{
  "id": "2001.09338",
  "version": "v1",
  "published": "2020-01-25T17:21:53.000Z",
  "updated": "2020-01-25T17:21:53.000Z",
  "title": "Left m-invertibility by the adjoint of Drazin inverse and m-selfadjointness of Hilbert space operators",
  "authors": [
    "B. P. Duggal",
    "I. H. Kim"
  ],
  "comment": "14 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "A Hilbert space operator $A\\in\\B$ is left $(X,m)$-invertible by $B\\in\\B$ (resp., $B\\in\\B$ is an $(X,m)$-adjoint of $A\\in\\B$) for some operator $X\\in\\B$ if $\\triangle_{B,A}^m(X)=\\sum_{j=0}^m(-1)^j\\left(\\begin{array}{clcr}m\\\\j\\end{array}\\right)B^{m-j}XA^{m-j}=0$ (resp., $\\delta_{B,A}^m(X)=\\sum_{j=0}^m(-1)^j\\left(\\begin{array}{clcr}m\\\\j\\end{array}\\right)B^{(m-j)}XA^j=0$). No Drazin invertible operator $A\\in\\B$, with Drazin inverse $A_d$, can be left $(I,m)$-invertible (equivalently, $m$-invertible) by its adjoint or its Drazin inverse or the adjoint of its Drazin inverse. For Drazin inverrtible operators $A$, it is seen that the existence of an $X$ acts as a conduit for implications $\\triangle_{B,A}(X)=0\\Longrightarrow \\delta^m_{C,A}(X)=0$, where the pair $(B,C)=$ either $(A,A_d)$ or $(A_d,A)$ or $(A^*,A^*_d)$ or $(A^*_d,A^*)$. Reverse implications fail. Assuming certain commutativity conditions, it is seen that $\\triangle_{A^*_d,A}^m(X)=0=\\triangle^n_{B^*_d,B}(Y)$ implies $\\delta^{m+n-1}_{A^*B^*,AB}(XY)=0=\\delta^{m+n-1}_{A^*+B^*,A+B}(XY)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-25T17:21:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A05",
      "47A55",
      "47A11",
      "47B47"
    ],
    "keywords": [
      "hilbert space operator",
      "drazin inverse",
      "left m-invertibility",
      "m-selfadjointness",
      "reverse implications fail"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}