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DOI: 10.46698/b8543-3760-0663-r

On \(b\)-Weakly Demicompact Operators on Banach Lattices

Benkhaled, H. , Jeribi, A.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4.
Abstract:
Aqzzouz and Elbour proved that an operator \(T\) on a Banach lattice \(E\) is \(b\)-weakly compact if and only if \(\|Tx_{n}\|\rightarrow 0\) as \(n\rightarrow \infty\) for each \(b\)-order bounded weakly null sequence \(\{x_{n}\}\) in \(E_{+}\). In this present paper, we introduce and study new concept of operators that we call \(b\)-weakly demicompact, use it to generalize known classes of operators which defined by \(b\)-weakly compact operators. An operator \(T\) on a Banach lattice \(E\) is said to be \(b\)-weakly demicompact if for every \(b\)-order bounded sequence \(\{x_{n}\}\) in \(E_{+}\) such that \(x_{n}\rightarrow 0\) in \(\sigma(E,E')\) and \(\|x_{n}-Tx_{n}\|\rightarrow 0\) as \(n\rightarrow \infty\), we have \(\|x_{n}\|\rightarrow 0\) as \(n\rightarrow \infty\). As consequence, we obtain a characterization of \(KB\)-spaces in terms of \(b\)-weakly demicompact operators. After that, we investigate the relationships between \(b\)-weakly demicompact operators and some other classes of operators on Banach lattices espaciallly their relationships with demi Dunford-Pettis operators and order weakly demicompact operators.
Keywords: Banach lattice, \(KB\)-space, \(b\)-weakly demicompact operator, order weakly demicompact operator, demi Dunford-Pettis operator
Language: English Download the full text  
For citation: Benkhaled, H. and Jeribi, A. On \(b\)-Weakly Demicompact Operators on Banach Lattices, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 20-28. DOI 10.46698/b8543-3760-0663-r
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