Weak group inverses and partial isometries in proper *-rings

Abstract

[EN] A weak group element is introduced in a proper ¿-ring. Several equivalent conditions of weak group elements are investigated. We prove that an element is pseudo core invertible if it is both partial isometry and weak group invertible. Reverse order law and additive property of the weak group inverse are presented. Finally, under certain assumption on a, equivalent conditions of aW a¿ = a¿aW are presented by using the normality of the group invertible part of an element in its group-EP decomposition