  In this paper we investigate the spectra and the ergodic properties of the
multiplication operators and the convolution operators acting on the Schwartz
space $\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions, i.e.,
operators of the form $M_h: \mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$,
$f \mapsto h f $, and $C_T\colon
\mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$, $f\mapsto T\star f$.
Precisely, we determine their spectra and characterize when those operators are
power bounded and mean ergodic.
