# Dieudonné operators on the space of Bochner integrable functions

• Volume: 92, Issue: 1, page 279-282
• ISSN: 0137-6934

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## Abstract

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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 ${i}_{\infty }:{L}^{\infty }\left(X\right)\to L¹\left(X\right)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T\circ {i}_{\infty }:{L}^{\infty }\left(X\right)\to Y$ is a weakly compact operator. Moreover, we obtain that if T: L¹(X) → Y is a bounded linear operator and $T\circ {i}_{\infty }:{L}^{\infty }\left(X\right)\to Y$ is weakly compact, then T is a Dieudonné operator.

## How to cite

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Marian Nowak. "Dieudonné operators on the space of Bochner integrable functions." Banach Center Publications 92.1 (2011): 279-282. <http://eudml.org/doc/281787>.

@article{MarianNowak2011,
abstract = {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.},
author = {Marian Nowak},
journal = {Banach Center Publications},
keywords = {Dieudonné operators; weakly completely continuous operators; weakly compact operators; conditional compactness},
language = {eng},
number = {1},
pages = {279-282},
title = {Dieudonné operators on the space of Bochner integrable functions},
url = {http://eudml.org/doc/281787},
volume = {92},
year = {2011},
}

TY - JOUR
AU - Marian Nowak
TI - Dieudonné operators on the space of Bochner integrable functions
JO - Banach Center Publications
PY - 2011
VL - 92
IS - 1
SP - 279
EP - 282
AB - 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.
LA - eng
KW - Dieudonné operators; weakly completely continuous operators; weakly compact operators; conditional compactness
UR - http://eudml.org/doc/281787
ER -

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