Weighted composition operators on non-locally convex weighted spaces

Let (A, tau) be a topological vector space, X and Y Hausdorff completely regular spaces and V and U Nachbin families on X and Y respectively. For a pair of maps phi : Y -> X and psi : Y -> L(A), L(A) being the vector space of continuous operators from A into itself, we study the conditions under which the corresponding weighted composition operator V)C,, assigning to each f E CV(X, A) the function y -> psi(y)(f circle phi(y)), maps a subspace E of CV(X, A) continuously into another given subspace F of CU(Y, A). We also examine when V)C, is bounded, (locally) equicontinuous or (locally) precompact from E into F.