Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)$
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arXiv
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| Format: | Preprint |
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2024
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| author | Asensio, Vicente Jordá, Enrique Kalmes, Thomas |
| author_facet | Asensio, Vicente Jordá, Enrique Kalmes, Thomas |
| contents | We characterize those pairs $(ψ,φ)$ of smooth mappings $ψ:\mathbb{R}^d\rightarrow\mathbb{C},φ:\mathbb{R}^d\rightarrow\mathbb{R}^d$ for which the corresponding weighted composition operator $C_{ψ,φ}f=ψ\cdot(f\circφ)$ 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 $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $ψ,φ$. Among other things, as an application of our results we show that for a univariate polynomial $φ$ with $\text{deg}(φ)\geq 2$, power boundedness of $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for every $ψ\in\mathscr{O}_M(\mathbb{R})$ only depends on $φ$ and that in this case power boundedness of $C_{ψ,φ}$ is equivalent to $(C_{ψ,φ}^n)_{n\in\mathbb{N}}$ converging to $0$ in $\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean ergodicity of $C_{ψ,φ}$. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for which neither the multiplication operator $f\mapsto ψf$ nor the composition operator $f\mapsto f\circφ$ acts on $\mathscr{S}(\mathbb{R})$. Our results complement and considerably extend various results of Fernández, Galbis, and the second named author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01018 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)$ Asensio, Vicente Jordá, Enrique Kalmes, Thomas Functional Analysis 47B33, 47A35 We characterize those pairs $(ψ,φ)$ of smooth mappings $ψ:\mathbb{R}^d\rightarrow\mathbb{C},φ:\mathbb{R}^d\rightarrow\mathbb{R}^d$ for which the corresponding weighted composition operator $C_{ψ,φ}f=ψ\cdot(f\circφ)$ 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 $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $ψ,φ$. Among other things, as an application of our results we show that for a univariate polynomial $φ$ with $\text{deg}(φ)\geq 2$, power boundedness of $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for every $ψ\in\mathscr{O}_M(\mathbb{R})$ only depends on $φ$ and that in this case power boundedness of $C_{ψ,φ}$ is equivalent to $(C_{ψ,φ}^n)_{n\in\mathbb{N}}$ converging to $0$ in $\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean ergodicity of $C_{ψ,φ}$. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator $C_{ψ,φ}$ on $\mathscr{S}(\mathbb{R})$ for which neither the multiplication operator $f\mapsto ψf$ nor the composition operator $f\mapsto f\circφ$ acts on $\mathscr{S}(\mathbb{R})$. Our results complement and considerably extend various results of Fernández, Galbis, and the second named author. |
| title | Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)$ |
| topic | Functional Analysis 47B33, 47A35 |
| url | https://arxiv.org/abs/2405.01018 |