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Dunford-Pettis operators on Banach lattices

Authors: C. D. Aliprantis and O. Burkinshaw
Journal: Trans. Amer. Math. Soc. 274 (1982), 227-238
MSC: Primary 47B55; Secondary 46B30, 47D15
MathSciNet review: 670929
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Abstract: Consider a Banach lattice $ E$ and two positive operators $ S,T:E \to E$ that satisfy $ 0 \leqslant S \leqslant T$. In $ [{\mathbf{2,3}}]$ we examined the case when $ T$ is a compact (or weakly compact) operator and studied what effect this had on an operator (such as $ S$) dominated by $ T$. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator $ T$, to ensure that $ S$ (or some power of $ S$) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters.

Theorem. Let $ E$ be a Banach lattice, and let $ S,T:E \to E$ be two positive operators such that $ 0 \leqslant S \leqslant T$. If $ T$ is compact then

(1) $ {S^3}$ is a compact operator (although $ {S^2}$ need not be compact);

(2) $ {S^2}$ is a Dunford-Pettis and weakly compact operator ( although $ S$ need not be );

(3) $ S$ is a weak Dunford-Pettis operator.

In another direction, our techniques and results will be related to the lattice stracture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.

References [Enhancements On Off] (What's this?)

  • [1] C. D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York and London, 1978. MR 0493242 (58:12271)
  • [2] -, Positive compact operators on Banach lattices, Math. Z. 174 (1980), 289-298. MR 593826 (81m:47053)
  • [3] -, On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), 573-578. MR 627695 (82j:47057)
  • [4] K. T. Andrews, Dunford-Pettis sets in the space of Bochner integrable functions, Math. Ann. 241 (1979), 35-41. MR 531148 (80f:46041)
  • [5] J. Bourgain, Dunford-Pettis operators on $ {L^1}$ and the Radon-Nikodým property, Israel J. Math. 37 (1980), 34-47. MR 599300 (82k:47047a)
  • [6] O. Burkinshaw, Weak compactness in the order dual of a vector lattice, Ph. D. thesis, Purdue University, 1972.
  • [7] J. J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys, No. 15, Amer. Math. Soc., Providence, R. I., 1977. MR 0453964 (56:12216)
  • [8] P. G Dodds $ o$-weakly compact mappings on Riesz spaces, Trans. Amer. Math. Soc. 214 (1975), 389-402. MR 0385629 (52:6489)
  • [9] P. G. Dodds and D. H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), 287-320. MR 570888 (81g:47037)
  • [10] M. Duhoux, $ o$-weakly compact mappings from a Riesz space to a locally convex space, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N. S.) 22 (1978), 371-378. MR 522533 (80a:47057)
  • [11] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974. MR 0423039 (54:11023)

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Keywords: Banach lattices, positive operators, compact operators, Dunford-Pettis operators
Article copyright: © Copyright 1982 American Mathematical Society

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