Multiplication operators on weighted spaces in the non-locally convex framework

Abstract

Let X be a completely regular Hausdorff space, E a topological vector space, V a Nachbin family of weights on X, and CV0(X,E) the weighted space of continuous E-valued functions on X. Let θ:X→C be a mapping, f∈CV0(X,E) and define Mθ(f)=θf (pointwise). In case E is a topological algebra, ψ:X→E is a mapping then define Mψ(f)=ψf (pointwise). The main purpose of this paper is to give necessary and sufficient conditions for Mθ and Mψ to be the multiplication operators on CV0(X,E) where E is a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption that E is locally convex.

Keywords

• Nachbin family of weights
• topological vector spaces
• vector-valued continuous functions
• weighted topology
• multiplication operators
• locally idempotent topological algebras.