{
  "id": "1403.5453",
  "version": "v1",
  "published": "2014-03-21T13:34:49.000Z",
  "updated": "2014-03-21T13:34:49.000Z",
  "title": "On operators which are adjoint to each other",
  "authors": [
    "Dan Popovici",
    "Zoltan Sebestyen"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Given two linear operators $S$ and $T$ acting between Hilbert spaces $\\mathscr{H}$ and $\\mathscr{K}$, respectively $\\mathscr{K}$ and $\\mathscr{H}$ which satisfy the relation \\begin{equation*} \\langle Sh, k\\rangle=\\langle h, Tk\\rangle, \\quad h\\in\\dom S, \\ k\\in\\dom T, \\end{equation*} i.e., according to the classical terminology of M.H. Stone, which are adjoint to each other, we provide necessary and sufficient conditions in order to ensure the equality between the closure of $S$ and the adjoint of $T.$ A central role in our approach is played by the range of the operator matrix $M_{S, T}=\\begin{pmatrix} 1_{\\dom S} & -T S & 1_{\\dom T} \\end{pmatrix}.$ We obtain, as consequences, several results characterizing skewadjointness, selfadjointness and essential selfadjointness. We improve, in particular, the celebrated selfadjointness criterion of J. von Neumann.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-21T13:34:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A05",
      "47A20",
      "47B25"
    ],
    "keywords": [
      "von neumann",
      "hilbert spaces",
      "celebrated selfadjointness criterion",
      "essential selfadjointness",
      "central role"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.5453P"
    }
  }
}