Application of ${\rm (L)$ sets to some classes of operators}

• Published in: Mathematica Bohemica
• Main Authors: Kamal El Fahri, Nabil Machrafi, Jawad H'Michane, Aziz Elbour
• Format: Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 2016-10-01
• Subjects: ${\rm (L)$ set; order $\rm(L)$-Dunford-Pettis operator; weakly sequentially continuous lattice operations; Banach lattice}
• Online Access: http://mb.math.cas.cz/full/141/3/mb141_3_3.pdf
• ISSN: 0862-7959; 2464-7136

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 ${\rm (L)$-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 $T X\rightarrow E$ 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.}