A note on mutiplication operators on Köthe-Bochner spaces

Abstract

Let (Ω, A, μ) is a finite measure space, E an order continuous Banach function space over μ, X a Banach space and E(X) the Köthe-Bochner space. A new simple proof is given of the result that a continuous linear operator T: E(X) ® E(X) is a multiplication operator (by a function in L¥) iff T(g < f, x* > x) =g < T(f), x* > x for every g Î L¥, f Î E(X), x Î X, x* Î X*.