Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 1047-1057, 2020
A note on L-Dunford-Pettis sets in a topological dual Banach space
Abderrahman Retbi
Received March 18, 2019. Published online April 30, 2020.
Abstract: The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between L-Dunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.
References: [1] C. D. Aliprantis, O. Burkinshaw: Positive Operators. Springer, Berlin (2006). DOI 10.1007/978-1-4020-5008-4 | MR 2262133 | Zbl 1098.47001
[2] B. Aqzzouz, K. Bouras: (L) sets and almost (L) sets in Banach lattices. Quaest. Math. 36 (2013), 107-118. DOI 10.2989/16073606.2013.779962 | MR 3043674 | Zbl 1274.46053
[3] B. Aqzzouz, K. Bouras, M. Moussa: Duality property for positive weak Dunford-Pettis operators. Int. J. Math. Math. Sci. 2011 (2011), Article ID 609287, 12 pages. DOI 10.1155/2011/609287 | MR 2821970 | Zbl 1262.47057
[4] K. Bouras, M. Moussa: Banach lattice with weak Dunford-Pettis property. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 5 (2011), 174-178.
[5] J. X. Chen, Z. L. Chen, G. X. Ji: Almost limited sets in Banach lattices. J. Math. Anal. Appl. 412 (2014), 547-553. DOI 10.1016/j.jmaa.2013.10.085 | MR 3145821 | Zbl 1353.46015
[6] P. G. Dodds, D. H. Fremlin: Compact operators in Banach lattices. Isr. J. Math. 34 (1979), 287-320. DOI 10.1007/BF02760610 | MR 0570888 | Zbl 0438.47042
[7] I. Ghenciu, P. Lewis: The Dunford-Pettis property, the Gelfand-Phillips property, and $L$-sets. Colloq. Math. 106 (2006), 311-324. DOI 10.4064/cm106-2-11 | MR 2283818 | Zbl 1118.46017
[8] P. Meyer-Nieberg: Banach Lattices. Universitext, Springer, Berlin (1991). DOI 10.1007/978-3-642-76724-1 | MR 1128093 | Zbl 0743.46015
[9] A. Retbi, B. El Wahbi: L-Dunford-Pettis property in Banach spaces. Methods Func. Anal. Topol. 22 (2016), 387-392. MR 3591087 | Zbl 1374.46025
[10] Y. Wen, J. Chen: Characterizations of Banach spaces with relatively compact Dunford-Pettis sets. Adv. Math., Beijing 45 (2014), 122-132. DOI 10.11845/sxjz.2014095b | MR 3483491 | Zbl 1374.47031
[11] W. Wnuk: Banach lattices with the weak Dunford-Pettis property. Atti Semin. Mat. Fis. Univ. Modena 42 (1994), 227-236. MR 1282338 | Zbl 0805.46023
Affiliations: Abderrahman Retbi, Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, Avenue de L'Université, B.P.133, Kénitra, Morocco, e-mail: abderrahmanretbi@hotmail.com
