{
  "id": "1711.08391",
  "version": "v1",
  "published": "2017-11-22T17:00:29.000Z",
  "updated": "2017-11-22T17:00:29.000Z",
  "title": "On the adjoint of Hilbert space operators",
  "authors": [
    "Zoltán Sebestyén",
    "Zsigmond Tarcsay"
  ],
  "comment": "19 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "In general, it is a non trivial task to determine the adjoint $S^*$ of an unbounded operator $S$ acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator $T$ to be identical with $S^*$. In our considerations, a central role is played by the operator matrix $M_{S,T}=\\left(\\begin{array}{cc} I & -T\\\\ S & I\\end{array}\\right)$. Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that $T^*T$ always has a positive selfadjoint extension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-22T17:00:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A05",
      "47B25"
    ],
    "keywords": [
      "hilbert space operators",
      "non trivial task",
      "complex hilbert spaces",
      "orthogonal projection operators",
      "positive selfadjoint extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}