The (b, c)-Inverse in Rings and in the Banach Context

Abstract

In this article, the (b, c)-inverse will be studied. Several equivalent conditions for the existence of the (b, c)-inverse in rings will be given. In particular, the conditions ensuring the existence of the (b, c)-inverse, of the annihilator (b, c)-inverse and of the hybrid (b, c)-inverse, will be proved to be equivalent, provided b and c are regular elements in a unitary ring \(\mathcal {R}\). In addition, the set of all (b, c)-invertible elements will be characterized and the reverse order law will be also studied. Moreover, the relationship between the (b, c)-inverse and the Bott–Duffin (p, q)-inverse will be considered. In the context of Banach algebras, integral, series and limit representations will be given. Finally, the continuity of the (b, c)-inverse will be characterized.