Generalized Hirano inverses in Banach algebras
Huanyin Chen, Marjan Sheibani · Filomat · 2019
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, a2-ab ? Aqnil. We prove that a ? A has generalized Hirano inverse if and only if a - a3 ? Aqnil, if and only if a is the sum of a tripotent and a quasinilpotent that commute. The Cline?s formula for generalized Hirano inverses is thereby obtained. Let a, b ? A have generalized Hirano inverses. If a2b = aba and b2a = bab, we prove that a + b has generalized Hirano inverse if and only if 1 + adb has generalized Hirano inverse. The generalized Hirano inverses of operator matrices on Banach spaces are also studied.