The g-Hirano inverse in Banach algebras
Huanyin Chen, Marjan Sheibani · Linear and Multilinear Algebra · 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. Additive results for the generalized Hirano inverse of Banach algebras are presented. The explicit formulas for the g-Hirano inverse of a+b are given. These extend the results on Drazin inverse of complex matrices of Dana and Yousefi [Formulas for the Drazin inverse of matrices with new conditions and its applications. Int J Appl Comput Math. 2018;4:697] and Ljubisavljevic and Cvetkovic-Ilic [Additive results for the Drazin inverse of block matrices and applications. J Comput Appl Math. 2011;235:3683–3690]. As an application we give some conditions under which a 2×2 operator matrix has generalized Hirano inverse.