{
  "abstract": "Let $1 \\leqslant p \\leqslant q < \\infty$, let $\\mathsf{X}$ be a separable $\\mathcal{L}_p$-space and let $\\mathsf{Y}$ be a separable $\\mathcal{L}_q$-space. We show that the space $\\mathfrak{L}(\\mathsf{X},\\mathsf{Y})$ of bounded linear operators from $\\mathsf{X}$ into $\\mathsf{Y}$ is isomorphic to $\\mathfrak{L}(\\ell_p,\\ell_q)$ if, and only if, $\\mathsf{Y}$ is complemented in its bidual. We also show that $\\mathfrak{L}(\\mathsf{Z},\\ell_{\\infty})$ is isomorphic to $\\mathfrak{L}(c_0,\\ell_{\\infty})$ for every separable $\\mathcal{L}_{\\infty}$-space $\\mathsf{Z}$. This extends and complements the earlier result of Arias and Farmer [On the structure of tensor products of $\\ell_p$-spaces, Pacific J. Math.]. Based on this general result, we give a completely different proof of the main result of Astashkin, Leśnik and Maligranda [Isomorphic structure of Cesàro and Tandori spaces, Canad. J. Math.], which states that the space $\\mathcal{K}$ and its sequence counterpart $\\mathcal{k}$ are isomorphic. Furthermore, we complete the picture by showing that the spaces $\\mathcal{k}$ and $\\mathcal{K}$, although isomorphic, are not isometric. Along the way, we show that for every infinite compact metrizable space $K$ the $\\ell_{\\infty}$-sum of copies of the space $\\mathscr{M}(K)$ of regular Borel measures on $K$ is isomorphic to $( \\bigoplus_{n=1}^{\\infty} L_1[0,1])_{\\ell_{\\infty}} \\approx \\mathfrak{L}(\\ell_1,L_1[0,1])$. This provides an abstract blueprint for another result of Astashkin, Leśnik and Maligranda, namely that $\\mathcal{K}$ is isomorphic to $( \\bigoplus_{n=1}^{\\infty} \\mathscr{M}[0,1])_{\\ell_{\\infty}}$. Finally, we show that there are models of set theory without the Axiom of Choice in which the spaces $\\mathcal{k}$ and $\\mathcal{K}$ are no longer isomorphic. This last part was inspired by Väth's paper [The dual space of $L_{\\infty}$ is $L_1$, Indag. Math. (N.S.)].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34421",
  "author_total": 2,
  "authors": [
    {
      "name": "Tomasz Kiwerski"
    },
    {
      "name": "Jakub Tomaszewski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7z859d4fe61fv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34421",
  "fulltext_url": null,
  "id": "7z859d4fe61fv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34421",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223973884",
  "pdf_url": "https://arxiv.org/pdf/2610.34421",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7z859d4fe61fv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34421",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34421"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the isomorphism of the spaces $\\mathcal{k}$ and $\\mathcal{K}$",
  "title_text": "On the isomorphism of the spaces k and 𝒦",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7z859d4fe61fv",
  "venue": "arXiv",
  "versions": []
}