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.
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