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Weakly precompact subsets of L₁(μ,X)

100%

Ghenciu I.

nr 1

133-143

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).

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

100%

Ghenciu I.

tom 63

nr 3

261-274

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.

The Embeddability of c₀ in Spaces of Operators

63%

Ghenciu I. , Lewis P.

nr 3

239-256

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.

The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets

63%

Ghenciu I. , Lewis P.

nr 2

311-324

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.