Rearrangement invariance of Rademacher multiplicator spaces

Let X be a rearrangement invariant function space on [0,1]. We consider the Rademacher multiplicator space Λ(R,X) of all measurable functions x such that x⋅h∈X for every a.e. converging series h=∑anrn∈X, where (rn) are the Rademacher functions. We study the situation when Λ(R,X) is a rearrangement invariant space different from L∞. Particular attention is given to the case when X is an interpolation space between the Lorentz space Λ(φ) and the Marcinkiewicz space M(φ). Consequences are derived regarding the behaviour of partial sums and tails of Rademacher series in function spaces.