Moore-Penrose inverse of Gram operator on Hilbert C*-modules
  Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^\dag$ 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 $t^{\dag} = (t^* t)^{\dag} t^*= t^* (t t^*)^{\dag}$ and $(t^*t)^{\dag}=t^{\dag}t^{* \dag}$ hold. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators. 
M. S. Moslehian, K. Sharifi, M. Forough and M. Chakoshi
math.FA math.OA