Weighted g-MPD inverse

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.