Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Abstract

Let X be a rearrangement invariant function space on [ 0 , 1 ] . We consider the subspace Radi X of X which consists of all functions of the form f = ∑ k = 1 ∞ x k r k , where x k are arbitrary independent functions from X and r k are usual Rademacher functions independent of { x k } . We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X ′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [ 0 , ∞ ) . This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)].