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Keywords:
Dunford-Pettis operator; order continuous norm; positive Schur property; KB-space
Dunford-Pettis operator; order continuous norm; positive Schur property; KB-space
Summary:
We establish sufficient conditions for the duality of regular Dunford-Pettis operators on Banach lattices and necessary conditions for the duality condition ``if the adjoint of a (positive) operator is Dunford-Pettis, then the operator itself is''. In particular, we show that if each operator $T\colon E\rightarrow F$ from a Banach lattice $E$ with an order continuous norm to another Banach lattice $F$ is Dunford-Pettis whenever its adjoint $T^{\prime }\colon F^{\prime }\rightarrow E^{\prime }$ is Dunford-Pettis, then $E$ has the Schur property or $F$ is a KB-space. As consequences, we deduce a characterization of the Schur property (and KB-spaces).
References:
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006). DOI 10.1007/978-1-4020-5008-4 | MR 2262133 | Zbl 1098.47001
[2] Aqzzouz, B., Bourass, K., Elbour, A.: Some generalizations on positive Dunford-Pettis operators. Result. Math. 54 (2009), 207-218. DOI 10.1007/s00025-009-0372-2 | MR 2534444 | Zbl 1191.47051
[3] Aqzzouz, B., Elbour, A., Wickstead, A. W.: Positive almost Dunford-Pettis operators and their duality. Positivity 15 (2011), 185-197. DOI 10.1007/s11117-010-0050-3 | MR 2803812 | Zbl 1232.46019
[4] Aqzzouz, B., Nouira, R., Zraoula, L.: On the duality problem of positive Dunford-Pettis operators on Banach lattices. Rend. Circ. Mat. Palermo (2) 57 (2008), 287-294. DOI 10.1007/s12215-008-0021-8 | MR 2452672 | Zbl 1166.47036
[5] Kalton, N. J., Saab, P.: Ideal properties of regular operators between Banach lattices. Ill. J. Math. 29 (1985), 382-400. DOI 10.1215/ijm/1256045630 | MR 0786728 | Zbl 0568.47030
[6] Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991). DOI 10.1007/978-3-642-76724-1 | MR 1128093 | Zbl 0743.46015
[7] Wickstead, A. W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems. Math. Proc. Camb. Philos. Soc. 120 (1996), 175-179. DOI 10.1017/S0305004100074752 | MR 1373356 | Zbl 0872.47018
[8] Wnuk, W.: Banach spaces with properties of the Schur type -- A survey. Conf. Semin. Mat. Univ. Bari 249 (1993), 25 pages. MR 1230964 | Zbl 0812.46010
[9] Zaanen, A. C.: Riesz Spaces II. North-Holland Mathematical Library 30. North-Holland, Amsterdam (1983). DOI 10.1016/s0924-6509(08)x7015-1 | MR 0704021 | Zbl 0519.46001
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