Multiplicationo perators and dynamicals ystems on weighted spaces of cross-sections

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.