Multiplication operators on the space of Rademacher series in rearrangement invariant spaces

Abstract

Let $E$ be a rearrangement invariant (r.i.) function space on ‘0, 1’. We consider the space $\Lambda({\cal R},E)$ of measurable functions $f$ such that $fg \in E$ for every a.e. converging series $g =\sum a_nr_n \in E$, where $(r_n)$ are the Rademacher functions. Curbera ‘4’ showed that, for a broad class of spaces $E$, the space $\Lambda({\cal R},E)$ is not order-isomorphic to a r.i. space. We study cases when $\Lambda({\cal R},E)$ is order-isomorphic to a r.i. space. We give conditions on $E$ so that $\Lambda({\cal R},E)$ is order-isomorphic to $L_{\infty}$. This includes certain classes of Lorentz and Marcinkiewicz spaces. We study further when $\Lambda({\cal R},E)$ is orderisomorphic to a r.i. space different from $L_{\infty}$. This occurs for the Orlicz spaces $E = L_{\Phi_q}$ with $\Phi_q(t)$ asymptotically equivalent to $\exp |t|^q-1$ and $0 < q < 2$.