Abstract
We introduce the class of weak Schur spaces, i.e., Banach lattices in which relatively weakly compact sets and almost order bounded sets coincide. There follows a detailed study of Banach lattices in which every semi-normalized, order bounded, weakly null sequence contains a subsequence satisfying a lower, resp. an upper, 2-estimate. From the previous results we obtain conditions under which non-Dunford-Pettis operators between certain classes of Banach lattices fix a copy of ℓ2.
Work on this paper was supported by Deutsche Forschungsgemeinschaft DFG.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D.J. Aldous and D.H. Fremlin, Colacunary sequences in L-spaces, Studia Math., 71 (1982), 297–304.
C.D. Aliprantis and O. Burkinshaw, “Locally Solid Riesz Spaces”, Academic Press, New York, San Francisco, London, 1978.
C.D. Aliprantis and O. Burkinshaw, “Positive Operators”, Academic Press, Orlando, London, 1985.
J. Bourgain, Dunford-Pettis operators on L 1 and the Radon-Nikodým property, Israel J. Math., 37 (1980), 34–47.
J. Bourgain, A characterization of non-Dunford-Pettis operators on L 1, Israel J. Math., 37 (1980), 48–53.
T. Figiel, W.B. Johnson and L. Tzafriri, On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approximation Theory, 13 (1975), 395–412.
N. Ghoussoub and H.P. Rosenthal, Martingales, G ∞-embeddings and quotients of L 1, Math. Ann., 264 (1983), 321–332.
W.B. Johnson, B. Maurey, G. Schechtman and L. Tzafriri, Symmetric structures in Banach spaces, Memoirs Amer. Math. Soc., 217 (1979).
B. Kühn, Orthogonalkompakte Teilmengen topologischer Vektorverbände, Manuscripta Math., 33 (1981), 217–226.
H.E. Lacey, “The Isometric Theory of Classical Banach Spaces”, Springer-Verlag, Berlin, Heidelberg, New York, 1974.
D.H. Leung, On the weak Dunford-Pettis property, Arch. Math., 52 (1989), 363–364.
J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I. Sequence Spaces”, Springer-Verlag, Berlin, Heidelberg, New York, 1977.
J. Lindenstrauss and L. Tzafriri, “Classical Banach Spaces II. Function Spaces”, Springer-Verlag, Berlin, Heidelberg, New York, 1979.
P. Meyer-Nieberg, Zur schwachen Kompaktheit in Banachverbänden, Math. Z., 134 (1973), 303–315.
F. Räbiger, Beiträge zur Strukturtheorie der Grothendieck-Räume, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, Math.-Naturwiss. Klasse, Jahrgang 1985, 4. Abh., Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.
F. Räbiger, Lower 2-estimates for sequences in Banach lattices, Proc. Amer. Math. Soc. 111 (1991), 81–83.
H.P. Rosenthal, Convolution by a biased coin, The Altgeld Book 1975/76, Univ. of Illinois, Functional Analysis Seminar.
H.H. Schaefer, “Banach Lattices and Positive Operators”, Springer-Verlag, New York, Heidelberg, Berlin, 1974.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag
About this paper
Cite this paper
Räbiger, F. (1991). Lower and upper 2-estimates for order bounded sequences and Dunford-Pettis operators between certain classes of Banach lattices. In: Odwell, E.E., Rosenthal, H.P. (eds) Functional Analysis. Lecture Notes in Mathematics, vol 1470. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090219
Download citation
DOI: https://doi.org/10.1007/BFb0090219
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54206-3
Online ISBN: 978-3-540-47493-7
eBook Packages: Springer Book Archive