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1
Content available remote

Mapping Properties of $c_0$

100%
Lewis P.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 2
235-244
EN
Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^*$ of a Banach space X, then $ℓ^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of $ℓ^1$ contains a copy of $ℓ^1$ that is complemented in $ℓ^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1 [0,1]$ contains a copy of $ℓ^1$ that is complemented in $L^1 [0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of $c_0$ which simultaneously yields extensions of the preceding theorems as corollaries. Additional classical mapping properties of $c_0$ are briefly discussed and applications are given.
2
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The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets

64%
Ghenciu I., Lewis P.
Colloquium Mathematicum
|
2006
|
tom 106
|
nr 2
311-324
EN
The Dunford-Pettis property and the Gelfand-Phillips property are studied in the context of spaces of operators. The idea of L-sets is used to give a dual characterization of the Dunford-Pettis property.
3
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The Embeddability of c₀ in Spaces of Operators

64%
Ghenciu I., Lewis P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2008
|
tom 56
|
nr 3
239-256
EN
Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.
4
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Almost Weakly Compact Operators

64%
Ghenciu I., Lewis P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 3
237-256
EN
Dunford-Pettis type properties are studied in individual Banach spaces as well as in spaces of operators. Bibasic sequences are used to characterize Banach spaces which fail to have the Dunford-Pettis property. The question of whether a space of operators has a Dunford-Pettis property when the dual of the domain and the codomain have the respective property is studied. The notion of an almost weakly compact operator plays a consistent and important role in this study.
5
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Completely Continuous operators

64%
Ghenciu I., Lewis P.
Colloquium Mathematicum
|
2012
|
tom 126
|
nr 2
231-256
EN
A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely continuous if and only if every bounded weakly precompact subset of X is a limited set. We show that in some cases, the projective and the injective tensor products of two spaces contain weakly precompact sets which are not limited. As a consequence, we deduce that for any infinite compact Hausdorff spaces K₁ and K₂, $C(K₁) ⊗_{π} C(K₂)$ and $C(K₁) ⊗_{ϵ} C(K₂)$ contain weakly precompact sets which are not limited.
6
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Evaluation maps, restriction maps, and compactness

51%
Bator E., Lewis P., Ochoa J.
Colloquium Mathematicum
|
1998
|
tom 78
|
nr 1
1-17
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