Additive results for the generalized n-strong Drazin inverse in Banach algebras

In this paper, we obtain some equivalent characterizations of the generalized [Formula: see text]-strong Drazin invertibility in Banach algebras. By utilizing Harte’s techniques, we demonstrate some additive results for two generalized [Formula: see text]-strong Drazin invertible elements under certain conditions. In particular, we prove that [Formula: see text] is generalized [Formula: see text]-strong Drazin invertible if and only if [Formula: see text] possesses the same property, provided that [Formula: see text] and [Formula: see text] commute in Banach algebras. Finally, we derive the common generalized [Formula: see text]-strong Drazin invertibility of two elements under certain intertwined conditions.