Order spectrum of the Ces\`aro operator in Banach lattice sequence spaces

The discrete Ces\`aro 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 $1 < p \leq \infty ,$ 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 $\sigma_{\rm o} (C)$ of $C$ coincides with its usual spectrum $\sigma ( C)$ when $C$ is considered as a continuous linear operator on the Banach space $X .$