Weighted Core–ep Inverse and Weighted Core–ep Pre-Orders in a -Algebra

Abstract

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 $C^{\ast }$ -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 $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

Weighted generalized core-EP inverse in a Banach algebra with involution

“…Since every bounded linear operator in B(X, Y ) can be regarded as a restriction of an operator in B(X Y ), it could be seen as an element in a C * -algebra. Recently, Mosić extended the weighted core-EP inverse of bounded linear operators on Hilbert spaces to elements of a C * -algebra and the weighted core-EP inverse in a C * -algebra was characterized by means of range projections (see [20]).…”

Generalized weighted core inverse in Banach *-algebras

“…This new generalized inverse was extensively investigated in [13,19]. Recently, Mosić extended core-EP inverse of bounded linear operators on Hilbert spaces to elements of a C * -algebra by means of range projections (see [16]). Surprisingly, the preceding result shows that the core-EP inverse of Mosić and Djordjević coincide with the generalized weighted core inverse with weight e = 1.…”

On generalized core invertibility in Banach algebras

“…This new generalized inverse was extensively investigated in [18,22,19]. Recently, Mosić extended core-EP inverse of bounded linear operators on Hilbert spaces to elements of a C * -algebra by means of range projections (see [21]).…”