{
  "id": "2011.09915",
  "version": "v1",
  "published": "2020-11-19T15:49:28.000Z",
  "updated": "2020-11-19T15:49:28.000Z",
  "title": "Subsymmetric bases have the factorization property",
  "authors": [
    "Richard Lechner"
  ],
  "comment": "17 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We show that every subsymmetric Schauder basis $(e_j)$ of a Banach space $X$ has the factorization property, i.e. $I_X$ factors through every bounded operator $T\\colon X\\to X$ with a $\\delta$-large diagonal (that is $\\inf_j |\\langle Te_j, e_j^*\\rangle| \\geq \\delta > 0$, where the $(e_j^*)$ are the biorthogonal functionals to $(e_j)$). Even if $X$ is a non-separable dual space with a subsymmetric weak$^*$ Schauder basis $(e_j)$, we prove that if $(e_j)$ is non-$\\ell^1$-splicing (there is no disjointly supported $\\ell^1$-sequence in $X$), then $(e_j)$ has the factorization property. The same is true for $\\ell^p$-direct sums of such Banach spaces for all $1\\leq p\\leq \\infty$. Moreover, we find a condition for an unconditional basis $(e_j)_{j=1}^n$ of a Banach space $X_n$ in terms of the quantities $\\|e_1+\\ldots+e_n\\|$ and $\\|e_1^*+\\ldots+e_n^*\\|$ under which an operator $T\\colon X_n\\to X_n$ with $\\delta$-large diagonal can be inverted when restricted to $X_\\sigma = [e_j : j\\in\\sigma]$ for a \"large\" set $\\sigma\\subset \\{1,\\ldots,n\\}$ (restricted invertibility of $T$; see Bourgain and Tzafriri [Israel J. Math. 1987, London Math. Soc. Lecture Note Ser. 1989). We then apply this result to subsymmetric bases to obtain that operators $T$ with a $\\delta$-large diagonal defined on any space $X_n$ with a subsymmetric basis $(e_j)$ can be inverted on $X_\\sigma$ for some $\\sigma$ with $|\\sigma|\\geq c n^{1/4}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-19T15:49:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B25",
      "46B26",
      "47A68",
      "46B07",
      "15A09"
    ],
    "keywords": [
      "subsymmetric basis",
      "factorization property",
      "banach space",
      "large diagonal",
      "subsymmetric schauder basis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}