Order spectrum of the Cesàro operator in Banach lattice sequence spaces

Abstract

The discrete Cesàro operator $C$ acts continuously in various classical Banach sequence spaces within ${\mathbb{C}}^{\mathbb{N}}.$ For the coordinatewise order, many such sequence spaces $X$ are also complex Banach lattices (eg. $c_0,{\ell }^p$ for $1<p\le \infty ,$ and $ces(p)$ for $p\in \{0\}\cup (1,\infty )).$ In such Banach lattice sequence spaces, $C$ is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on $X.$ The purpose of this note is to show, for every $X$ belonging to the above list of Banach lattice sequence spaces, that the order spectrum ${\sigma }_{\mathrm{o}}(C)$ of $C$ coincides with its usual spectrum $\sigma (C)$ when $C$ is considered as a continuous linear operator on the Banach space $X.$

9 pages