Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces
Volume 193 / 2009
Studia Mathematica 193 (2009), 269-283
MSC: Primary 46E30.
DOI: 10.4064/sm193-3-4
Abstract
If $G$ is the closure of $L_\infty$ in $\exp L_{2}$, it is proved that the inclusion between rearrangement invariant spaces $E\subset F$ is strictly singular if and only if it is disjointly strictly singular and $E\not\supset G$. For any Marcinkiewicz space $M(\varphi) \subset G$ such that $M(\varphi) $ is not an interpolation space between $L_{\infty}$ and $G$ it is proved that there exists another Marcinkiewicz space $M(\psi)\subsetneq M(\varphi)$ with the property that the $M(\psi)$ and $ M(\varphi)$ norms are equivalent on the Rademacher subspace. Applications are given and a question of Milman answered.
