Dieudonné operators on the space of Bochner integrable functions

• Autorzy: Marian Nowak
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 46E40: Spaces of vector- and operator-valued functions
• 47B38: Operators on function spaces (general)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 2011
• Tom: 92
• Numer: 1
• Strony: 279-282

Daty

wydano

2011

Twórcy

autor

Marian Nowak

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Prof. Szafrana 4a, 65-516 Zielona Góra, Poland

Identyfikatory

DOI

10.4064/bc92-0-19