Subspaces Generated by Translations in Rearrangement Invariant Spaces

In the setting of rearrangement invariant (r.i.) Banach function spaces E on [0, ∞) we study the complementability of subspaces Qa generated by sequences of translations of functions a∈E[0, 1). An r.i. function space E is said to be nice (in short, E∈N) if every subspace of type Qa is complemented. We give necessary and sufficient conditions for an r.i. function space to be nice. We determinate the Orlicz, Lorentz and Marcinkiewicz spaces belonging to the class N. As an application we obtain a new characterization of the Lp-spaces, 1<p<∞, among the class of r.i. function spaces.