Resumo:  We present in this talk additive properties for the $g$-Drazin inverse in a complex Banach algebra. Some recent papers deal with the problem of finding an explicit expression for the $g$-Drazin inverse of $a+b$ in terms of $a$, $b$ $a^D$, and $b^D$. We will provide representations of $(a+b)^D$ under conditions $a^2b =ab^2=0$, and  when the weaker conditions $a^Db=0$,  $a^2 ba^{\pi}=ab^2a^{\pi}=0$ are assumed. The auxiliary results used in our development involve the resolvent of a $2\times 2$ matrix with entries in a Banach algebra, and the square of its $g$-Drazin inverse. Our results recover the case $ab=0$ studied by Hartwig et al.  We will comment on the application of the additive results to obtain representations of the Drazin inverse of a $2\times2$  complex block matrix  in terms of the individual blocks.