Weighted g-MPD inverse

Abstract

The significance of weighted generalized inverses inspired us to extend the concepts of the g-MPD and g-DMP inverses.
In particular, we define weighted versions of the g-MPD and g-DMP inverses solving two systems of operator equations.
Characterizations and representations of weighted g-MPD and g-DMP inverses are presented.
The operator matrix forms of the weighted g-MPD inverse are given. Consequently, we get new results for
g-MPD and g-DMP inverses and extend some known properties of g-MPD and g-DMP inverses.
Applying the weighted g-MPD and g-DMP inverses, we verify solvability of some systems of linear equations and minimizations problems.