Convolution operators in discrete Cesàro spaces

Abstract

The discrete Cesaro spaces $${{\text {ces(p )}}}, 1< p < \infty ,$$ are well known and arise from the classical sequence spaces $$\ell ^p,1<p<\infty $$ , via the process of averaging. It is known that a sequence $$b\in {\mathbb {C}}^{\mathbb {N}}$$ convolves $${{\text {ces(p )}}}$$ into itself if and only if $$b\in \ell ^1$$ , which is a very different situation than for convolution in the spaces $$\ell ^p$$ . The purpose of this note is to determine the spectrum of the convolution operator $$a \mapsto b*a$$ , for $$a\in {{\text {ces(p )}}}$$ , whenever $$b\in \ell ^1$$ . It is then possible to describe the mean ergodic properties of such a convolution operator.