Additive Property of Pseudo Drazin Inverse of Elements in Banach Algebras

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 $ab = \lambda ba$, we also show that $a -b$ is pseudo Drazin invertible if and only if $aa^\ddag (a-b)bb^\ddag$ is pseudo Drazin invertible.