Application of $\left(\mathrm{L}\right)$ sets to some classes of operators

El Fahri, Kamal, et al. "Application of $\rm (L)$ sets to some classes of operators." Mathematica Bohemica 141.3 (2016): 327-338. <http://eudml.org/doc/286849>.

@article{ElFahri2016,
abstract = {The paper contains some applications of the notion of $Ł$ 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 $T\colon X\rightarrow E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $|T(x_\{n\})|\rightarrow 0$ for $\sigma (E,E^\{\prime \})$ for every weakly null sequence $(x_\{n\})\subset X$. We also investigate relationships between order $Ł$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T\colon E\rightarrow F$ 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.},
author = {El Fahri, Kamal, Machrafi, Nabil, H'michane, Jawad, Elbour, Aziz},
journal = {Mathematica Bohemica},
keywords = {$\rm (L)$ set; order $\rm (L)$-Dunford-Pettis operator; weakly sequentially continuous lattice operations; Banach lattice},
language = {eng},
number = {3},
pages = {327-338},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Application of $\rm (L)$ sets to some classes of operators},
url = {http://eudml.org/doc/286849},
volume = {141},
year = {2016},
}

TY - JOUR
AU - El Fahri, Kamal
AU - Machrafi, Nabil
AU - H'michane, Jawad
AU - Elbour, Aziz
TI - Application of $\rm (L)$ sets to some classes of operators
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 3
SP - 327
EP - 338
AB - The paper contains some applications of the notion of $Ł$ 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 $T\colon X\rightarrow E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $|T(x_{n})|\rightarrow 0$ for $\sigma (E,E^{\prime })$ for every weakly null sequence $(x_{n})\subset X$. We also investigate relationships between order $Ł$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T\colon E\rightarrow F$ 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.
LA - eng
KW - $\rm (L)$ set; order $\rm (L)$-Dunford-Pettis operator; weakly sequentially continuous lattice operations; Banach lattice
UR - http://eudml.org/doc/286849
ER -