The Cesàro operator on power series spaces
Volume 240 / 2018
Studia Mathematica 240 (2018), 47-68
MSC: Primary 47A10, 47B37; Secondary 46A04, 47A16, 46A45.
DOI: 10.4064/sm8590-2-2017
Published online: 23 June 2017
Abstract
The discrete Cesàro operator $\mathsf {C}$ is investigated in the class of power series spaces $\varLambda _0(\alpha )$ of finite type. Of main interest is its spectrum, which is distinctly different in the cases when $\varLambda _0(\alpha )$ is nuclear and when it is not. Actually, the nuclearity of $\varLambda _0(\alpha )$ is characterized via certain properties of the spectrum of $\mathsf {C}$. Moreover, $\mathsf {C}$ is always power bounded, uniformly mean ergodic, and whenever $\varLambda _0(\alpha )$ is nuclear, also $(I-\mathsf {C})^m(\varLambda _0(\alpha ))$ is closed in $\varLambda _0(\alpha )$ for each $m\in {\mathbb N}$.
