Generalized group inverse in Banach *-algebras

Abstract

In this paper, we introduce the notion of the generalized group inverse in a *-Banach algebra. This is a natural generalization of weak group inverse for a complex matrix and bounded linear operator over a Hilbert space. We present polar-like property for the generalized group inverse and characterize it by the generalized Drazin inverse. Furthermore, the relations between generalized group inverse and generalized core-EP inverse are investigated. 2020 Mathematics Subject Classification. 15A09, 16U90, 46H05.

Weighted Generalized Group Inverse in Banach *-Algebras

“…If x satisfies the system of conditions mentioned above, then wx = (wa) g . Therefore x = a(wx) 2 = a[(wa) g ] 2 , as asserted.…”

“…As a generalization of weak group inverse mentioned above, the author introduced and studied generalized group inverse (see [2]). An element a ∈ A has generalized group inverse if there exists x ∈ A such that…”

“…Theorem 1. (see [2] Theorem 2.2, Theorem 4.1 and Theorem 5.1) Let A be a Banach *-algebra, and let a ∈ A. Then the following are equivalent:…”

On generalized core invertibility in Banach algebras

“…In [3,4], the authors introduced and studied generalized core-EP inverse and generalized group inverse for an element in a Banach *-algebra. An element a ∈ A has generalized core-EP inverse if there exists a x ∈ A such that…”

“…Such x is unique if it exists, and denote it by a g ❖ (see [4]). An element a ∈ A has More-Penrose inverse provided that there exists some x ∈ A such that a = axa, x = xax, (ax) * = ax, (xa) * = xa.…”