Symmetric kernel of Rademacher multiplicator spaces

Let X be a rearrangement invariant (r.i.) function space on [0,1]. We consider the Rademacher or multiplicator space Lambda(R, X) of measurable functions x such that xh is an element of X for every a.e. converging series h=Sigma a(n)r(n) is an element of X, where (r(n)) are the Rademacher functions. We show that for a broad class of r.i. spaces X, the space Lambda(R, X) is not r.i. In this case, we identify the symmetric kernel Syrn (R, X) of the Rademacher multiplicator space and study when Syrn (R, X) reduces to L-infinity. In the opposite direction, we find new examples of r.i. spaces for which Lambda(R, X) is r.i. We consider in detail the case when X is a Marcinkiewicz or an exponential Orlicz space. (c) 2005 Elsevier Inc. All rights reserved.