Additive property of pseudo Drazin inverse of elements in a Banach algebra

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 $1+a^{\mathrm{‡}}b$ is pseudo Drazin invertible. Moreover, the formula of $(a+b{)}^{\mathrm{‡}}$ is presented . When the commutative condition is weaken to $ab=\lambda ba(\lambda \mathrm{\ne }0)$, we also show that $a-b$ is pseudo Drazin invertible if and only if $a{a}^{\mathrm{‡}}(a-b)b{b}^{\mathrm{‡}}$ is pseudo Drazin invertible.