Drazin Inverses of Operators With Rational Resolvent

Christoph Schmoeger

Let $A$ be a bounded linear operator on a Banach space such that the resolvent of $A$ is rational. If $0$ is in the spectrum of $A$, then it is well known that $A$ is Drazin invertible. We investigate spectral properties of the Drazin inverse of $A$. For example we show that the Drazin inverse of $A$ is a polynomial in $A$.