Dieudonné operators on the space of Bochner integrable functions

Marian Nowak

Banach Center Publications, Tome 95 (2011), p. 279-282 / Harvested from The Polish Digital Mathematics Library

Résumé

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{∞}: L^{∞}(X) → L¹(X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T∘i_{∞}: L^{∞}(X) → Y$ is a weakly compact operator. Moreover, we obtain that if T: L¹(X) → Y is a bounded linear operator and $T∘i_{∞}: L^{∞}(X) → Y$ is weakly compact, then T is a Dieudonné operator.

Détail

Publié le : 2011-01-01

Zbl 1260.47032

EUDML-ID : urn:eudml:doc:281787