The Ces\`aro operator in weighted $\ell_1$ spaces

Abstract

Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Ces\`aro 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.