Moore–Penrose inverses of Gram operators on Hilbert $C^*$-modules
Volume 210 / 2012
Studia Mathematica 210 (2012), 189-196
MSC: Primary 46L08; Secondary 47A05, 15A09, 46L05.
DOI: 10.4064/sm210-2-6
Abstract
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.
