{
  "abstract": "Let $\\mathcal{H}$ be a complex Hilbert space, and let $T=H+iK\\in\\mathcal{B}(\\mathcal{H})$ be the Cartesian decomposition of $T$, where $H=H^{*}$ and $K=K^{*}$. We introduce the $L^2$-Cartesian numerical radius and study its basic properties. We compare it with the classical numerical radius $ω(T)$. In particular, we establish the sharp estimate $ \\frac{1}{\\sqrt{2}}ω(T)\\le \\wcart{2}(T)\\le ω(T), $ and establish lower and upper bounds in terms of the Cartesian parts $H,K$. We also give an equality characterization and explicit sharpness examples. As a consequence, we obtain $\\frac{1}{4}\\left\\|T^{*}T+TT^{*}\\right\\|\\le \\wcart{2}(T)^2\\le ω(T)^2. $ More generally, we study the $L^p$-Cartesian numerical radius and prove the sharp comparison $ c_p\\,ω(T)\\le \\wcart{p}(T)\\le ω(T),\\qquad 1\\le p<\\infty. $ Finally, we present a block-operator analogue of the classical refinement of the triangle inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04844",
  "author_total": 4,
  "authors": [
    {
      "name": "Wasim Audeh"
    },
    {
      "name": "Fuad Kittaneh"
    },
    {
      "name": "Hamid Reza Moradi"
    },
    {
      "name": "Satyajit Sahoo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0h233bcr389hs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04844",
  "fulltext_url": null,
  "id": "0h233bcr389hs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04844",
  "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": "W7220666902",
  "pdf_url": "https://arxiv.org/pdf/2610.04844",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0h233bcr389hs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.04844",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04844"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The $L^2$-Cartesian Numerical Radii of Hilbert Space Operators",
  "title_text": "The L^2-Cartesian Numerical Radii of Hilbert Space Operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/0h233bcr389hs",
  "venue": "arXiv",
  "versions": []
}