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Semi-compactness of positive Dunford-Pettis operators on Banach lattices. (English) Zbl 1152.47012
An operator \(T\) from a Banach space \(E\) into an ordered Banach space \(F\) is called semicompact if for each \(\varepsilon > 0\), there exists \( 0\leq u\) in \(F^+\) such that \(T(B_E)\subseteq [-u, u ]+ \varepsilon B_E\), where \(B_E\) and \(B_F\) are the closed unit balls of \(E\) and \(F\). An operator \(T\) between Banach spaces \(E\) and \(F\) is called a Dunford-Pettis operator if \(T\) maps each weakly compact subset of \(E\) into a compact subset of \(F\). A Dunford-Pettis operator need not be semicompact.
The authors give necessary and sufficient conditions for a regular Dunford-Pettis operator to be semicompact. They also study the converse of this result and give sufficient conditions for a semicompact operator to be Dunford-Pettis. They also study the relation between semicompact operators and weakly compact operators. They give many useful consequences of their results among which they recapture a theorem of Aliprantis and Burkinshaw on compactness of Dunford-Pettis operators and a result of Wickstead on compactness of an operator \(S\) satisfying \(0\leq S\leq T\), where \(T\) is compact.

MSC:
47B07 Linear operators defined by compactness properties
47B60 Linear operators on ordered spaces
46A40 Ordered topological linear spaces, vector lattices
46B40 Ordered normed spaces
46B42 Banach lattices
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[1] C. D. Aliprantis and O. Burkinshaw, Dunford-Pettis operators on Banach lattices, Trans. Amer. Math. Soc. 274 (1982), no. 1, 227 – 238. · Zbl 0498.47013
[2] 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. · Zbl 1043.46003
[3] Charalambos D. Aliprantis and Owen Burkinshaw, Positive operators, Springer, Dordrecht, 2006. Reprint of the 1985 original. · Zbl 1098.47001
[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). · Zbl 1073.47025 · doi:10.1016/j.crma.2004.11.015 · doi.org
[5] 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. · Zbl 1112.47028 · doi:10.1016/j.jmaa.2005.10.083 · doi.org
[6] 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). · Zbl 1099.46016
[7] N. J. Kalton and Paulette Saab, Ideal properties of regular operators between Banach lattices, Illinois J. Math. 29 (1985), no. 3, 382 – 400. · Zbl 0568.47030
[8] Anthony W. Wickstead, Extremal structure of cones of operators, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 126, 239 – 253. · Zbl 0431.47024 · doi:10.1093/qmath/32.2.239 · doi.org
[9] A. W. Wickstead, Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 175 – 179. · Zbl 0872.47018 · doi:10.1017/S0305004100074752 · doi.org
[10] A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. · Zbl 0519.46001
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