On the semi-continuity of generalized inverses in Banach algebras

The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore–Penrose inverse in a C*-algebra and therefore, derive some new and well-known criteria in operator theory.