Multiplication operators on vector measure Orlicz spaces ✩

Let m be a countably additive vector measure with values in a real Banach space X, and let L1(m) and Lw(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces LΦ(m) and LΦw(m) and establish that the Banach space of multiplication operators going from W = LΦ(m) into Y = L1 (m) is M = LΨw (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = LΦw(m), Y = L1(m) we have M = LΨw (m), and when W = LΦw(m), Y = L1(m) we have M = LΨ (m).