Functional Analysis
The Banach–Saks index of rearrangement invariant spaces on [0,1]
Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 397-401.

The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L2, for p∈(1,2) this is Lp,∞0. We compute the Banach–Saks index for the families of Lorentz spaces L p,q ,1<p<, 1⩽q⩽∞, and Lorentz–Zygmund spaces L(p,α), 1p<,α, extending the classical results of Banach–Saks and Kadec–Pelczynski for Lp-spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods.

L'ensemble des espaces invariants par réarrangement sur [0,1] qui possèdent la propriété de p-Banach–Saks admet un unique élément maximal pour p∈(1,2]. Pour p=2 c'est L2 ; pour p∈(1,2) c'est L0p,∞. Nous calculons l'indice de Banach–Saks de la famille des espaces de Lorentz L p,q ,1<p<, 1⩽q⩽∞, et des espaces de Lorentz–Zygmund L(p,α), 1p<,α, généralisant ainsi les résultats classiques de Banach–Saks et Kadec–Pelczynski pour les espaces Lp. Nous montrons que l'ensemble des espaces invariants par réarrangement qui ont p∈(1,2] indice de Banach–Saks n'est pas stable par interpolation réelle ou complexe.

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DOI: 10.1016/j.crma.2003.07.003
Semenov, E.M. 1; Sukochev, Fyodor A. 2

1 Department of Mathematics, Voronezh State University, Universitetskaya pl. 1, Voronezh 394693, Russia
2 School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, Australia
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Semenov, E.M.; Sukochev, Fyodor A. The Banach–Saks index of rearrangement invariant spaces on [0,1]. Comptes Rendus. Mathématique, Volume 337 (2003) no. 6, pp. 397-401. doi : 10.1016/j.crma.2003.07.003. http://www.numdam.org/articles/10.1016/j.crma.2003.07.003/

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