The Cesàro operator on power series spaces

Abstract

The discrete Cesàro 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 Fréchet 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 $m\in \mathbb{N}$.

This article is accepted for publication in Studia Math

Cited by in corpus (1)

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