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

Abstract

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

22 pages; comments welcome; minor editorial changes; accepted for publication in Journal of Functional Analysis

References in corpus (5)

• Power bounded weighted composition operators on function spaces defined by local properties
• Mean ergodic composition operators in spaces of homogeneous polynomials
• Non power bounded generators of strongly continuous semigroups
• Topologizable and power bounded weighted composition operators on spaces of distributions
• Mean ergodic composition operators on spaces of smooth functions and distributions