The Cesàro operator in weighted ${\ell }_1$ spaces

Abstract

Unlike for ${\ell }_p$, $1<p\le \infty$, the discrete Cesàro operator $C$ does not map ${\ell }_1$ into itself. We identify precisely those weights $w$ such that $C$ does map ${\ell }_1(w)$ continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of $C$ are presented. It is also possible to identify all $w$ such that $C$ is a compact operator in ${\ell }_1(w)$. The final section investigates the mean ergodic properties of $C$ in ${\ell }_1(w)$. Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.