Positive semi-definite matrices of adjointable operators on Hilbert C∗-modules

Let A be a C∗-algebra, H,K be two Hilbert A-modules, and B be an adjointable operator from H to K. In this paper, we prove that the Moore–Penrose inverse of B exists if and only if B has closed range. In addition, some known results about the Moore–Penrose inverses acting on Hilbert spaces, as well as in C∗-algebras are extended in the context of Hilbert C∗-modules. As an application, a characterization of positive semi-definite matrices of adjointable operators with respect to an orthogonal pair of a Hilbert C∗-module is given.