Bounded weighted composition operators on functional quasi-Banach spaces and stability of dynamical systems

Abstract

In this paper, we investigate the boundedness of weighted composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We show that the boundedness of a weighted composition operator strongly limits the dynamics of the original map, and it provides us an effective method to investigate properties of weighted composition operators via the theory of dynamical system. As a result, we prove that only an affine map can induce a bounded composition operator on an arbitrary quasi-Banach space continuously included in the space of entire functions of one-variable. We also obtain the same result for bounded weighted composition operators on infinite dimensional quasi-Banach space under certain condition for weights. We also prove that any polynomial automorphism except an affine transform cannot induce a bounded weighted composition operator with non-vanishing weight on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under certain conditions.Comment: We generalize the previous version to the weighted cas