Weak group inverses and partial isometries in proper *-rings

Abstract

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 a W a * = a * a W are presented by using the normality of the group invertible part of an element in its group-EP decomposition.

Generalized group elements and their orders

“…The involution * is proper if x * x = 0 =⇒ x = 0 for any x ∈ A, e.g., in a C * -algebra, the involution is always proper. In [26], Zou et al extend the notion of weak group inverse from complex matrices to elements in a ring with proper involution. An element a ∈ A has weak group inverse provided that there exist x ∈ A and k ∈ N such that x = ax 2 , (a * a 2 x) * = a * a 2 x, a k = xa k+1 .…”

“…If such x exists, it is unique, and denote it by a W ❖ . We refer the reader for weak group inverse in [8,9,12,18,19,20,22,23,26,27,28].…”

Generalized group inverse in Banach *-algebras

“…By virtue of Corollary 6.5, we see thatthe generalized group inverse and weak group inverse for any complex matrix coincide with each other. Recall that an element a ∈ A is partial isometry provided that a * = a † (see [22,27]). We come now to characterize the generalized group inverse for partial isometry.…”