Spectra and ergodic properties of multiplication and convolution operators on the space 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 S ( R ) of rapidly decreasing functions, i.e., operators of the form M h : S ( R ) → S ( R ) , f ↦ h f , and C T : S ( R ) → S ( R ) , f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.