WEIGHTED CORE-EP INVERSE AND WEIGHTED CORE-EP PRE-ORDERS IN A C*-ALGEBRA

We define extensions of the weighted core-EP inverse and weighted core-EP pre-orders of bounded linear operators on Hilbert spaces to elements of a C*-algebra. Some properties of the weighted core-EP inverse and weighted core-EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core-EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all wg-Drazin invertible elements of a C*-algebra. Applying these results, we introduce and characterize new partial orders which extend the core-EP pre-order to a partial order.