The Properties of the Generalized G-zhou Inverses and the Generalization of the Weighted Generalized Inverses Based on the W* Operator

Abstract In this paper, we have investigated the generalized inverse form of a class of elements in a ring, and propose a new generalized inverses generalized G-zhou inverses. An element a ∈ R is called generalized G-zhou invertible if there exists an element x ∈ R such that x ∈ comm 2 ( a ), xax = x , and a n + a n +1 f ( a ) – ab ∈ R qnil . And fulfill any x ∈ R , (1 + af ( a )) x ∈ R qnil ⇔ x ∈ R qnil . A few fundamental characteristics of the inverse are examined. For the inverse, we demonstrate expansions of Clines formula. We prove the generality of the generalized G-zhou inverse, which is a more general definition without the generalized inverse itself and an equivalent proof. Weighted Mosic-Abyzov inverses and Weight zhou inverses are defined and characterized in a Banach algebra. The equivalence of the w* operator on Banach algebras and ordinary operations is studied.