Multipliers on the Set of Rademacher Series in Symmetric Spaces

Abstract

Let E be a symmetric space on [0,1]. Let Λ(ℛ,E) be the space of measurable functions f such that fg∈ E for every almost everywhere convergent series g=∑bnrn∈ E, where (rn) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space Λ(ℛ,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for Λ(ℛ,E) to be order isomorphic to L∞. This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which Λ(ℛ,E) is order isomorphic to a symmetric space that differs from L∞. The answer is positive for the Orlicz spaces E=LΦq with Φq(t)=exp|t|q-1 and 0<q<2.