The Ces\`aro operator on power series spaces

Abstract

The discrete Ces\`aro operator $\mathsf{C}$ 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 when the underlying Fr\'echet space $\Lambda_0(\alpha)$ is nuclear as for the case when it is not. Actually, the nuclearity of $\Lambda_0(\alpha)$ is characterized via certain properties of the spectrum of $\mathsf{C}$. Moreover, $\mathsf{C}$ is always power bounded, uniformly mean ergodic and, whenever $\Lambda_0(\alpha)$ is nuclear, also has the property that the range $(I-\mathsf{C})^m(\Lambda_0(\alpha))$ is closed in $\Lambda_0(\alpha)$, for each $m\in\mathbb{N}$.