Some converse problems on the g-Drazin invertibility in Banach algebras

The main purpose of this paper is to investigate the converse problems of some well-known results related to the generalized Drazin (g-Drazin for short) inverse in Banach algebras. Let \({\mathcal {A}}\) $\mathcal{A}$ be a Banach algebra and \(a,b\in {\mathcal {A}}\) $a,b\in \mathcal{A}$ . First, we give the relationship between the Drazin (g-Drazin, group) invertibility of a, b and that of the sum \(a+b\) $a+b$ under certain conditions. Then, for a given polynomial \(f(x)\in {\mathbb {C}}[x]\) $f(x)\in \mathbb{C}[x]$ , the g-Drazin invertibility of f(a), \(f(a^{d})\) $f(a^d)$ , f(ab), \(f(1-ab)\) $f(1-ab)$ and \(f(a+b)\) $f(a+b)$ are investigated.