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# Limited and Dunford-Pettis operators on Banach lattices

### 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.

### 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},
}

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