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

Abstract

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 *(tt *) ↑ and (t *t) ↑ = t ↑t* ↑. As an application, we get some results for densely defined closed operators on Hilbert C *-modules over C *-algebras of compact operators. © 2012 Instytut Matematyczny PAN.