Multiplication operators and dynamical systems

Let J be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CV_b(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CV_b(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.