Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)$

TL;DR

This paper characterizes when weighted composition operators act continuously, are power bounded, or topologize on Schwartz space, providing new criteria and examples, especially for polynomial symbols with degree at least two.

Contribution

It offers new characterizations of boundedness and power boundedness of weighted composition operators on Schwartz space, including conditions for special cases and polynomial symbols.

Findings

Characterization of continuous weighted composition operators on Schwartz space.

Conditions for power boundedness and topologizability in terms of symbols.

Example of a power bounded operator not decomposable into simpler operators.

Abstract

We characterize those pairs $(\psi,\varphi)$ of smooth mappings $\psi:\mathbb{R}^d\rightarrow\mathbb{C},\varphi:\mathbb{R}^d\rightarrow\mathbb{R}^d$ for which the corresponding weighted composition operator $C_{\psi,\varphi}f=\psi\cdot(f\circ\varphi)$ acts continuously on $\mathscr{S}(\mathbb{R}^d)$. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $\psi,\varphi$. Among other things, as an application of our results we show that for a univariate polynomial $\varphi$ with $\text{deg}(\varphi)\geq 2$, power boundedness of $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for every $\psi\in\mathscr{O}_M(\mathbb{R})$ only depends on $\varphi$ and that in this case power boundedness of $C_{\psi,\varphi}$ is equivalent to $(C_{\psi,\varphi}^n)_{n\in\mathbb{N}}$ converging to $0$ in $\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean ergodicity of $C_{\psi,\varphi}$. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for which neither the multiplication operator $f\mapsto \psi f$ nor the composition operator $f\mapsto f\circ\varphi$ acts on $\mathscr{S}(\mathbb{R})$. Our results complement and considerably extend various results of Fern\'andez, Galbis, and the second named author.