{
  "id": "2305.08182",
  "version": "v1",
  "published": "2023-05-14T15:20:09.000Z",
  "updated": "2023-05-14T15:20:09.000Z",
  "title": "On $g-$Fusion Frames Representations via Linear Operators",
  "authors": [
    "S. Jahedi",
    "F. Javadi",
    "M. J. Mehdipour"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\{\\frak{M} _k \\} _{ k \\in \\mathbb{Z}} $ be a sequence of closed subspaces of Hilbert space $H$, and let $\\{\\Theta_k\\}_{k \\in \\mathbb{Z}}$ be a sequence of linear operators from $H$ into $\\frak{M}_k$, $k \\in \\mathbb{Z}$. In the definition of fusion frames, we replace the orthogonal projections on $\\frak{M} _k$ by $\\Theta_k$ and find a slight generalization of fusion frames. In the case where, $\\Theta_k$ is self-adjoint and $\\Theta_k(\\frak{M} _k)= \\frak{M} _k$ for all $k \\in \\mathbb{Z}$, we show that if a $g-$fusion frame $\\{(\\frak{M} _k, \\Theta_k)\\}_{k \\in \\mathbb{Z}}$ is represented via a linear operator $T$ on $\\hbox{span} \\{\\frak{M} _k\\}_{ k \\in \\mathbb{Z}}$, then $T$ is bounded; moreover, if $\\{(\\frak{M} _k, \\Theta_k)\\}_{k \\in \\mathbb{Z}}$ is a tight $g-$fusion frame, then $T$ is not invertible. We also study the perturbation and the stability of these fusion frames. Finally, we give some examples to show the validity of the results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-14T15:20:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear operator",
      "fusion frames representations",
      "hilbert space",
      "slight generalization",
      "orthogonal projections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}