Pelczynski's property (V) onC(?,E) spaces

Cembranos, Pilar; Kalton, N. J.; Saab, Elias; Saab, Paulette

doi:10.1007/bf01455797

Cited by 25 publications

(23 citation statements)

References 11 publications

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“…In the case of X = C{K) there are some papers devoted to the study of this question (see [2,[6][7][8][9]), but nothing seems to be known in the present setting; we observe that the theorems proved in the paper extend all of the above-quoted results, but their proofs make use of the results in [2,8], so that they may be considered interesting complements to those theorems. Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader.…”

Section: Introductionmentioning

confidence: 71%

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Remarks on weak compactness of operators defined on certain injective tensor products

Emmanuele¹

1992

Proc. Amer. Math. Soc.

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Abstract.We show that if A" is a -2oo -space with the Dieudonné property and y is a Banach space not containing l\ , then any operator T: X®e Y -> Z , where Z is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained.Let A be a Jz^-space (see [1] for this notion and some useful results on Jz^-spaces) and Y be a Banach space not containing l\ . We consider the injective tensor product X ®£ Y (see [3]), and we investigate the following problem: when is any operator T : X £ Y -> Z, where Z is a Banach space, weakly compact?In the case of X = C{K) there are some papers devoted to the study of this question (see [2,[6][7][8][9]), but nothing seems to be known in the present setting; we observe that the theorems proved in the paper extend all of the above-quoted results, but their proofs make use of the results in [2,8], so that they may be considered interesting complements to those theorems. Because the proofs of our results are similar, we give the proof of Theorem 2 only and leave the others to the reader. We need the following definition: a Banach space E has the Dieudonné property if any weakly completely continuous (or Dieudonné) operator defined on it is weakly compact [8].We recall that C{K) spaces have the Dieudonné property.

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Section: Introductionmentioning

confidence: 71%

“…A similar proof (making use of the main result of the paper [2]) gives the following theorem (for the definitions of Pelczynski's properties (V) and (u) we refer to [2]). Proof.…”

mentioning

confidence: 87%

Remarks on weak compactness of operators defined on certain injective tensor products

Emmanuele¹

1992

Proc. Amer. Math. Soc.

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“…The proof of the following theorem follows the steps of Theorem 3 in [4]. The following sketch is given for the reader's convenience.…”

Section: Mo = M/(0)-mentioning

confidence: 99%

“…As in [4], let G be the representing measure of T, let A be its control measure and let h : fi -» E* be a weak*-Borel A-measurable map from fi -► E* such that for each / G C{Q,E), (f,T*x*) = /n(/(w),/i(w)> dA(w) and \\T'x*\\ = /n \\h{iü)\\dX. Now for each w G fi, let ^«(oj) = 6n(h(u)), where 9n are the maps obtained in Lemma 3.…”

Section: Mo = M/(0)-mentioning

confidence: 99%

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On the Dieudonné property for 𝐶(Ω,𝐸)

Kalton¹,

Saab²,

Saab³

1986

Proc. Amer. Math. Soc.

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ABSTRACT.In a recent paper, F. Bombai and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(iï,E), the Banach space of continuous functions from a compact Hausdorff space fi to E, has the Dieudonné property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of fi. In this paper we give a positive answer to their question.As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonné property if and only if C(fi, E) has the same property.

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Pełczyński's property () of order p and its quantification

Chen

Chávez-Domínguez

2017

Mathematische Nachrichten

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We introduce the concepts of Pełczyński's property ( ) of order and Pełczyński's property ( * ) of order . It is proved that, for each 1 < < ∞, the James -spaces enjoys Pełczyński's property ( * ) of order and the James * -spaces * (where * denotes the conjugate number of ) enjoys Pełczyński's property ( ) of order . We prove that both 1 ( ) ( a finite positive measure) and 1 enjoy a quantitative version of Pełczyński's property ( * ). K E Y W O R D SPełczyński's property ( ) of order , Pełczyński's property ( * ) of order , quantitative Pełczyński's property ( * ) of order M S C ( 2 0 1 0 ) 46-B

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Pelczynski's property (V) onC(?,E) spaces

Cited by 25 publications

References 11 publications

Remarks on weak compactness of operators defined on certain injective tensor products

Remarks on weak compactness of operators defined on certain injective tensor products

On the Dieudonné property for 𝐶(Ω,𝐸)

Pełczyński's property () of order p and its quantification

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