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

Let 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= (( a b) (c d) )(p) relative to the idempotent p, with a is generalized Drazin invertible such that a(d) is its generalized Drazin inverse in pAp, under the more general case of the generalized Schur complement s=d-ca(d)b being generalized Drazin invertible.