  We study properties of pseudo Drazin inverse in a Banach algebra with unity
1. If $ab=ba$ and $a,b$ are pseudo Drazin invertible, we prove that $a+b$ is
pseudo Drazin invertible if and only if $1+a^\ddag b$ is pseudo Drazin
invertible. Moreover, the formula of $(a+b)^\ddag$ is presented . When the
commutative condition is weaken to $ab=\lambda ba ~(\lambda \neq 0)$, we also
show that $a-b$ is pseudo Drazin invertible if and only if
$aa^\ddag(a-b)bb^\ddag$ is pseudo Drazin invertible.
