Weighted composition operators on non locally convex weighted spaces with operator-valued weights

In this paper, we propose a general framework for the study of the weighted spaces of vector-valued continuous functions. We consider operator-valued weights and assume no local convexity condition. Precisely, for a separated topological vector space A, a Hausdorff completely regular space X, and a family V of mappings (called weights) v from X into the algebra L(A) of all continuous operators on A, we consider the generalized weighted spaces CV (X, A) = {f : X -> A continuous : vf is bounded on X, v is an element of V} and CV0(X, A) := {f is an element of CV (X, A) : vf vanishes at infinity, v is an element of V}, endowed with a linear topology tau(V) associated with V. We then present some of their properties. We further investigate the weighted composition operators psi C-phi from a subspace E of CV (X, A) into CU (Y, A) or CU0(Y, A), where Y is a Hausdorff completely regular space, U is a family of weights on Y, psi : Y -> L(A) and phi : Y -> X are mappings, and psi C-phi(f)(y) := psi(y)[f(phi(y))], y is an element of Y and f is an element of E. In particular, we give necessary and sufficient conditions for such an operator to be continuous, bounded, or locally equicontinuous.