On partial orders of Hilbert space operators

First, an overview of partial orders defined on bounded linear operators on an infinite-dimensional Hilbert space is presented. A definition for the core inverse of operators on a Hilbert space is proposed. Extensions of the sharp and the core partial orders are considered. An explicit formula for the core inverse of matrices is obtained using a full-rank factorization. Relationships between all these partial orders and formula for generalized inverses of differences of operators, when they are related with respect to these partial orders, are investigated.