{
  "abstract": "A long-standing open problem asks whether there exists an infinite-dimensional Banach space on which every bounded linear operator attains its norm. With $\\mathrm{NA}_1(X,Y)$ and $\\mathrm{NA}_2(X,Y)$ denoting the classes of operators whose adjoints and second adjoints, respectively, attain their norms, we prove that \\[ \\mathrm{NA}_2(c_0,c_0)=\\mathcal{L}(c_0,c_0) \\qquad\\text{and}\\qquad \\mathrm{NA}_2(\\ell_1,\\ell_1)=\\mathcal{L}(\\ell_1,\\ell_1), \\] providing, to the best of our knowledge, the first known infinite-dimensional spaces on which every operator has a norm-attaining second adjoint. Building on the result for $c_0$, we undertake a systematic study of this equality within a natural family of $\\ell_1$-preduals given by hyperplanes of $c$, obtaining a complete characterization in this setting. In particular, we prove that \\[ \\mathrm{NA}_2(c,c)\\neq \\mathcal{L}(c,c), \\qquad\\text{whereas}\\qquad \\mathrm{NA}_3(c,c)=\\mathcal{L}(c,c). \\] We also establish Holub--Mujica-type theorems for the classes $\\mathrm{NA}_1$ and $\\mathrm{NA}_2$. More precisely, under suitable separability and approximation property assumptions, the identity $\\mathcal L(X,Y)=\\mathrm{NA}_1(X,Y)$ forces every operator from $X$ into $Y$ to be compact, whereas $\\mathcal L(X,Y)=\\mathrm{NA}_2(X,Y)$ forces every weakly compact operator from $X$ into $Y$ to be compact. Finally, strengthening a construction of Ostrovskii, we show that every infinite-dimensional Banach space admits an equivalent norm and a projection whose second adjoint does not attain its norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00794",
  "author_total": 3,
  "authors": [
    {
      "name": "Sheldon Dantas"
    },
    {
      "name": "Mingu Jung"
    },
    {
      "name": "Miguel Martín"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2m6spbqr2ecnb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00794",
  "fulltext_url": null,
  "id": "2m6spbqr2ecnb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00794",
  "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",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7206175829",
  "pdf_url": "https://arxiv.org/pdf/2609.00794",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2m6spbqr2ecnb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00794",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00794"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On operators whose adjoints or second adjoints attain their norms",
  "title_text": "On operators whose adjoints or second adjoints attain their norms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/2m6spbqr2ecnb",
  "venue": "arXiv",
  "versions": []
}