Additive properties for the pseudo core inverse of morphisms in an additive category

In this paper, given a morhism [Formula: see text] with its pseudo core inverse [Formula: see text] and a morphism [Formula: see text] such that [Formula: see text] is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely [Formula: see text] holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

The W-weighted Drazin-star matrix and its dual

After decades studying extensively two generalized inverses, namely Moore--Penrose inverse and Drazin inverse, currently, we found immersed in a new generation of generalized inverses (core inverse, DMP inverse, etc.). The main aim of this paper is to introduce and investigate a matrix related to these new generalized inverses defined for rectangular matrices. We apply our results to the solution of linear systems.

Further characterizations of the weak core inverse of matrices and the weak core matrix

<abstract><p>The present paper is devoted to characterizing the weak core inverse and the weak core matrix using the core-EP decomposition. Some new characterizations of the weak core inverse are presented by using its range space, null space and matrix equations. Additionally, we give several new representations and properties of the weak core inverse. Finally, we consider several equivalent conditions for a matrix to be a weak core matrix.</p></abstract>