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 and two positive operators that satisfy . In we examined the case when is a compact (or weakly compact) operator and studied what effect this had on an operator (such as ) dominated by . In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator , to ensure that (or some power of ) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters.

Theorem. *Let* *be a Banach lattice, and let* *be two positive operators such that* . *If* *is compact then*

(1) *is a compact operator (although* *need not be compact);*

(2) *is a Dunford-Pettis and weakly compact operator ( although* *need not be );*

(3) *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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DOI:
https://doi.org/10.1090/S0002-9947-1982-0670929-1

Keywords:
Banach lattices,
positive operators,
compact operators,
Dunford-Pettis operators

Article copyright:
© Copyright 1982
American Mathematical Society