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

TL;DR

This paper characterizes weights for which the Cesàro operator acts boundedly and compactly on weighted  spaces, describes its spectrum and eigenvalues, and studies its ergodic properties.

Contribution

It provides a complete characterization of weights ensuring boundedness, compactness, and spectral properties of the Cesro operator on weighted  spaces, extending understanding beyond classical cases.

Findings

Identified weights for boundedness of C on (w)

Described the spectrum and eigenvalues of C in these spaces

Characterized weights for which C is compact and analyzed ergodic properties

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.