Multiplication Operators and Dynamical Systems on Weighted Spaces of Cross-Sections

Abstract

Abstract. Let Y be a Hausdorff topological space, let V be a system of weights on Y, and let LVq{Y) and LVb{Y) be the weighted locally convex spaces of cross-sections with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper, we characterise the multiplication operators on the spaces LV0{Y) and LVb(Y) induced by the scalar-valued, the vector-valued, and the operator-valued mappings. A (linear) dynamical system on the weighted spaces of cross-sections is obtained as an application of the theory of the multiplication operators. Many examples are given to illustrate the theory.