{
  "id": "1309.1219",
  "version": "v2",
  "published": "2013-09-05T01:46:58.000Z",
  "updated": "2013-09-15T20:16:12.000Z",
  "title": "Frames of subspaces in Hilbert spaces with $W$-metrics",
  "authors": [
    "Primitivo Acosta-Humánez",
    "Kevin Esmeral",
    "Osmin Ferrer"
  ],
  "comment": "15 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "If $\\left(\\h,\\langle\\cdot,\\cdot\\rangle\\right)$ is a Hilbert space and on it we consider the sesquilinear form $\\langle\\,W\\cdot,\\cdot\\rangle$ so-called $W$-metric, where $W^{*}=W\\in\\BH$, and $\\ker\\,W=\\{0\\}$, then the space $\\left(\\h,\\langle\\,W\\cdot,\\cdot\\rangle\\right)$ is called Hilbert space with $W$-metric or simply $W$-space. In this paper we investigate the dynamic of frames of subspace on these spaces, where the sense of dynamics refers to the behavior of frames of subspace in $\\h_{W}$ (the completion of $\\left(\\h,\\langle\\,W\\cdot,\\cdot\\rangle\\right)$) comparing with $\\h$ and vice versa. This work is based on the study made in \\cite{KEFER,GMMM} on frames in Krein spaces. In a similar way, Casazza and Kutyniok obtained some results in the context of Hilbert spaces, see \\cite{CG}. We take tools of theory of $C^{*}$-algebra, and properties of $\\BH$, to show that every Hilbert space with $W$-metric $\\h_{W}$ with $0\\in\\sigma(W)$ has a decomposition $$\\h_{W}=\\bigoplus_{n\\in\\N\\cup\\{\\infty\\}}\\h_{\\psi_{n}}^{W},$$ where $\\h_{\\psi_{n}}^{W}\\simeq \\Ele(\\sigma(W),x\\,d\\mu_{n}(x))$ are Krein spaces, for every $n\\in\\N\\cup\\{\\infty\\}$. Moreover, we investigate the dynamics of frames of subspace when the self-adjoint operator $W$ is unbounded.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-15T20:16:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C15",
      "46C20",
      "47B50",
      "47B15"
    ],
    "keywords": [
      "hilbert space",
      "krein spaces",
      "vice versa",
      "self-adjoint operator",
      "sesquilinear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.1219A"
    }
  }
}