On the generalized n-strong Drazin inverses and block matrices in Banach algebras

Let \(\mathcal {A}\) $\mathcal{A}$ be a complex unital Banach algebra. The purpose of this paper is to give a new characterization of generalized n-strong Drazin invertible elements by means of their spectra. Consequently, we address key results in relation with the problem of existence and representations of the generalized n-strong Drazin inverse of the block matrix \(x=\left( \begin{array}{cc}a&{}b\\ c&{}d\end{array}\right) _{p}\) $x={\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)}_p$ relative to the idempotent p, with a is generalized Drazin invertible such that \(a^{d}\) $a^d$ is its generalized Drazin inverse in \(p \mathcal {A}p\) $p\mathcal{A}p$ , under the more general case of the generalized Schur complement \(s=d-ca^{d}b\) $s=d-c{a}^{d}b$ being generalized Drazin invertible.