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

We characterize those pairs (ψ,φ) of smooth mappings ψ:Rᵈ,φ:Rᵈᵈ for which the corresponding weighted composition operator Cψ,φf=ψ·(f∘φ) acts continuously on S(Rᵈ). 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 Cψ,φ on S(Rᵈ) in terms of ψ,φ. Among other things, as an application of our results we show that for a univariate polynomial φ with deg(φ)≥ 2, power boundedness of Cψ,φ on S(R) for every ψM(R) only depends on φ and that in this case power boundedness of Cψ,φ is equivalent to (Cψ,φⁿ)ₙ converging to $0$ in Lb(S(R)) as well as to the uniform mean ergodicity of Cψ,φ. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator Cψ,φ on S(R) for which neither the multiplication operator f↦ ψ f nor the composition operator f↦ f∘φ acts on S(R). Our results complement and considerably extend various results of Fern\'andez, Galbis, and the second named author.