MATHEMATICA BOHEMICA, Vol. 141, No. 3, pp. 327-338, 2016

Kamal El Fahri, Nabil Machrafi, Université Ibn Tofail, Faculté des Sciences, Département de Mathématiques, B. P. 133, Av. université, 14000 Kénitra, Maroc, e-mail: kamalelfahri@gmail.com, nmachrafi@gmail.com; Jawad H'Michane, Université Moulay Ismail, Faculté des Sciences, Département de Mathématiques, B. P. 11201, Av. Zitoune, 50000 Meknès, Maroc, e-mail: hm1982jad@gmail.com; Aziz Elbour, Université Moulay Ismail, Faculté des Sciences et Techniques, Département de Mathématiques, B. P. 509, Av. Boutalamine, 52000 Errachidia, Maroc, e-mail: azizelbour@hotmail.com

Abstract: The paper contains some applications of the notion of $L$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $ (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 $T X\rightarrow E$ from a Banach space into a Banach lattice is order $L$-Dunford-Pettis, if and only if $|T(x_n)|\rightarrow0$ for $\sigma(E,E')$ for every weakly null sequence $(x_n)\subset X$. We also investigate relationships between order $L$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T E\rightarrow F$ 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.

Keywords: $ (L)$ set; order $\rm(L)$-Dunford-Pettis operator; weakly sequentially continuous lattice operations; Banach lattice

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Application of $ (L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H'Michane, Aziz Elbour

Received December 5, 2014. First published June 16, 2016.