Operator-valued multiplication operators on weighted function spaces

Let X be a completely regular Hausdorff space, E a Hausdorff topological vector space, V a Nachbin family of weights on X, and CVi,(X,E) the weighted space of continuous /^-valued functions on X. Let B(E) be the vector space of all continuous linear mappings from E into itself, endowed with the topology of uniform convergence on bounded sets. If φ : X —* B(E) is a continuous mapping and / € CV^(X, E), let Μψ(}) = %l>f, where (ifif)(x) — ip(x)(f(x)) (χ ζ X). In this paper we give a necessary and sufficient condition for Μφ to be the multiplication operator (i.e. continuous self-mapping) on CV¡,(X, E), where E is a general space or a locally bounded space. These results extend recent work of Singh and Manhas to a non-locally convex setting and that of the authors where φ has been considered to be a complex or ^-valued map.