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

Abstract

[EN] We characterize those pairs of smooth mappings for which the corresponding weighted composition operator acts continuously on . 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 on in terms of . Among other things, as an application of our results we show that for a univariate polynomial ¿ with , power boundedness of on for every only depends on ¿ and that in this case power boundedness of is equivalent to converging to 0 in as well as to the uniform mean ergodicity of . Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator on for which neither the multiplication operator nor the composition operator acts on . Our results complement and considerably extend various results of Fernández, Galbis, and the second named author.