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

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