The Cesàro operator on weighted l_1 spaces

Abstract

[EN] Unlike for l(p), 1 < p <= infinity, the discrete Cesaro operator C does not map l(1) into itself. We identify precisely those weights w such that C does map l(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 l(1)(w). The final section investigates the mean ergodic properties of C in l(1)(w). Many examples are presented in order to supplement the results and to illustrate the phenomena that occur.