Almost demi Dunford--Pettis operators on Banach lattices

We introduce new concept of almost demi Dunford--Pettis operators. Let $E$ be a Banach lattice. An operator $T$ from $E$ into $E$ is said to be almost demi Dunford--Pettis if, for every 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 $n\rightarrow \infty$. In addition, we study some properties of this class of operators and its relationships with others known operators.