Compact and strictly singular operators in rearrangement invariant spaces and Rademacher functions

We refine some earlier results by Flores, Hernandez, Semenov, and Tradacete on compactness of the square of strictly singular endomorphisms and identifying general Banach lattices with the Kato property in the setting of rearrangement invariant spaces on [0, 1]. A Banach space X is said to have the Kato property if every strictly singular operator acting in X is compact. We show that each strictly singular operator bounded in a disjointly homogeneous rearrangement invariant space with the non-trivial Boyd indices has compact square, and that the Kato property is shared by a 2-disjointly homogeneous rearrangement invariant space X whenever X superset of G, where G is the closure of L infinity in the Orlicz space, generated by the function eu2-1. Moreover, a partial converse to the latter result is given under the assumption that X subset of Llog1/2L. As an application we find rather sharp conditions, under which a Lorentz space ?(2,psi) possesses the Kato property. In particular, ?(2,log-alpha(e/u))\, with 0<alpha <= 1 is a 2-DH and 2-convex rearrangement invariant space, which does not have the Kato property. This gives a negative answer to the question posed by Hernandez, Semenov, and Tradacete.