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 , 1⩽q⩽∞, and Lorentz–Zygmund spaces L(p,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 , 1⩽q⩽∞, et des espaces de Lorentz–Zygmund L(p,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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E.M. Semenov 1; Fyodor A. Sukochev 2
@article{CRMATH_2003__337_6_397_0, author = {E.M. Semenov and Fyodor A. Sukochev}, title = {The {Banach{\textendash}Saks} index of rearrangement invariant spaces on [0,1]}, journal = {Comptes Rendus. Math\'ematique}, pages = {397--401}, publisher = {Elsevier}, volume = {337}, number = {6}, year = {2003}, doi = {10.1016/j.crma.2003.07.003}, language = {en}, }
E.M. Semenov; Fyodor A. Sukochev. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.07.003/
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