zbMATH — the first resource for mathematics

On the duality problem of positive Dunford–Pettis operators on Banach lattices. (English) Zbl 1166.47036
Summary: We give some sufficient and necessary conditions for that a positive Dunford–Pettis operator admits a dual operator which is also Dunford–Pettis, and conversely.

47B65 Positive linear operators and order-bounded operators
46A40 Ordered topological linear spaces, vector lattices
46B40 Ordered normed spaces
46B42 Banach lattices
Full Text: DOI
[1] Abramovich, Y., Lozanovsky, G. Ja.: Some numerical characteristics of normed lattices and on one result of Shimogaki, Mat. Zametki, 14 (1973), 723–732
[2] Albiac F., Kalton, N. J.: Topics in Banach space theory, Graduate Texts in Mathematics, 233. Springer, New York, (2006) · Zbl 1094.46002
[3] Aliprantis, C.D., Burkinshaw, O.: Dunford-Pettis operators on Banach lattices, Trans. Amer. Math. Soc., 274,1 (1982), 227–238 · Zbl 0498.47013 · doi:10.1090/S0002-9947-1982-0670929-1
[4] Aqzzouz, B., Nouira R., Zraoula L.: La compacité des opérateurs de Dunford-Pettis positifs sur les treillis de Banach, C. R. Math. Acad. Sc. Paris, 340 (2005), 37–42 · Zbl 1073.47025
[5] Aqzzouz, B., Nouira, R., Zraoula, L.: Les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts, Proc. Amer. Math. Soc., 134 (2006), 1161–1165 · Zbl 1099.46016 · doi:10.1090/S0002-9939-05-08083-4
[6] Aqzzouz, B., Nouira, R., Zraoula, L.: About positive Dunford-Pettis operators on Banach lattices, J. Math. Anal. Appl., 324 (2006), 49–59 · Zbl 1112.47028 · doi:10.1016/j.jmaa.2005.10.083
[7] Aqzzouz, B., Nouira, R., Zraoula, L.: The duality problem for the class of AM-compact operators on Banach lattices, Canad. Math. Bull., 51 2008), 15–20 · Zbl 1149.47030 · doi:10.4153/CMB-2008-002-0
[8] Kalton, N.J., Saab, P.: Ideal properties of regular operators between Banach lattices, Illinois Journal of Math., 29 (1985), 382–400 · Zbl 0568.47030
[9] Schaefer, H.H.: Banach lattices and positive operators, Berlin and New York, Springer-Verlag (1974) · Zbl 0296.47023
[10] Wickstead, A.W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Camb. Phil. Soc., 120 (1996), 175–179 · Zbl 0872.47018 · doi:10.1017/S0305004100074752
[11] Zaanen, A.C.: Riesz spaces II, North Holland Publishing Company, (1983) · Zbl 0519.46001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.