# Dieudonné operators on the space of Bochner integrable functions

Banach Center Publications (2011)

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

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topMarian 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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