The Cesàro operator on power series spaces

The discrete Cesàro operator $C$ is investigated in the class of power series spaces $\Lambda_0(\alpha)$ of finite type. Of main interest is its spectrum, which is distinctly different in the cases when $\Lambda_0(\alpha)$ is nuclear and when it is not. Actually, the nuclearity of $\Lambda_0(\alpha)$ is characterized via certain properties of the spectrum of $C$. Moreover, $C$ is always power bounded, uniformly mean ergodic, and whenever $\Lambda_0(\alpha)$ is nuclear, also $(\lambda- C)^m(\Lambda_0(\alpha))$ is closed in $\Lambda_0(\alpha)$ for each $m\in\mathbb{N}$.