Banach lattices with the subsequence splitting property

Weis, Lutz W.

doi:10.1090/S0002-9939-1989-0937853-X

Banach lattices with the subsequence splitting property
HTML articles powered by AMS MathViewer

by Lutz W. Weis PDF

Proc. Amer. Math. Soc. 105 (1989), 87-96 Request permission

Abstract:

A Banach lattice $X$ has SSP if every bounded sequence in $X$ has a subsequence that splits into a $X$-equi-integrable sequence and a sequence with pairwise disjoint support. We characterize such lattices in terms of uniform order continuity conditions and ultrapowers. This implies that rearrangement invariant function spaces with the Fatou-property have SSP.

References

Mustafa A. Akcoglu and Louis Sucheston, On ergodic theory and truncated limits in Banach lattices, Measure theory, Oberwolfach 1983 (Oberwolfach, 1983) Lecture Notes in Math., vol. 1089, Springer, Berlin, 1984, pp. 241–262. MR 786702, DOI 10.1007/BFb0072619

Equi-measurable rearrangements of functions

D. Dacunha-Castelle and J.-L. Krivine, Sous-espaces de $L^{1}$, Israel J. Math. 26 (1977), no. 3-4, 320–351 (French, with English summary). MR 626845, DOI 10.1007/BF03007651
Didier Dacunha-Castelle and Michel Schreiber, Techniques probabilistes pour l’étude de problèmes d’isomorphismes entre espaces de Banach, Ann. Inst. H. Poincaré Sect. B (N.S.) 10 (1974), 229–277 (French, with English summary). MR 0380915
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
P. G. Dodds and D. H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), no. 4, 287–320 (1980). MR 570888, DOI 10.1007/BF02760610
T. Figiel, N. Ghoussoub, and W. B. Johnson, On the structure of nonweakly compact operators on Banach lattices, Math. Ann. 257 (1981), no. 3, 317–334. MR 637954, DOI 10.1007/BF01456502

Factoring operators through Banach lattices not containing

Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
W. B. Johnson, B. Maurey, G. Schechtman, and L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19 (1979), no. 217, v+298. MR 527010, DOI 10.1090/memo/0217

Bases, lacunary sequences and complemented subspaces in the spaces

21

Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367

Measures of non-compactness of operators in Banach lattices

Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
L. W. Weis, On the computation of some quantities in the theory of Fredholm operators, Proceedings of the 12th winter school on abstract analysis (Srní, 1984), 1984, pp. 109–133. MR 781944