L-weakly and M-weakly Demicompact Operators On Banach Lattices

Abstract

In the present paper, we introduce and study new concepts of  L-weakly and M-weakly demicompact operators. Let $E$ be a Banach lattice. An operator $T:E\longrightarrow E$ is called L-weakly demicompact if  for every norm bounded sequence $(x_{n})$ in $\mathcal{B}_{E}$ such that $\{x_{n}-Tx_{n}, \hskip0.2cm n  \in \mathbb{N} \}$ is L-weakly compact subset of $E$, we have $\{x_{n}, \hskip0.2cm n  \in \mathbb{N} \}$ is L-weakly compact subset of $E$, and an operator $T:E\longrightarrow E$ is called M-weakly demicompact if  for every norm bounded disjoint sequence $(x_{n})$ in $E$ such that $\|x_{n}-Tx_{n}\|\rightarrow 0$, we have $\|x_{n}\|\rightarrow 0$. L-weakly (Resp. M-weakly) demicompact operators generalize known classes of operators which defined by  L-weakly  (Resp. M-weakly) compact operators.