Weighted G-Outer Inverse of Banach Spaces Operators

In order to extend the notions of G-outer inverses and weighted G-Drazin inverses, we firstly define and characterize the (M, N)-weighted outer inverses for Banach spaces operators. Using the (M, N)-weighted outer inverse, beside the (M, N)-weighted outer equivalence relation, we create and study the (M, N)-weighted G-outer inverse of an operator between two Banach spaces. This new inverse presents generalization of the G-outer inverse and the W-weighted G-Drazin inverse. By means of (M, N)-weighted G-outer inverse, we also introduce and investigate the (M, N)-weighted G-outer relation which is a preorder on corresponding set. Thus, we extend the notions of the G-outer partial order and the W-weighted G-Drazin preorder for Banach spaces operators.