Weighted core–EP inverse of an operator between Hilbert spaces

Abstract

We introduce and study the weighted core–EP inverse of an operator between two Hilbert spaces as a generalization of the weighted core–EP inverse for a rectangular matrix. Several new properties of weighted core–EP inverses are given and some known results are extended. Using a weighted operator, the core–EP pre-order and the minus partial order of corresponding operators, we define new pre-orders on the set of all Wg–Drazin invertible operators between two Hilbert spaces. As consequences of our results, we present a new characterization and new representations of the core–EP inverse, new characterizations of the core–EP pre-order and extend the core–EP pre-order to a partial order.