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 ∗ .