×

Upper \(\ell_ p\)-estimates in vector sequence spaces, with some applications. (English) Zbl 0806.46007

Let \(\lambda\) be a Banach sequence space with a monotone basis. For a Banach space \(X\) the space \(\lambda(X):= \{(x_ n)\mid x_ n\in X,\;\sum_ k\| x_ k\| e_ k\in \lambda\}\) is a Banach space with the norm \(\|(x_ n)\|:= \|\sum_ k \| x_ k\| e_ k\|\). A sequence \((x_ n)\subset X\) is said to be weakly \(p\)- convergent \((1\leq p\leq \infty)\) to \(x\in X\) if there is a constant \(C>0\) such that \(\sup_ n\|\sum^ n_{k=1} \alpha_ k(x_ k- x)\|\leq C\|(\alpha_ n)\|_{\ell^{p^*}}\) for any sequence \((\alpha_ n)\in \ell_{p^*}\) \((1/p+ 1/p^*= 1)\).
A Banach space \(X\) is said to have
1) the property \(W_ p\) if every bounded sequence admits a weakly-\(p\)- convergent subsequence and
2) the hereditary Dunford-Pettis property if weakly compact operators defined on \(X\) are compact.
The authors prove, that if \(X\) and \(\lambda\) have the \(W_ p\) property [the hereditary Dunford-Pettis property], then \(\lambda(X)\) has this property.
Reviewer: T.Leiger (Tartu)

MSC:

46A45 Sequence spaces (including Köthe sequence spaces)
46B45 Banach sequence spaces
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cembranos, Illinois J. Math. 31 pp 365– (1987)
[2] DOI: 10.2307/2001402 · Zbl 0704.46033 · doi:10.2307/2001402
[3] Beauzamy, Mod?les ?tal?s des Espaces de Banach (1984)
[4] DOI: 10.1007/BF02761374 · Zbl 0486.46021 · doi:10.1007/BF02761374
[5] Partington, Math. Proc. Cambridge Philos. Soc. 82 pp 369– (1977)
[6] DOI: 10.2307/2047951 · Zbl 0833.46007 · doi:10.2307/2047951
[7] DOI: 10.1007/BFb0090216 · doi:10.1007/BFb0090216
[8] Jaramillo, Proc. Amer. Math. Soc.
[9] Johnson, Compositio Math. 34 pp 69– (1977)
[10] Diestel, Contemporary Mathematics 2 pp 15– (1980)
[11] Knaust, Israel J. Math. 67 pp 153– (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.