The Characterizations and Representations for the Generalized Inverses With Prescribed Idempotents in Banach Algebras

Jianbing Cao, Yifeng Xue

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.