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1
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On Some Classes of Operators on C(K,X)

100%
Ghenciu I.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2015
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tom 63
|
nr 3
261-274
EN
Suppose X and Y are Banach spaces, K is a compact Hausdorff space, Σ is the σ-algebra of Borel subsets of K, C(K,X) is the Banach space of all continuous X-valued functions (with the supremum norm), and T:C(K,X) → Y is a strongly bounded operator with representing measure m:Σ → L(X,Y). We show that if T is a strongly bounded operator and T̂:B(K,X) → Y is its extension, then T is limited if and only if its extension T̂ is limited, and that T* is completely continuous (resp. unconditionally converging) if and only if T̂* is completely continuous (resp. unconditionally converging). We prove that if K is a dispersed compact Hausdorff space and T is a strongly bounded operator, then T is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) whenever m(A):X → Y is limited (resp. weakly precompact, has a completely continuous adjoint, has an unconditionally converging adjoint) for each A ∈ Σ.
2
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Weakly precompact subsets of L₁(μ,X)

100%
Ghenciu I.
Colloquium Mathematicum
|
2012
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tom 129
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nr 1
133-143
EN
Let (Ω,Σ,μ) be a probability space, X a Banach space, and L₁(μ,X) the Banach space of Bochner integrable functions f:Ω → X. Let W = {f ∈ L₁(μ,X): for a.e. ω ∈ Ω, ||f(ω)|| ≤ 1}. In this paper we characterize the weakly precompact subsets of L₁(μ,X). We prove that a bounded subset A of L₁(μ,X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fₙ) in A, there exists a sequence (gₙ) with $gₙ ∈ co{f_i: i ≥ n}$ for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly Cauchy in X. We also prove that if A is a bounded subset of L₁(μ,X), then A is weakly precompact if and only if for every ϵ >0, there exist a positive integer N and a weakly precompact subset H of NW such that A ⊆ H + ϵB(0), where B(0) is the unit ball of L₁(μ,X).
3
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Weak precompactness and property (V*) in spaces of compact operators

100%
Ghenciu I.
Colloquium Mathematicum
|
2015
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tom 138
|
nr 2
255-269
EN
We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w*}(E*,F)$ (resp. $K_{w*}(E*,F)$) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w*}(E*,F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w*}(E*,F)$ has property (wV*). Suppose that $L_{w*}(E*,F) = K_{w*}(E*,F)$. Then $K_{w*}(E*,F)$ has property (V*) if and only if E and F have property (V*).
4
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On the Dunford-Pettis property of tensor product spaces

100%
Ghenciu I.
Colloquium Mathematicum
|
2011
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tom 125
|
nr 2
221-231
EN
We give sufficient conditions on Banach spaces E and F so that their projective tensor product $E ⊗ _π F$ and the duals of their projective and injective tensor products do not have the Dunford-Pettis property. We prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:E* → F** is completely continuous, then $(E ⊗ _ϵ F)*$ does not have the DPP. We also prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T: F** → E* is completely continuous, then $(E ⊗ _πF)* ≃ L(E,F*)$ does not have the DPP.
5
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The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets

64%
Ghenciu I., Lewis P.
Colloquium Mathematicum
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2006
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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.
6
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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
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2008
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tom 56
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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.
7
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Almost Weakly Compact Operators

64%
Ghenciu I., Lewis P.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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tom 54
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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.
8
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Completely Continuous operators

64%
Ghenciu I., Lewis P.
Colloquium Mathematicum
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2012
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tom 126
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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.
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