The characterizations and representations for the generalized inverses with prescribed idempotents in Banach algebras

In this paper, we investigate the various different generalized inverses in a Banach algebra with respect to prescribed two idempotents $p$ and $q$. Some new characterizations and explicit representations for these generalized inverses, such as $a_{p,q}^{(2)}$, $a_{p,q}^{(1,2)}$ and $a_{p,q}^{(2,l)}$ will be presented. The obtained results extend and generalize some well–known results for matrices or operators.