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.

在本文中，我们研究了作用于快速递减函数的施瓦茨空间\({\mathcal S}({\mathbb R})\)的乘法算子和卷积算子的谱和遍历特性，即形式\(M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})\)、\(f \mapsto hf \)和\(C_T:{\ mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})\) , \(f\mapsto T\star f\)。准确地说，我们确定它们的频谱并表征这些算子何时是幂有界和平均遍历的。