  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 $a^{(2)}_{p,q}$, $a^{(1,2)}_{p,q}$ and $a^{(2,l)}_{p,q}$ will be
presented. The obtained results extend and generalize some well--known results
for matrices or operators.
