A note on mutiplication operators on K the-Bochner spaces

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 K the-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 L￥, f E(X), x X, x* X*.