ACTA UNIVERSITATIS APULENSIS No 19/2009 ON PRECOMPACT MULTIPLICATION OPERATORS ON WEIGHTED FUNCTION SPACES Hamed H. Alsulami, Saud M. Alsulami and Liaqat Ali Khan Abstract. Let X be a completely regular Hausdorff space, E a Hausdorff topo- logical vector space, CL(E) the algebra of continuous operators on E,V a Nachbin family on X and F⊆ CV b (X, E) a topological vector space (for a given topol- ogy). If π : X → CL(E) is a mapping, consider the induced multiplication operator M π : F→F given by M π (f )(x) := π(x)f (x), f ∈ G, x ∈ coz (F ). In this paper we give necessary and sufficient conditions for the induced linear map- ping M π to be (1) an equicontinuous operator, (2) a precompact operator and (3) a bounded operator on a subspace F of CV b (X, E) in the non-locally convex setting. 2000 Mathematics Subject Classification : 47B38, 46E40, 46A16 1. Introduction The fundamental work on weighted spaces of continuous scalar-valued functions has been done mainly by Nachbin [21, 22] in the 1960’s. Since then it has been studied extensively for a variety of problems by Bierstedt [2, 3], Summers [35, 36], Prolla [25, 26], Ruess and Summers [27], Khan [10, 11], Singh and Summers [34], Nawrocki [23], Khan and Oubbi [12] and many others. The multiplication operators M π on the Weighted spaces CV b (X, E) and CV o (X, E) were first considered by Singh and Manhas in [29] in the cases of π : X → C and π : X → E and later in [30] in the case of π : X → CL(E), E a locally convex space. This class form a special class of the more general notion of weighted composition operators W π,ϕ , where ϕ : X → X [8, 33, 34]. The extension of these results to non-locally convex setting have been given later in [13, 14, 20]. The compactness of weighted composition operators W π,ϕ and various other types of operators on spaces of continuous functions have also been studied exten- sively in recent years by many authors; see, e.g. [7, 8, 33, 6, 37, 38, 5, 16, 31, 32, 19, 125