The Ces\`aro operator in weighted ${\mathrm{\ell }}_1$ spaces

Unlike for ${\mathrm{\ell }}_p$, $1<p\le \mathrm{\infty }$, the discrete Ces`aro operator $C$ does not map ${\mathrm{\ell }}_1$ into itself. We identify precisely those weights $w$ such that $C$ does map ${\mathrm{\ell }}_1(w)$ 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 ${\mathrm{\ell }}_1(w)$. The final section investigates the mean ergodic properties of $C$ in ${\mathrm{\ell }}_1(w)$. Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.