Moore-Penrose inverse of Gram operator on Hilbert C*-modules

Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^†$ 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 $t^† = (t^* t)^† t^*= t^* (t t^*)^†$ and $(t^*t)^†=t^†t^{* †}$ hold. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators.

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