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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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.

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