{
  "id": "1712.07638",
  "version": "v1",
  "published": "2017-12-20T18:48:14.000Z",
  "updated": "2017-12-20T18:48:14.000Z",
  "title": "joint spreading models and uniform approximation of bounded operators",
  "authors": [
    "S. A. Argyros",
    "A. Georgiou",
    "A. -R. Lagos",
    "P. Motakis"
  ],
  "comment": "34 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We investigate the following property for Banach spaces. A Banach space $X$ satisfies the Uniform Approximation on Large Subspaces (UALS) if there exists $C>0$ such that for $A\\in\\mathcal{L}(X)$, $W\\subset\\mathcal{L}(X)$ with $W$ convex and compact such that for some $\\varepsilon>0$ and for every $x\\in X$, $x\\neq0$, there exists $B\\in W$ with $\\|A(x)-B(x)\\|<\\varepsilon\\|x\\|$ then there exists a subspace $Y$ of $X$ of finite codimension and a $B\\in W$ with $\\|(A-B)|_Y\\|_{\\mathcal{L}(Y,X)}<C\\varepsilon$. We prove that a class of separable Banach spaces including $\\ell_p$, for $1\\le p< \\infty$, and $c_0$ satisfy the UALS. On the other hand every $L_p[0,1]$, for $1\\le p\\le \\infty$ and $p\\neq2$, fails the property and the same holds for $C(K)$, where $K$ is an uncountable metrizable compact space. Our sufficient conditions for UALS are based on joint spreading models, a multidimensional extension of the classical concept of spreading model, introduced and studied in the present paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-20T18:48:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B03",
      "46B06",
      "46B25",
      "46B28",
      "46B45"
    ],
    "keywords": [
      "joint spreading models",
      "uniform approximation",
      "bounded operators",
      "separable banach spaces",
      "large subspaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}