{
  "id": "1705.07037",
  "version": "v1",
  "published": "2017-05-19T15:04:16.000Z",
  "updated": "2017-05-19T15:04:16.000Z",
  "title": "Solutions of the system of operator equations $BXA=B=AXB$ via $*$-order",
  "authors": [
    "Mehdi Vosough",
    "Mohammad Sal Moslehian"
  ],
  "comment": "13 pages, to appear in Electron. J. Linear Algebra (ELA)",
  "categories": [
    "math.FA",
    "math.OA"
  ],
  "abstract": "In this paper, we establish some necessary and sufficient conditions for the existence of solutions to the system of operator equations $ BXA=B=AXB $ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions we prove that an operator $X$ is a solution of $ BXA=B=AXB $ if and only if $B \\stackrel{*}{ \\leq} AXA$, where the $*$-order $C\\stackrel{*}{ \\leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover we present the general solution of the equation above. Finally, we present some characterizations of $C \\stackrel{*}{ \\leq} D$ via other operator equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-19T15:04:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A24",
      "15B48",
      "47A62",
      "46L05"
    ],
    "keywords": [
      "operator equations",
      "bounded linear operators",
      "sufficient conditions",
      "hilbert space",
      "unknown operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}