Generalized group elements and their orders

Abstract

In this paper, we present reverse order of generalized group inverse in a proper Banach algebra. We characterize generalized group elements and determine when an element and its generalized group inverse commute each other. The properties of the generalized group orders are investigated. As applications, new properties of the core-EP order of two complex matrices and Hilbert space operators are thereby extended to wider cases. 2020 Mathematics Subject Classification. 15A09, 16U99, 46H05.

m-generalized group inverse in Banach algebras

“…Such x is unique if it exists, denoted by a g , and called the generalized group inverse of a (see [2,3]). We list some characterizations of generalized group inverse.…”