H’Michane, J.; El Fahri, K. On the domination of limited and order Dunford-Pettis operators. (English. French summary) Zbl 1350.46018 Ann. Math. Qué. 39, No. 2, 169-176 (2015). A bounded subset \(A\) of a Banach space \(X\) is called limited in \(X\) if, for every weak\(^*\) null sequence \((x_n')\) in \(X'\), we have \(x_n'(x)\to 0\) uniformly for \(x\in A\). An operator \(T:X\to Y\) between Banach spaces is said to be limited whenever the set \(T(B_X)\) is limited in \(Y\). A bounded subset \(A\) of a Banach space is said to be a Dunford-Pettis set whenever every weakly compact operator from \(X\) to another Banach space carries \(A\) to a norm totally bounded set. An operator \(T:E\to X\) is said to be an order Dunford-Pettis operator whenever \(T\) carries order bounded subsets of \(E\) to Dunford-Pettis sets in \(X\).In this paper, the authors consider the so-called “domination problem” for limited (resp.order limited, order Dunford-Pettis) operators: suppose that positive operators \(S,T:E\to F\) between Banach lattices \(E\) and \(F\) satisfy \(0\leq S\leq T\). If \(T\) is limited (resp.order limited, order Dunford-Pettis), does it follow that \(S\) is also limited (resp.order limited, order Dunford-Pettis)? The authors prove that \(S\) is an order Dunford-Pettis operator whenever \(T\) is. However, they provide an example of a rank-one operator \(T:\ell_1\to c\) which dominates a nonlimited operator. Under additional assumptions, when \(F\) is Dedekind \(\sigma\)-complete and \(E'\) is order continuous, the authors prove that \(S\) is limited whenever \(T\) is limited. They also prove that whenever \(F\) is \(\sigma\)-Dedekind complete and \(T\) is order limited, then \(S\) is order limited. Reviewer: Marko Kandić (Ljubljana) Cited in 6 Documents MSC: 46B42 Banach lattices 47B60 Linear operators on ordered spaces 47B65 Positive linear operators and order-bounded operators Keywords:order-Dunford-Pettis operator; limited operator; order limited operator; order continuous norm; weak\(^*\) sequentially continuous lattice operations; Dedekind \(\sigma\)-complete Banach lattice PDFBibTeX XMLCite \textit{J. H'Michane} and \textit{K. El Fahri}, Ann. Math. Qué. 39, No. 2, 169--176 (2015; Zbl 1350.46018) Full Text: DOI References: [1] Aliprantis, C.D., Burkinshaw, O.: Dunford-Pettis operators on Banach lattices. Trans. Am. Math. Soc. 274(1), 227-238 (1982) · Zbl 0498.47013 · doi:10.1090/S0002-9947-1982-0670929-1 [2] Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Reprint of the 1985 Original. Springer, Dordrecht (2006) · Zbl 1098.47001 [3] Andrews, K.T.: Dunford-Pettis sets in the space of Bochner integrable functions. Math. Ann. 241, 35-41 (1979) · Zbl 0398.46025 · doi:10.1007/BF01406706 [4] Aqzzouz, B., Bouras, K.: Dunford-Pettis sets in Banach lattices. Acta Math. Univ. Comen. New Ser. 81, 185-196 (2012) · Zbl 1274.46051 [5] Aqzzouz, B., Nouira, R., Zraoula, L.: Compactness properties for dominated by AM-compact operators. Proc. Am. Math. Soc. 135, 1151-1157 (2007) · Zbl 1118.47029 · doi:10.1090/S0002-9939-06-08585-6 [6] Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachrichten 119, 55-58 (1984) · Zbl 0601.47019 · doi:10.1002/mana.19841190105 [7] Dodds, P.G., Fremlin, D.H.: Compact operators on Banach lattices. Israel J. Math. 34, 287-320 (1979) · Zbl 0438.47042 · doi:10.1007/BF02760610 [8] El Kaddouri, A., Moussa, M.: About the class of ordered limited operators. Acta Univ. Carolinae-Math. Physica 54(1), 37-43 (2013) · Zbl 1307.46008 [9] Flores, J., Hernández, F.L., Tradacete, P.: Domination problems for strictly singular operators and other related classes. Positivity 15(4), 595-616 (2011) · Zbl 1250.47042 · doi:10.1007/s11117-010-0100-x [10] Kalton, N.J., Saab, P.: Ideal properties of regular operators between Banach lattices. Ill. J. Math. 29(3), 382-400 (1985) · Zbl 0568.47030 [11] Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer, Berlin (1991) · Zbl 0743.46015 · doi:10.1007/978-3-642-76724-1 [12] Wickstead, A.W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems. Math. Proc. Camb. Philos. Soc. 120, 175-179 (1996) · Zbl 0872.47018 · doi:10.1017/S0305004100074752 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.