Smoothness of generalized inverses

The paper is largely expository. It is shown that if a(x)a(x) is a smooth unital Banach algebra valued function of a parameter xx, and if a(x)a(x) has a locally bounded generalized inverse in the algebra, then a generalized inverse of a(x)a(x) exists which is as smooth as a(x)a(x) is. Smoothness is understood in the sense of having a certain number of continuous derivatives, being real-analytic, or complex holomorphic. In the complex holomorphic case, the space of parameters is required to be a Stein manifold. Local formulas for the generalized inverses are given. In particular, the Moore–Penrose and the generalized Drazin inverses are studied in this context.