Multiplicator space and complemented subspaces of rearrangement invariant space

We show that the multiplicator space M(X) of an rearrangement invariant (r.i.) space X on [0,1] and the nice part N0(X) of X, that is, the set of all a∈X for which the subspaces generated by sequences of dilations and translations of a are uniformly complemented, coincide when the space X is separable. In the general case, the nice part is larger than the multiplicator space. Several examples of descriptions of M(X) and N0(X) for concrete X are presented