| Title:
|
Application of $\rm (L)$ sets to some classes of operators (English) |
| Author:
|
El Fahri, Kamal |
| Author:
|
Machrafi, Nabil |
| Author:
|
H'michane, Jawad |
| Author:
|
Elbour, Aziz |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
141 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
327-338 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $\rm (L)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\rm (L)$ sets. As a sequence characterization of such operators, we see that an operator $T\colon X\rightarrow E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $|T(x_{n})|\rightarrow 0$ for $\sigma (E,E')$ for every weakly null sequence $(x_{n})\subset X$. We also investigate relationships between order $Ł$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T\colon E\rightarrow F$ between Banach lattices is Dunford-Pettis whenever it is both order $\rm (L)$-Dunford-Pettis and weak* Dunford-Pettis, if and only if $E$ has the Schur property or the norm of $F$ is order continuous. (English) |
| Keyword:
|
$\rm (L)$ set |
| Keyword:
|
order $\rm (L)$-Dunford-Pettis operator |
| Keyword:
|
weakly sequentially continuous lattice operations |
| Keyword:
|
Banach lattice |
| MSC:
|
46B42 |
| MSC:
|
46B50 |
| MSC:
|
47B65 |
| idZBL:
|
Zbl 06644017 |
| idMR:
|
MR3557583 |
| DOI:
|
10.21136/MB.2016.0076-14 |
| . |
| Date available:
|
2016-10-01T16:00:43Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145897 |
| . |
| Reference:
|
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators.Springer, Dordrecht (2006). Zbl 1098.47001, MR 2262133 |
| Reference:
|
[2] Aqzzouz, B., Bouras, K.: Weak and almost Dunford-Pettis operators on Banach lattices.Demonstr. Math. 46 (2013), 165-179. Zbl 1280.46010, MR 3075506 |
| Reference:
|
[3] Aqzzouz, B., Bouras, K.: Dunford-Pettis sets in Banach lattices.Acta Math. Univ. Comen., New Ser. 81 (2012), 185-196. Zbl 1274.46051, MR 2975284 |
| Reference:
|
[4] Dodds, P. G., Fremlin, D. H.: Compact operators in Banach lattices.Isr. J. Math. 34 (1979), 287-320. Zbl 0438.47042, MR 0570888, 10.1007/BF02760610 |
| Reference:
|
[5] Kaddouri, A. El, Moussa, M.: About the class of ordered limited operators.Acta Univ. Carol. Math. Phys. 54 (2013), 37-43. Zbl 1307.46008, MR 3222749 |
| Reference:
|
[6] Emmanuele, G.: A dual characterization of Banach spaces not containing $\ell ^{1}$.Bull. Pol. Acad. Sci. Math. 34 (1986), 155-160. MR 0861172 |
| Reference:
|
[7] Ghenciu, I., Lewis, P.: The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets.Colloq. Math. 106 (2006), 311-324. Zbl 1118.46017, MR 2283818, 10.4064/cm106-2-11 |
| Reference:
|
[8] Meyer-Nieberg, P.: Banach Lattices.Universitext. Springer, Berlin (1991). Zbl 0743.46015, MR 1128093 |
| . |
