Abstract
Using the duality between Dunford-Pettis operators onL 1 and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL 1 intoL 1 form a lattice. We show also that a Banach spaceX has the Radon-Nikodým property provided the Dunford-Pettis members of ℒ(L 1,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bourgain,Sets with the Radon-Nikodým property in conjugate Banach spaces, Studia Math.663 (1978).
J. Bourgain,A non-dentable set without the tree-property, Studia Math.682 (1980), to appear.
J. Bourgain and H. P. Rosenthal,Martingales valued in certain subspaces of L 1, Israel J. Math.37 (1980), 54–75.
J. Diestel,Topics in Geometry of Banach Spaces, No. 485, Springer Verlag, 1975.
J. Diestel and J. J. Uhl,The Theory of Vector Measures, Mathematical Surveys, Vol. 15, Amer. Math. Soc., 1977.
K. Musial,The weak Radon-Nikodým property in Banach spaces, to appear.
H. Rosenthal,Convolution by a biased coin, The Altgeld Book 1975/76, University of Illinois, Functional Analysis Seminar.
J. J. Uhl,Applications of Radon-Nikodým theorems to martingale convergence, Trans. Amer. Math. Soc.145 (1969), 271–285.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgain, J. Dunford-pettis operators onL 1 and the Radon-Nikodym property. Israel J. Math. 37, 34–47 (1980). https://doi.org/10.1007/BF02762866
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02762866