{
  "id": "2408.14893",
  "version": "v1",
  "published": "2024-08-27T09:08:26.000Z",
  "updated": "2024-08-27T09:08:26.000Z",
  "title": "Using the Baire Category Theorem to Explore Lions Problem for Quasi-Banach Spaces",
  "authors": [
    "A. G. Aksoy",
    "J. M. Almira"
  ],
  "categories": [
    "math.FA",
    "math.CA"
  ],
  "abstract": "Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire Category Theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple $(A_0, A_1)$. Lions problem, posed in 1960's, is to prove that different parameters $(\\theta,p)$ produce different interpolation spaces $(A_0, A_1)_{\\theta, p}$. We first establish conditions on $A_0$ and $A_1$ so that interpolation spaces of this couple are strictly intermediate spaces between $A_0+A_1$ and $A_0\\cap A_1$. This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if $p\\neq p^*$ the operator ideals $\\mathcal{L}^{(a)}_{p,q}(X,Y)$, $\\mathcal{L}^{(a)}_{p^*,q^*}(X,Y)$ generated by approximation numbers are distinct. Moreover, for any fixed $p$, either all operator ideals $\\mathcal{L}^{(a)}_{p,q}(X,Y)$ collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch's problem for operator ideals based on general $s$-numbers. However, the BCT can be used to prove a lethargy result for arbitrary $s$-numbers which guarantees that, under very minimal conditions on $X,Y$, the space $\\mathcal{L}^{(s)}_{p,q}(X,Y)$ is strictly embedded into $\\mathcal{L}^{\\mathcal{A}}(X,Y)$. The paper is dedicated to the memory of Prof. A. Pietsch, who passed away recently.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T09:08:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "baire category theorem",
      "lions problem",
      "quasi-banach spaces",
      "operator ideals",
      "interpolation spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}