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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.

Weighted G-Outer Inverse of Banach Spaces Operators