The Cesàro operator in weighted $\ell_1$ spaces

Unlike for $\ell_p$, $1 < \leq\infty$, the discrete Cesàro operator $$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $$ does map $\ell_1(w)$ continuously into itself. For these weights a complete description of the eigenvalues and the spectrum of $$ are presented. It is also possible to identify all $w$ such that $$ is a compact operator in $\ell_1(w)$. The final section investigates the mean ergodic properties of $$ in $\ell_1(w)$. Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.