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

Let \(\mathcal {A}\) 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 \(x=\left( \begin{array}{cc}a&{}b\\ c&{}d\end{array}\right) _{p}\) relative to the idempotent p, with a is generalized Drazin invertible such that \(a^{d}\) is its generalized Drazin inverse in \(p \mathcal {A}p\), under the more general case of the generalized Schur complement \(s=d-ca^{d}b\) being generalized Drazin invertible.