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A note on the positive Schur property. (English) Zbl 0694.46020
The author proves that for a Banach lattice $$(E,\| \cdot \|)$$ the following statements are equivalent:
(i) E is $$\sigma$$-Dedekind complete and any operator T: $$E\to C_ 0$$ is a Dunford-Pettis operator iff T is regular;
(ii) E has the positive Schur property, i.e. $$(x_ n)\subset E_+$$ and $$x_ n\to 0$$ weakly imply $$\| x_ n\| \to 0$$.