×

On spreading models in \(L^ 1(E)\). (English) Zbl 0579.46012

We construct a Banach space E which has the Schur property (hence \(\ell^ 1\) is its only spreading model) but such that for each family \((a_{n,k})\) with \(a_{n,k}\geq 1\), \(\lim_{n}a_{n,k}=+\infty\), there is a sequence \((f_ n)\) in \(L^ 1(E)\) for which \(\| \sum_{k\leq i\leq n}\pm f_ i\| \leq a_{n,k}\). In particular, \(L^ 1(E)\) has a spreading model isomorphic to \(c_ 0({\mathbb{N}})\).

MSC:

46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
46E40 Spaces of vector- and operator-valued functions
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] J. HAGLER , A Counterexample to Several Questions About Banach Spaces (Studia Math., Vol. 60, 1977 , pp. 289-308). MR 56 #1032 | Zbl 0387.46015 · Zbl 0387.46015
[2] S. KWAPIEN , On Banach Spaces Containing c0 (Studia Math., Vol. 22, 1974 , pp. 188-189). MR 50 #8627 | Zbl 0295.60003 · Zbl 0295.60003
[3] H. P. ROSENTHAL , A Characterization of Banach Spaces Containing l1 (Proc. Nat. Acad. Sc. U.S.A., Vol. 71, 1974 , pp. 2411-2413). MR 50 #10773 | Zbl 0297.46013 · Zbl 0297.46013 · doi:10.1073/pnas.71.6.2411
[4] M. TALAGRAND , Sur la propriété de Dunford-Pettis dans C ([0, 1], E) et L1 (E) , Israel, J. of Math. 44, 1983 , pp. 317-321. MR 84j:46065 | Zbl 0523.46015 · Zbl 0523.46015 · doi:10.1007/BF02761990
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.