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