Multipliers between Orlicz sequence space

Let $ M,N $ be Orlicz functions and let $ D(l_M,l_N) $ be the space of all diagonal operators (that is multipliers) acting between the Orlicz sequence spaces $ l_M $ and $ l_N $. We prove that the space of multipliers $ D(l_M,l_N) $ coincides with (and is isommorphic to) the Orlicz sequence space $ l_{M_{N}^*} $, where $ M_{N}^* $ is the Orlicz function defined by $ M_N^*( \lambda )=\text{sup} \lbrace N( \lambda x)-M(x),x\in(0,1) \rbrace $