Semi-compactness of positive Dunford–Pettis operators on Banach lattices

Aqzzouz, Belmesnaoui; Nouira, Redouane; Zraoula, Larbi

doi:10.1090/S0002-9939-08-09032-1

Semi-compactness of positive Dunford–Pettis operators on Banach lattices
HTML articles powered by AMS MathViewer

by Belmesnaoui Aqzzouz, Redouane Nouira and Larbi Zraoula PDF

Proc. Amer. Math. Soc. 136 (2008), 1997-2006 Request permission

Abstract:

We investigate Banach lattices on which each positive Dunford–Pettis operator is semi-compact and the converse. As an interesting consequence, we obtain Theorem 2.7 of Aliprantis–Burkinshaw and an element of Theorem 1 of Wickstead.

References

C. D. Aliprantis and O. Burkinshaw, Dunford-Pettis operators on Banach lattices, Trans. Amer. Math. Soc. 274 (1982), no. 1, 227–238. MR 670929, DOI 10.1090/S0002-9947-1982-0670929-1
Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, 2nd ed., Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, Providence, RI, 2003. MR 2011364, DOI 10.1090/surv/105
Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Springer, Dordrecht, 2006. Reprint of the 1985 original. MR 2262133, DOI 10.1007/978-1-4020-5008-4
Belmesnaoui Aqzzouz, Redouane Nouira, and Larbi Zraoula, Compacité des opérateurs de Dunford-Pettis positifs sur les treillis de Banach, C. R. Math. Acad. Sci. Paris 340 (2005), no. 1, 37–42 (French, with English and French summaries). MR 2112038, DOI 10.1016/j.crma.2004.11.015
Belmesnaoui Aqzzouz, Redouane Nouira, and Larbi Zraoula, About positive Dunford-Pettis operators on Banach lattices, J. Math. Anal. Appl. 324 (2006), no. 1, 49–59. MR 2262455, DOI 10.1016/j.jmaa.2005.10.083
Belmesnaoui Aqzzouz, Redouane Nouira, and Larbi Zraoula, Sur les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts, Proc. Amer. Math. Soc. 134 (2006), no. 4, 1161–1165 (French, with English and French summaries). MR 2196052, DOI 10.1090/S0002-9939-05-08083-4
N. J. Kalton and Paulette Saab, Ideal properties of regular operators between Banach lattices, Illinois J. Math. 29 (1985), no. 3, 382–400. MR 786728, DOI 10.1215/ijm/1256045630
Anthony W. Wickstead, Extremal structure of cones of operators, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 126, 239–253. MR 615198, DOI 10.1093/qmath/32.2.239
A. W. Wickstead, Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 175–179. MR 1373356, DOI 10.1017/S0305004100074752
A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4