On some new characterizations of weakly compact sets in Banach spaces

Lixin Cheng; Qingjin Cheng; Zhenghua Luo

Displaying similar documents to “On some new characterizations of weakly compact sets in Banach spaces”

Weakly precompact subsets of L₁(μ,X)

Ioana Ghenciu (2012)

Colloquium Mathematicae

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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 $g ₙ \in c o f_{i} : i \geq n$ for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly...

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

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We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

Completely Continuous operators

Ioana Ghenciu, Paul Lewis (2012)

Colloquium Mathematicae

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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...

Weakly amenable groups and the RNP for some Banach algebras related to the Fourier algebra

Edmond E. Granirer (2013)

Colloquium Mathematicae

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It is shown that if G is a weakly amenable unimodular group then the Banach algebra $A_{p}^{r} (G) = A_{p} \cap L^{r} (G)$ , where $A_{p} (G)$ is the Figà-Talamanca-Herz Banach algebra of G, is a dual Banach space with the Radon-Nikodym property if 1 ≤ r ≤ max(p,p’). This does not hold if p = 2 and r > 2.

Factorization of vector measures and their integration operators

José Rodríguez (2016)

Colloquium Mathematicae

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Let X be a Banach space and ν a countably additive X-valued measure defined on a σ-algebra. We discuss some generation properties of the Banach space L¹(ν) and its connection with uniform Eberlein compacta. In this way, we provide a new proof that L¹(ν) is weakly compactly generated and embeds isomorphically into a Hilbert generated Banach space. The Davis-Figiel-Johnson-Pełczyński factorization of the integration operator $I_{ν} : L ¹ (ν) \to X$ is also analyzed. As a result, we prove that if $I_{ν}$ is both completely...

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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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 $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

A new class of weakly $K$ -analytic Banach spaces

Sophocles Mercourakis, E. Stamati (2006)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we define and investigate a new subclass of those Banach spaces which are $K$ -analytic in their weak topology; we call them strongly weakly $K$ -analytic (SWKA) Banach spaces. The class of SWKA Banach spaces extends the known class of strongly weakly compactly generated (SWCG) Banach spaces (and their subspaces) and it is related to that in the same way as the familiar classes of weakly $K$ -analytic (WKA) and weakly compactly generated (WCG) Banach spaces are related. We show...

Banach spaces which admit a norm with the uniform Kadec-Klee property

S. Dilworth, Maria Girardi, Denka Kutzarova (1995)

Studia Mathematica

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Several results are established about Banach spaces Ӿ which can be renormed to have the uniform Kadec-Klee property. It is proved that all such spaces have the complete continuity property. We show that the renorming property can be lifted from Ӿ to the Lebesgue-Bochner space $L_{2} (Ӿ)$ if and only if Ӿ is super-reflexive. A basis characterization of the renorming property for dual Banach spaces is given.

Projections from $L (X, Y)$ onto $K (X, Y)$

Kamil John (2000)

Commentationes Mathematicae Universitatis Carolinae

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Generalization of certain results in [Sap] and simplification of the proofs are given. We observe e.g.: Let $X$ and $Y$ be Banach spaces such that $X$ is weakly compactly generated Asplund space and $X^{*}$ has the approximation property (respectively $Y$ is weakly compactly generated Asplund space and $Y^{*}$ has the approximation property). Suppose that $L (X, Y) \neq K (X, Y)$ and let $1 < λ < 2$ . Then $X$ (respectively $Y$ ) can be equivalently renormed so that any projection $P$ of $L (X, Y)$ onto $K (X, Y)$ has the sup-norm greater or equal to $λ$ . ...

Factorization of weakly compact operators between Banach spaces and Fréchet or (LB)-spaces

José Bonet, J. D. Maitland Wright (2012)

Matematički Vesnik

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An indecomposable Banach space of continuous functions which has small density

Rogério Augusto dos Santos Fajardo (2009)

Fundamenta Mathematicae

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Using the method of forcing we construct a model for ZFC where CH does not hold and where there exists a connected compact topological space K of weight $ω ₁ < 2^{ω}$ such that every operator on the Banach space of continuous functions on K is multiplication by a continuous function plus a weakly compact operator. In particular, the Banach space of continuous functions on K is indecomposable.

On super-weakly compact sets and uniformly convexifiable sets

Lixin Cheng, Qingjin Cheng, Bo Wang, Wen Zhang (2010)

Studia Mathematica

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This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James'...

A generalized notion of $n$ -weak amenability

Abasalt Bodaghi, Behrouz Shojaee (2014)

Mathematica Bohemica

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In the current work, a new notion of $n$ -weak amenability of Banach algebras using homomorphisms, namely $(ϕ, ψ)$ - $n$ -weak amenability is introduced. Among many other things, some relations between $(ϕ, ψ)$ - $n$ -weak amenability of a Banach algebra $𝒜$ and $M_{m} (𝒜)$ , the Banach algebra of $m \times m$ matrices with entries from $𝒜$ , are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra $L^{1} (G)$ is ( $ϕ, ψ$ )- $n$ -weakly amenable...

Some properties of weakly countably determined Banach spaces

M. Valdivia (1989)

Studia Mathematica

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