{
  "id": "2307.09174",
  "version": "v1",
  "published": "2023-07-15T21:28:55.000Z",
  "updated": "2023-07-15T21:28:55.000Z",
  "title": "On characterizations of a some classes of Schauder frames in Banach spaces",
  "authors": [
    "Rafik Karkri",
    "Samir Kabbaj",
    "Hamad Sidi Lafdal"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "In this paper, we prove the following results. There exists a Banach space without basis which has a Schauder frame. There exists an universal Banach space $B$ (resp. $\\tilde{B}$) with a basis (resp. an unconditional basis) such that, a Banach $X$ has a Schauder frame (resp. an unconditional Schauder frame ) if and only if $X$ is isomorphic to a complemented subspace of $B$ (resp. $\\tilde{B}$). For a weakly sequentially complete Banach space, a Schauder frame is unconditional if and only if it is besselian. A separable Banach space $X$ has a Schauder frame if and only if it has the bounded approximation property. Consequenty, The Banach space $\\mathcal{L}(\\mathcal{H},\\mathcal{H})$ of all bounded linear operators on a Hilbert space $\\mathcal{H}$ has no Schauder frame. Also, if $X$ and $Y$ are Banach spaces with Schauder frames then, the Banach space $ X\\widehat{\\otimes}_{\\pi}Y$ (the projective tensor product of $X$ and $Y$) has a Schauder frame. From the Faber$-$Schauder system we construct a Schauder frame for the Banach space $C[0,1]$ (the Banach space of continuous functions on the closed interval $ [0,1]$) which is not a Schauder basis of $C[0,1]$. Finally, we give a positive answer to some open problems related to the Schauder bases (In the Schauder frames setting).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-15T21:28:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B10",
      "46B15",
      "46B25",
      "46B45"
    ],
    "keywords": [
      "schauder basis",
      "characterizations",
      "weakly sequentially complete banach space",
      "unconditional schauder frame",
      "universal banach space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}