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In this paper we investigate the spectra and the ergodic properties of themultiplication operators and the convolution operators acting on the Schwartzspace $\mathcal{S}(\mathbb{R})$ 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 arepower bounded and mean ergodic.

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