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

TL;DR

This paper analyzes the spectral and ergodic properties of multiplication and convolution operators on the Schwartz space, providing explicit spectra and conditions for power boundedness and mean ergodicity.

Contribution

It offers a detailed spectral analysis and ergodic characterization of these operators on Schwartz space, a topic not extensively covered before.

Findings

Determined the spectra of multiplication and convolution operators.

Characterized when these operators are power bounded.

Identified conditions for mean ergodicity.

Abstract

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 $M_h: \mathcal{S}(\mathbb{R})\to\mathcal{S}(\mathbb{R})$, $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 are power bounded and mean ergodic.