Operators and Dynamical Systems on Weighted Function Spaces

Let X be a completely regular Hausdorff space, let V be a system of weights on X and let E be a locally convex Hausdorff space. Let CV₀(X, E) and CV_b(X, E) be the weighted locally convex spaces of vector‐valued continuous functions with a topology generated by seminorms which are weighted analogue of the supremum norm. In the present paper, we characterize multiplication operators and weighted composition operators on the spaces CV₀(X, E) and CV_b(X, E) induced by scalar‐valued and vector‐valued mappings. A (linear) dynamical system on these weighted spaces is obtained as an application of the theory of multiplication operators.