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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 \(\| x_ n\| \to 0\).
Reviewer: S.S.Kutateladze

MSC:
46B42 Banach lattices
47B60 Linear operators on ordered spaces
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References:
[1] Gretsky, Glasgow Math. J. 28 pp 113– (1986)
[2] Buhvalov, Mat. Meh. Astronom. Vyp. 2 pp 11– (1973)
[3] Aliprantis, Positive operators (1985)
[4] Aliprantis, Locally solid Riesz spaces (1978)
[5] Groenewegen, Indag. Math. 48 pp 313– (1986) · doi:10.1016/1385-7258(86)90017-X
[6] Varshavskaya, Soobšč. Akad. Nauk Gruzin. SSR 120 pp 21– (1985)
[7] DOI: 10.1007/BF01228263 · Zbl 0275.46005 · doi:10.1007/BF01228263
[8] Holub, Glasgow Math. J. 29 pp 271– (1987)
[9] Zaanen, Riesz spaces II (1983)
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