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Keywords:
M-weakly compact operator; L-weakly compact operator; Dunford-Pettis operator; weakly compact operator; semi-compact operator; compact operator; order continuous norm; discrete Banach lattice; positive Schur property
Summary:
We establish some sufficient conditions under which the subspaces of Dunford-Pettis operators, of M-weakly compact operators, of L-weakly compact operators, of weakly compact operators, of semi-compact operators and of compact operators coincide and we give some consequences.
References:
[1] D., Aliprantis C., O., Burkinshaw: Locally Solid Riesz Spaces. Academic Press, Providence, RI (1978). MR 0493242 | Zbl 0402.46005
[2] D., Aliprantis C., O., Burkinshaw: Dunford-Pettis operators on Banach lattices. Trans. Amer. Math. Soc. 274 (1982), 227-238. DOI 10.1090/S0002-9947-1982-0670929-1 | MR 0670929 | Zbl 0498.47013
[3] D., Aliprantis C., O., Burkinshaw: Positive Operators. Pure and Applied Mathematics, 119. Academic Press, Inc., Orlando, FL (1985). MR 0809372 | Zbl 0608.47039
[4] D., Aliprantis C., O., Burkinshaw: On the ring ideal generated by a positive operator. J. Funct. Anal. 67 (1986), 60-72. DOI 10.1016/0022-1236(86)90043-1 | MR 0842603 | Zbl 0588.47044
[5] B., Aqzzouz, R., Nouira, L., Zraoula: Les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts. Proc. Amer. Math. Soc. 134 (2006), 1161-1165. DOI 10.1090/S0002-9939-05-08083-4 | MR 2196052 | Zbl 1099.46016
[6] B., Aqzzouz, R., Nouira, L., Zraoula: About positive Dunford-Pettis operators on Banach lattices. J. Math. Anal. Appl. 324 (2006), 49-59. DOI 10.1016/j.jmaa.2005.10.083 | MR 2262455 | Zbl 1112.47028
[7] B., Aqzzouz, R., Nouira, L., Zraoula: Semi-compactness of positive Dunford-Pettis operators. Proc. Amer. Math. Soc. 136 (2008), 1997-2006. DOI 10.1090/S0002-9939-08-09032-1 | MR 2383506 | Zbl 1152.47012
[8] B., Aqzzouz, R., Nouira, L., Zraoula: On the duality problem of positive Dunford-Pettis operators on Banach lattices. Rend. Circ. Mat. Palermo 57 (2008), 287-294. DOI 10.1007/s12215-008-0021-8 | MR 2452672 | Zbl 1166.47036
[9] L., Chen Z., W., Wickstead A.: L-weakly and M-weakly compact operators. Indag. Math. (N.S.) 10 (1999), 321-336. DOI 10.1016/S0019-3577(99)80025-1 | MR 1819891 | Zbl 1028.47028
[10] N., Cheng, L., Chen Z., Y., Feng: L and M-weak compactness of positive semi-compact operators. Rend. Circ. Mat. Palermo 59 101-105 (2010). DOI 10.1007/s12215-010-0006-2 | MR 2639440 | Zbl 1202.47041
[11] G., Dodds P., H., Fremlin D.: Compact operators on Banach lattices. Israel J. Math. 34 (1979), 287-320. DOI 10.1007/BF02760610 | MR 0570888
[12] J., Kalton N., P., Saab: Ideal properties of regular operators between Banach lattices. Ill. J. Math. 29 (1985), 382-400. DOI 10.1215/ijm/1256045630 | MR 0786728
[13] Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991). MR 1128093
[14] W., Wickstead A.: Converses for the Dodds-Fremlin and Kalton-Saab Theorems. Math. Proc. Camb. Phil. Soc. 120 (1996), 175-179. DOI 10.1017/S0305004100074752 | MR 1373356 | Zbl 0872.47018
[15] W., Wnuk: A note on the positive Schur property. Glasgow Math. J. 31 (1989), 169-172. DOI 10.1017/S0017089500007692 | MR 0997812 | Zbl 0694.46020
[16] C., Zaanen A.: Riesz Spaces II. North Holland, Amsterdam (1983). MR 0704021 | Zbl 0519.46001
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