Power boundedness and related properties for weighted composition operators on S(Rd)

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