{
  "abstract": "In this article, we study Foguel-type operators $\\EuScript{F}=\\begin{bmatrix} V_1^{*} & T\\\\ 0 & V_2\\end{bmatrix}$ on $H_1\\oplus H_2$, where each $H_j$ is a separable Hilbert space, $V_j\\in \\mathcal{B}(H_j)$ is an isometry for $j=1,2$, and $T\\in \\mathcal{B}(H_2,H_1)$. We examine how the norm and minimum attainment properties of $\\EuScript{F}$ depend on the corner entry $T$ and on the defect spaces of the isometries. The norm formula $\\|\\EuScript{F}\\|=\\frac{\\|T\\|+\\sqrt{\\|T\\|^{2}+4}}{2}$ holds for every $T$ when $V_1,V_2$ are isometries, and more generally when $V_2$ is a contraction that is isometric on the range of $T^{*}$. Moreover, $\\EuScript{F}$ attains its norm if and only if $T$ does. We characterize the positive $2\\times 2$ block operator matrices that are absolutely norm attaining and absolutely minimum attaining, and use this to prove that $\\EuScript{F}$ is absolutely norm attaining if and only if it is absolutely minimum attaining if and only if $\\dim N(V_1^{*})<\\infty$ and $T$ has finite rank. The closure of the absolutely norm attaining class is described by $\\dim N(V_1^{*})<\\infty$ and compactness of $T$. Further, we establish that the reduced minimum modulus is $γ(\\EuScript{F})=\\frac{\\sqrt{\\|T\\|^{2}+4}-\\|T\\|}{2}=\\frac{1}{\\|\\EuScript{F}\\|}$, and $\\EuScript{F}$ attains it if and only if $T$ is norm attaining. We also show that $\\EuScript{F}$ is Fredholm if and only if $\\dim N(V_j^{*})<\\infty$ for $j=1,2$, with index $\\dim N(V_1^{*})-\\dim N(V_2^{*})$, and we give its Moore--Penrose inverse. Finally, for $V_1=V_2=T_z$ and more generally $V_j=T_{θ_j}$ with $θ_j$ inner, and Toeplitz or Hankel corner entries, the results become explicit conditions on the symbol.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42102",
  "author_total": 2,
  "authors": [
    {
      "name": "Puspendu Nag"
    },
    {
      "name": "Ramesh Golla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2a9ydywstmw8j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42102",
  "fulltext_url": null,
  "id": "2a9ydywstmw8j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42102",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42102",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2a9ydywstmw8j.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.42102",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42102"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Norm and Minimum Attainment of Foguel-Type Operators",
  "title_text": "Norm and Minimum Attainment of Foguel-Type Operators",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/2a9ydywstmw8j",
  "venue": "arXiv",
  "versions": []
}