{
  "id": "2209.01767",
  "version": "v1",
  "published": "2022-09-05T05:17:32.000Z",
  "updated": "2022-09-05T05:17:32.000Z",
  "title": "On the strong subdifferentiability of the homogeneous polynomials and (symmetric) tensor products",
  "authors": [
    "Sheldon Dantas",
    "Mingu Jung",
    "Martin Mazzitelli",
    "Jorge Tomás Rodríguez"
  ],
  "comment": "38 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\\mathcal{P}(^N X, Y^*)$, $X \\hat{\\otimes}_\\pi \\cdots \\hat{\\otimes}_\\pi X$ and $\\hat{\\otimes}_{\\pi_s,N} X$. Among other results, we characterize when the norms of the spaces $\\mathcal{P}(^N \\ell_p, \\ell_{q}), \\mathcal{P}(^N l_{M_1}, l_{M_2})$, and $\\mathcal{P}(^N d(w,p), l_{M_2})$ are strongly subdifferentiable. Analogous results for multilinear mappings are also obtained. Since strong subdifferentiability of a dual space implies reflexivity, we improve some known results on the reflexivity of spaces of $N$-homogeneous polynomials and $N$-linear mappings. Concerning the projective (symmetric) tensor norms, we provide positive results on the subsets $U$ and $U_s$ of elementary tensors on the unit spheres of $X \\hat{\\otimes}_\\pi \\cdots \\hat{\\otimes}_\\pi X$ and $\\hat{\\otimes}_{\\pi_s,N} X$, respectively. Specifically, we prove that $\\hat{\\otimes}_{\\pi_s,N} \\ell_2$ and $\\ell_2 \\hat{\\otimes}_\\pi \\cdots \\hat{\\otimes}_\\pi \\ell_2$ are uniformly strongly subdifferentiable on $U_s$ and $U$, respectively, and that $c_0 \\hat{\\otimes}_{\\pi_s} c_0$ and $c_0 \\hat{\\otimes}_\\pi c_0$ are strongly subdifferentiable on $U_s$ and $U$, respectively, in the complex case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-05T05:17:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "46M05",
      "46G25",
      "46B04",
      "46B07"
    ],
    "keywords": [
      "strong subdifferentiability",
      "homogeneous polynomials",
      "tensor products",
      "dual space implies reflexivity",
      "strongly subdifferentiable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}