\\), where \\(A\\) and \\(B\\) are the real and imaginary parts of \\(T\\). A central result is the exact formula \\[ ω(T)=\\fracπ{2}\\sup_{\\|x\\|=1}\\int_0^1 \\left|\\left\\langle (A\\cos πt+B\\sin πt)x,x\\right\\rangle\\right|\\,dt . \\] For every \\(n\\in\\mathbb N\\), we also prove the power form \\[ ω^{2n}(T)=\\frac{4^n}{\\binom{2n}{n}} \\sup_{\\|x\\|=1}\\int_0^1 \\left\\langle (A\\cos πt+B\\sin πt)x,x\\right\\rangle^{2n}\\,dt . \\] This yields sharp upper bounds; for \\(n=1\\), together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality \\[ \\frac14\\|T^*T+TT^*\\|\\le ω^2(T)\\le \\frac12\\|T^*T+TT^*\\|. \\] Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a \\(3\\times3\\) nilpotent matrix and longer alternating paths. The factor \\(π/2\\) is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15242",
  "author_total": 3,
  "authors": [
    {
      "name": "Wasim Audeh"
    },
    {
      "name": "Fuad Kittaneh"
    },
    {
      "name": "Hamid Reza Moradi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6s1wdzp1b2n9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15242",
  "fulltext_url": null,
  "id": "6s1wdzp1b2n9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15242",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7221787647",
  "pdf_url": "https://arxiv.org/pdf/2610.15242",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6s1wdzp1b2n9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15242",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15242"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators",
  "title_text": "Integral Representations and Sharp Numerical Radius Inequalities for Hilbert Space Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/6s1wdzp1b2n9c",
  "venue": "arXiv",
  "versions": []
}