Spectra and ergodic properties of multiplication and convolution operators on the space $\mathcal{S}(\mathbb{R})$

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