The Ces\`aro operator on power series spaces

TL;DR

This paper studies the spectral properties of the Cesàro operator on power series spaces, revealing how nuclearity influences its spectrum and ergodic behavior, with implications for the structure of these spaces.

Contribution

It characterizes the spectrum of the Cesàro operator on power series spaces and links nuclearity to spectral and ergodic properties, providing new insights into these functional spaces.

Findings

Spectrum varies with nuclearity of the space

Cesàro operator is always power bounded and mean ergodic

Nuclearity implies closedness of certain operator ranges

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 $m\in\mathbb{N}$.