Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let $E$ be a symmetric space on $[0,1]$. Let $\Lambda(\mathcal{R},E)$ be the space of measurable functions $f$ such that $fg\in E$ for every almost everywhere convergent series $g=\sum b_nr_n\in E$, where $(r_n)$ are the Rademacher functions. In [G. P. Curbera, Proc. Edinb. Math. Soc., 40, No. 1, 119–126 (1997)] it was shown that, for a broad class of spaces $E$, the space $\Lambda(\mathcal{R},E)$ 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 $\Lambda(\mathcal{R},E)$ to be order isomorphic to $L_\infty$. This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which $\Lambda(\mathcal{R},E)$ is order isomorphic to a symmetric space that differs from $L_\infty$. The answer is positive for the Orlicz spaces $E=L_{\Phi_q}$ with $\Phi_q(t)=\exp|t|^q-1$ and $0<q<2$.