Positive semi-definite matrices of adjointable operators on Hilbert C*-modules Original Research Article

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.