Multiplication operators in Köthe–Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe–Bochner space E(X) is a multiplication operator (by a function in L∞(μ)) if and only if the equality T(g〈f,x∗〉x)=g〈T(f),x∗〉x holds for every g∈L∞(μ), f∈E(X), x∈X and x∗∈X∗.