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Keywords:
weak Banach-Saks operator; weakly compact operator; {\rm L}-weakly compact operator; {\rm M}-weakly compact operator; order continuous norm, positive Schur property; reflexive Banach space
weak Banach-Saks operator; weakly compact operator; {\rm L}-weakly compact operator; {\rm M}-weakly compact operator; order continuous norm, positive Schur property; reflexive Banach space
Summary:
We establish necessary and sufficient conditions under which weak Banach-Saks operators are weakly compact (respectively, {\rm L}-weakly compact; respectively, {\rm M}-weakly compact). As consequences, we give some interesting characterizations of order continuous norm (respectively, reflexive Banach lattice).
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] Alpay, S., Altin, B., Tonyali, C.: On property (b) of vector lattices. Positivity 7 (2003), 135-139. DOI 10.1023/A:1025840528211 | MR 2028377 | Zbl 1036.46018
[3] Aqzzouz, B., Aboutafail, O., Belghiti, T., H'Michane, J.: The $b$-weak compactness of weak Banach-Saks operators. Math. Bohem. 138 (2013), 113-120. DOI 10.21136/MB.2013.143283 | MR 3099302 | Zbl 1289.46027
[4] Aqzzouz, B., Elbour, A., H'Michane, J.: Some properties of the class of positive Dunford-Pettis operators. J. Math. Anal. Appl. 354 (2009), 295-300. DOI 10.1016/j.jmaa.2008.12.063 | MR 2510440 | Zbl 1167.47033
[5] Aqzzouz, B., Elbour, A., H'Michane, J.: On some properties of the class of semi-compact operators. Bull. Belg. Math. Soc. - Simon Stevin 18 (2011), 761-767. DOI 10.36045/bbms/1320763136 | MR 2918181 | Zbl 1250.47020
[6] Aqzzouz, B., H'Michane, J., Aboutafail, O.: Weak compactness of AM-compact operators. Bull. Belg. Math. Soc. - Simon Stevin 19 (2012), 329-338. DOI 10.36045/bbms/1337864276 | MR 2977235 | Zbl 1253.46027
[7] Baernstein, A.: On reflexivity and summability. Stud. Math. 42 (1972), 91-94. DOI 10.4064/sm-42-1-91-94 | MR 0305044 | Zbl 0228.46014
[8] Beauzamy, B.: Propriété de Banach-Saks et modèles étalés. Séminaire sur la Géométrie des Espaces de Banach (1977-1978) École Polytech., Palaiseau (1978), 16 pages French. MR 0520205 | Zbl 0386.46017
[9] Chen, Z. L., Wickstead, A. W.: $L$-weakly and $M$-weakly compact operators. Indag. Math., New Ser. 10 (1999), 321-336. DOI 10.1016/S0019-3577(99)80025-1 | MR 1819891 | Zbl 1028.47028
[10] Ghoussoub, N., Johnson, W. B.: Counterexamples to several problems on the factorization of bounded linear operators. Proc. Am. Math. Soc. 92 (1984), 233-238. DOI 10.1090/S0002-9939-1984-0754710-8 | MR 0754710 | Zbl 0615.46022
[11] Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991). DOI 10.1007/978-3-642-76724-1 | MR 1128093 | Zbl 0743.46015
[12] Nishiura, T., Waterman, D.: Reflexivity and summability. Stud. Math. 23 (1963), 53-57. DOI 10.4064/sm-23-1-53-57 | MR 0155167 | Zbl 0121.09402
[13] Rosenthal, H. P.: Weakly independent sequences and the weak Banach-Saks property. Proceedings of the Durham Symposium on the Relations Between Infinite Dimensional and Finite-Dimentional Convexity Duke University, Durham (1975), 26 pages.
[14] Wnuk, W.: Banach Lattices with Order Continuous Norms. Advanced Topics in Mathematics. Polish Scientific Publishers, Warsaw (1999). Zbl 0948.46017
[15] Zaanen, A. C.: Introduction to Operator Theory in Riesz Spaces. Springer, Berlin (1997). DOI 10.1007/978-3-642-60637-3 | MR 1631533 | Zbl 0878.47022
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