Weighted composition operators on nonlocally convex weighted spaces of continuous functions

Let V be a system of weights on a completely regular Hausdorff space X and let B(E) be the topological vector space of all continuous linear operators on a general topological vector space E. Let CV₀(X, E) and CV_b(X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operators W_{π, Φ} on CV₀(X, E) (or CV_b(X, E)) induced by the operator-valued mappings π : X → B(E) (or the vector-valued mappings π : X → E, where E is a topological algebra) and the self-map Φ of X. Also, we characterize the mappings π : X → B(E) (or π : x → E) and π : X → X which induce the compact weighted composition operators on these weighted spaces of continuous functions.