| Title:
|
On the class of order Dunford-Pettis operators (English) |
| Author:
|
Bouras, Khalid |
| Author:
|
El Kaddouri, Abdelmonaim |
| Author:
|
H'michane, Jawad |
| Author:
|
Moussa, Mohammed |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
138 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
289-297 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T'$ from $F'$ into $E'$ if, and only if, the norm of $E'$ is order continuous or $F'$ has the Schur property. As a consequence we show that, if $E$ and $F$ are two Banach lattices such that $E$ or $F$ has the Dunford-Pettis property, then each order Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint $T'\colon F'\longrightarrow E'$ which is Dunford-Pettis if, and only if, the norm of $E'$ is order continuous or $F'$ has the Schur property. (English) |
| Keyword:
|
Dunford-Pettis operator |
| Keyword:
|
weak Dunford-Pettis operator |
| Keyword:
|
order Dunford-Pettis operator |
| Keyword:
|
order continuous norm |
| Keyword:
|
Schur property |
| MSC:
|
46B40 |
| MSC:
|
46B42 |
| MSC:
|
47B60 |
| idZBL:
|
Zbl 06260034 |
| idMR:
|
MR3136498 |
| DOI:
|
10.21136/MB.2013.143438 |
| . |
| Date available:
|
2013-09-14T11:47:42Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143438 |
| . |
| Reference:
|
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| Reference:
|
[2] Andrews, K. T.: Dunford-Pettis sets in the space of Bochner integrable functions.Math. Ann. 241 (1979), 35-41. Zbl 0398.46025, MR 0531148, 10.1007/BF01406706 |
| Reference:
|
[3] Aqzzouz, B., Bouras, K.: Weak and almost Dunford-Pettis operators on Banach lattices.Demonstr. Math. 46 165-179 (2013). MR 3075506 |
| Reference:
|
[4] Aqzzouz, B., Bouras, K.: Dunford-Pettis sets in Banach lattices.Acta Math. Univ. Comen., New Ser. 81 185-196 (2012). MR 2975284 |
| Reference:
|
[5] Aqzzouz, B., Bouras, K., Moussa, M.: Duality property for positive weak Dunford-Pettis operators.Int. J. Math. Math. Sci. 2011, Article ID 609287 12 p (2011). Zbl 1262.47057, MR 2821970 |
| Reference:
|
[6] Dodds, P. G., Fremlin, D. H.: Compact operators on Banach lattices.Isr. J. Math. 34 (1979), 287-320. MR 0570888, 10.1007/BF02760610 |
| Reference:
|
[7] Meyer-Nieberg, P.: Banach Lattices.Universitext. Springer, Berlin (1991). Zbl 0743.46015, MR 1128093 |
| . |
