The $(b,c)$-inverse in rings and in the Banach context

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 $R$. 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 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