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

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

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 bound...

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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
Series:Mathematica Bohemica
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
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author Kamal El Fahri
Nabil Machrafi
Jawad H'Michane
Aziz Elbour
author_facet Kamal El Fahri
Nabil Machrafi
Jawad H'Michane
Aziz Elbour
author_sort Kamal El Fahri
collection DOAJ
description 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.}
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spelling doaj.art-9508ae04f9a44d0ebebb00a4f5aebb292022-12-21T22:37:49ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362016-10-01141332733810.21136/MB.2016.0076-14MB.2016.0076-14Application of ${\rm (L)$ sets to some classes of operators}Kamal El FahriNabil MachrafiJawad H'MichaneAziz ElbourThe 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.}http://mb.math.cas.cz/full/141/3/mb141_3_3.pdf ${\rm (L)$ set order $\rm(L)$-Dunford-Pettis operator weakly sequentially continuous lattice operations Banach lattice}
spellingShingle Kamal El Fahri
Nabil Machrafi
Jawad H'Michane
Aziz Elbour
Application of ${\rm (L)$ sets to some classes of operators}
Mathematica Bohemica
${\rm (L)$ set
order $\rm(L)$-Dunford-Pettis operator
weakly sequentially continuous lattice operations
Banach lattice}
title Application of ${\rm (L)$ sets to some classes of operators}
title_full Application of ${\rm (L)$ sets to some classes of operators}
title_fullStr Application of ${\rm (L)$ sets to some classes of operators}
title_full_unstemmed Application of ${\rm (L)$ sets to some classes of operators}
title_short Application of ${\rm (L)$ sets to some classes of operators}
title_sort application of rm l sets to some classes of operators
topic ${\rm (L)$ set
order $\rm(L)$-Dunford-Pettis operator
weakly sequentially continuous lattice operations
Banach lattice}
url http://mb.math.cas.cz/full/141/3/mb141_3_3.pdf
work_keys_str_mv AT kamalelfahri applicationofrmlsetstosomeclassesofoperators
AT nabilmachrafi applicationofrmlsetstosomeclassesofoperators
AT jawadhmichane applicationofrmlsetstosomeclassesofoperators
AT azizelbour applicationofrmlsetstosomeclassesofoperators

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