Strictly singular inclusions of rearrangement invariant spaces and Rademacher spaces

Sergei V. Astashkin¹ ; Francisco L. Hernández² ; Evgeni M. Semenov³

¹ Department of Mathematics Samara State University Samara 443029, Russia
² Department of Mathematical Analysis Madrid Complutense University 28040 Madrid, Spain
³ Department of Mathematics Voronezh State University Voronezh 394006, Russia

Studia Mathematica, Tome 193 (2009) no. 3, pp. 269-283

Résumé

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.

Détail

DOI : 10.4064/sm193-3-4

Keywords: closure infty exp proved inclusion between rearrangement invariant spaces subset strictly singular only disjointly strictly singular supset marcinkiewicz space varphi subset varphi interpolation space between infty proved there exists another marcinkiewicz space psi subsetneq varphi property psi varphi norms equivalent rademacher subspace applications given question milman answered