Geodesic
Application of $\rm (L)$ sets to some classes of operators
Mathematica Bohemica, Tome 141 (2016) no. 3, pp. 327-338
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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 $\sigma (E,E')$ 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.
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 $\sigma (E,E')$ 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.
DOI : 10.21136/MB.2016.0076-14
Classification : 46B42, 46B50, 47B65
Keywords: $\rm (L)$ set; order $\rm (L)$-Dunford-Pettis operator; weakly sequentially continuous lattice operations; Banach lattice 
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El Fahri, Kamal; Machrafi, Nabil; H'michane, Jawad; Elbour, Aziz. Application of $\rm (L)$ sets to some classes of operators. Mathematica Bohemica, Tome 141 (2016) no. 3, pp. 327-338. doi: 10.21136/MB.2016.0076-14

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