Generalized Hirano inverses in Banach algebras

Abstract

Let A be a Banach algebra. An element a ∈ A has generalized Hirano inverse if there exists b ∈ A such that b = bab, ab = ba, a 2 − ab ∈ A qnil .We prove that a ∈ A has generalized Hirano inverse if and only if a has g-Drazin inverse and a− a 3 ∈ A qnil , if and only if there exists p 3 = p ∈ comm(a) such that a − p ∈ A qnil . The Cline's formula for generalized Hirano inverses are thereby obtained. Let a, b ∈ A have generalized Hirano inverse. If a 2 b = aba and b 2 a = bab, we prove that a + b has generalized Hirano inverse if and only if 1 + a d b has generalized Hirano inverse. Hirano inverses of operator matrices over Banach spaces are also studied.2010 Mathematics Subject Classification. 15A09, 32A65, 16E50.

Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras

“…In detail, the generalized n-strong Drazin inverse for n = 1, namely, the generalized strong Drazin inverse, was studied by Gürgün et al in [19,24]. The generalized n-strong Drazin inverse for n = 2 (so-called the generalized Hirano inverses) was investigated by Chen et al in [8,11]. The first purpose of the paper is to illustrate the generalized n-strong Drazin inverse for n ∈ N in terms of Jacobson's lemma.…”

Generalized group elements and their orders

“…Such x is unique, if it exists, and we denote it by a d ❖ (see [4]). It was proved that a ∈ A has generalized core-EP inverse if and only if it has generalized core-EP decomposition, i.e., there exist x, y ∈ A such that a = x + y, x * y = yx = 0, x ∈ A # ❖ , y ∈ A qnil (see [4,Theorem 1.2]).…”

New characterizations of G-Drazin inverse in Banach algebra

“…We shall prove that regular condition "x = xax" can be dropped from the definition of g-Drazin inverse. An element a ∈ A has strongly Drazin inverse if it is the sum of an idempotent and a quasinilpotent that commute (see [4]). We begin with a characterization of strongly Drazin inverse.…”