Moore-Penrose inverse, generalized Schur complement and inertia

In this article, a generalized Schur complement defined by the Moore-Penrose inverse and *congruence to a Hermitian block matrix are studied for a generalization of Haynsworth’s theorem. As an application, an alternative proof of Albert’s characterization of positivity is obtained [A. Albert, SIAM J. Appl. Math. (1969)]. Some of our results of matrices are extended to bounded linear operators.