Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

Bu, Qingying; Diestel, Joe; Dowling, Patrick N.; Oja, Eve

doi:10.1215/ijm/1258138106

Cited by 10 publications

(9 citation statements)

References 30 publications

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“…We will now consider a special type of substitution space that was used by Bu et al [2] to analyze Radon-Nikodym properties of the projective tensor product of Banach spaces [2]. Let U be a Banach space with a normalized 1-unconditional basis (e i ) i .…”

Section: Preliminaries and Resultsmentioning

confidence: 99%

On convexity properties of ψ-direct sums of Banach spaces

Dowling

2003

Journal of Mathematical Analysis and Applications

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Section: Preliminaries and Resultsmentioning

confidence: 99%

On convexity properties of ψ-direct sums of Banach spaces

Dowling

2003

Journal of Mathematical Analysis and Applications

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“…An immediate consequence of this theorem that follows from the techniques developed in [2] is the following stability result for projective tensor products of Banach spaces.…”

Section: Theorem 12 Let X Be a Banach Space Having A Boundedly Complmentioning

confidence: 92%

On the near differentiability property of Banach spaces

Dowling¹,

Robdera²

2006

Journal of Mathematical Analysis and Applications

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Let μ be a scalar measure of bounded variation on a compact metrizable abelian group G. Suppose that μ has the property that for any measure σ whose Fourier-Stieltjes transformσ vanishes at ∞, the measure μ * σ has Radon-Nikodým derivative with respect to λ, the Haar measure on G. Then L. Pigno and S. Saeki showed that μ itself has Radon-Nikodým derivative. Such property is not shared by vector measures in general. We say that a Banach space X has the near differentiability property if every X-valued measure of bounded variation shares the above property. We prove that Banach spaces with the Radon-Nikodým property have the near differentiability property, while Banach spaces with the near differentiability property enjoy the near Radon-Nikodým property. We also show that the Banach spaces L 1 [0, 1] and L 1 /H 1 0 have the near differentiability property. Lastly, we show that Banach spaces with the near differentiability property have type II-Λ-Radon-Nikodým property, whenever Λ is a Riesz subset of type 0 of G.

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“…In this case, it is easy to see that ϕ (X) has an unconditional and boundedly complete Schauder decomposition. It follows from [3,Theorem 4] that ϕ (X) has the Radon-Nikodym property if X has the RadonNikodym property. It is well-known (see [1]) that if there is a semi-embedding from a separable Banach space X to a Banach space with the Radon-Nikodym property, then X has the Radon-Nikodym property.…”

Section: The Rnp For the Fremlin Projective Tensor Productmentioning

confidence: 99%

The Radon-Nikodym Property for Tensor Products of Banach Lattices II

Buskes

Lai

2007

Positivity

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Let ϕ be an Orlicz function that has a complementary function ϕ * and let ϕ be an Orlicz sequence space. We prove two results in this paper. Result 1: ϕ⊗F X, the Fremlin projective tensor product of ϕ with a Banach lattice X, has the Radon-Nikodym property if and only if both ϕ and X have the Radon-Nikodym property. Result 2: ϕ⊗iX , the Wittstock injective tensor product of ϕ with a Banach lattice X, has the Radon-Nikodym property if and only if both ϕ and X have the Radon-Nikodym property and each positive continuous linear operator from hϕ * to X is compact.

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Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

Cited by 10 publications

References 30 publications

On convexity properties of ψ-direct sums of Banach spaces

On convexity properties of ψ-direct sums of Banach spaces

On the near differentiability property of Banach spaces

The Radon-Nikodym Property for Tensor Products of Banach Lattices II

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