| Title:
|
Some properties of weak Banach-Saks operators (English) |
| Author:
|
Aboutafail, Othman |
| Author:
|
Zraoula, Larbi |
| Author:
|
Hafidi, Noufissa |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
146 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
407-418 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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). (English) |
| Keyword:
|
weak Banach-Saks operator |
| Keyword:
|
weakly compact operator |
| Keyword:
|
{\rm L}-weakly compact operator |
| Keyword:
|
{\rm M}-weakly compact operator |
| Keyword:
|
order continuous norm, positive Schur property |
| Keyword:
|
reflexive Banach space |
| MSC:
|
46A40 |
| MSC:
|
46B40 |
| MSC:
|
46B42 |
| idZBL:
|
Zbl 07442510 |
| idMR:
|
MR4336547 |
| DOI:
|
10.21136/MB.2020.0055-18 |
| . |
| Date available:
|
2021-11-08T16:19:20Z |
| Last updated:
|
2021-12-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149257 |
| . |
| Reference:
|
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators.Springer, Berlin (2006). Zbl 1098.47001, MR 2262133, 10.1007/978-1-4020-5008-4 |
| Reference:
|
[2] Alpay, S., Altin, B., Tonyali, C.: On property (b) of vector lattices.Positivity 7 (2003), 135-139. Zbl 1036.46018, MR 2028377, 10.1023/A:1025840528211 |
| Reference:
|
[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. Zbl 1289.46027, MR 3099302, 10.21136/MB.2013.143283 |
| Reference:
|
[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. Zbl 1167.47033, MR 2510440, 10.1016/j.jmaa.2008.12.063 |
| Reference:
|
[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. Zbl 1250.47020, MR 2918181, 10.36045/bbms/1320763136 |
| Reference:
|
[6] Aqzzouz, B., H'Michane, J., Aboutafail, O.: Weak compactness of AM-compact operators.Bull. Belg. Math. Soc. - Simon Stevin 19 (2012), 329-338. Zbl 1253.46027, MR 2977235, 10.36045/bbms/1337864276 |
| Reference:
|
[7] Baernstein, A.: On reflexivity and summability.Stud. Math. 42 (1972), 91-94. Zbl 0228.46014, MR 0305044, 10.4064/sm-42-1-91-94 |
| Reference:
|
[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. Zbl 0386.46017, MR 0520205 |
| Reference:
|
[9] Chen, Z. L., Wickstead, A. W.: $L$-weakly and $M$-weakly compact operators.Indag. Math., New Ser. 10 (1999), 321-336. Zbl 1028.47028, MR 1819891, 10.1016/S0019-3577(99)80025-1 |
| Reference:
|
[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. Zbl 0615.46022, MR 0754710, 10.1090/S0002-9939-1984-0754710-8 |
| Reference:
|
[11] Meyer-Nieberg, P.: Banach Lattices.Universitext. Springer, Berlin (1991). Zbl 0743.46015, MR 1128093, 10.1007/978-3-642-76724-1 |
| Reference:
|
[12] Nishiura, T., Waterman, D.: Reflexivity and summability.Stud. Math. 23 (1963), 53-57. Zbl 0121.09402, MR 0155167, 10.4064/sm-23-1-53-57 |
| Reference:
|
[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. |
| Reference:
|
[14] Wnuk, W.: Banach Lattices with Order Continuous Norms.Advanced Topics in Mathematics. Polish Scientific Publishers, Warsaw (1999). Zbl 0948.46017 |
| Reference:
|
[15] Zaanen, A. C.: Introduction to Operator Theory in Riesz Spaces.Springer, Berlin (1997). Zbl 0878.47022, MR 1631533, 10.1007/978-3-642-60637-3 |
| . |
