### Abstract

This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:

1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.

2) The topological dual $E^{\prime }$is discrete and its norm is
order continuous.

3) The norm of $E^{\prime }$ is order continuous and the lattice
operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.

4) The norms of $E$ and of $E^{\prime }$ are order continuous.

**Key words:** Limited operator, Dunford-Pettis operator, Banach lattice, order continuous norm.

### Full Text

### Article Information

Title | Limited and Dunford-Pettis operators on Banach lattices |

Source | Methods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 205-210 |

MathSciNet |
MR4016207 |

Milestones | Received 12/02/2019; Revised 04/04/2019 |

Copyright | The Author(s) 2019 (CC BY-SA) |

### Authors Information

*Abdennabi EL Aloui*

Department of Mathematics, Faculty Polydisciplinary of Larache, Abdelmalek Essaadi University, P.O. Box 745, Larache 92004, Morocco

*Khalid Bouras*

Department of Mathematics, Faculty Polydisciplinary of Larache, Abdelmalek Essaadi University, P.O. Box 745, Larache 92004, Morocco

*Aziz Elbour*

Department of Mathematics, Faculty of Science and Technology, Moulay Ismail University, P.O. Box 509, Boutalamine 52000, Errachidia, Morocco

### Citation Example

Khalid Bouras, Abdennabi EL Aloui, and Aziz Elbour, *Limited and Dunford-Pettis operators on Banach lattices*, Methods Funct. Anal. Topology **25**
(2019), no. 3, 205-210.

### BibTex

@article {MFAT1205,
AUTHOR = {Khalid Bouras and Abdennabi EL Aloui and Aziz Elbour},
TITLE = {Limited and Dunford-Pettis operators on Banach lattices},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {3},
PAGES = {205-210},
ISSN = {1029-3531},
MRNUMBER = {MR4016207},
URL = {http://mfat.imath.kiev.ua/article/?id=1205},
}