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.

We define and investigate the e-core inverse and f-dual core inverse in a ring with involution as extensions of the core and dual core inverse, respectively. Using these definitions, we present and characterize the e-core partial order and the f-dual core partial order. We describe the sets of all elements a Rickart ∗-ring which are below a given element under the e-core and f-dual core partial order. New characterizations of weighted-EP elements are given too.

Using the Moore-Penrose inverse and the core-EP inverse, we define a new generalized inverse (called the MPCEP inverse) for a Hilbert space operator. Several equivalent conditions for a Hilbert space operator to be the MPCEP inverse are presented. We also give a canonical form for the MPCEP inverse. Various kinds of generalized inverses are integrated by MPCEP inverse and many equivalent relations among these inverses are investigated using appropriate idempotents. Also, we develop maximal classes of operators for which the representation of the MPCEP inverse is still valid. We introduce and study a new binary relation associated with the MPCEP inverse.

We introduce the e–core–EP inverse and the core–EP inverse for elements of Banach algebras. Several characterizations and representations for these inverses are presented using idempotents and Drazin inverses. Then, we define the e–core inverse of elements in Banach algebras and give its characterizations, generalizing the related results of the core inverse.

A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the...

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.