Moore–Penrose inverses of Gram operators on Hilbert $C^*$-modules

Tom 210 / 2012

M. S. Moslehian, K. Sharif, M. Forough, M. Chakoshi Studia Mathematica 210 (2012), 189-196 MSC: Primary 46L08; Secondary 47A05, 15A09, 46L05. DOI: 10.4064/sm210-2-6


Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^\dagger$ be its Moore–Penrose inverse. We investigate the Moore–Penrose invertibility of the Gram operator $t^*t$. More precisely, we study some conditions ensuring that $t^{ \dagger} = (t^* t)^{ \dagger} t^*= t^*(t t^*)^{ \dagger}$ and $(t^*t)^{\dagger}=t^{ \dagger}t^{* \dagger}$. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators.


  • M. S. MoslehianDepartment of Pure Mathematics
    Center of Excellence in Analysis
    on Algebraic Structures (CEAAS)
    Ferdowsi University of Mashhad
    P.O. Box 1159, Mashhad 91775, Iran
  • K. SharifDepartment of Mathematics
    Shahrood University of Technology
    P.O. Box 3619995161-316, Shahrood, Iran
    School of Mathematics
    Institute for Research
    in Fundamental Sciences (IPM)
    P.O. Box 19395-5746, Tehran, Iran
  • M. ForoughDepartment of Mathematics
    Ferdowsi University of Mashhad
    P.O. Box 1159, Mashhad, Iran
  • M. ChakoshiTusi Mathematical Research Group
    P.O. Box 1113, Mashhad, Iran

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