Abstract.

To extend the notions of the GDMP inverse and its dual for a square complex matrix, we introduce two new generalized inverses, called the GOMP and MPGO inverses for an operator between two Banach spaces. We develop different characterizations and representations of these new inverses and describe the sets of all the GOMP and MPGO inverses. The GOMP and MPGO partial orders are defined by means of the GOMP and MPGO inverses, respectively. Using these results, we present and characterize the GDMP partial order as a particular case of the GOMP partial order. Applying GOMP and MPGO inverses, we solve some equations and especially the so-called normal equation.

Keywords:

G-outer inverse, G-Drazin inverse, GDMP inverse, minus partial order.