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Keywords:
almost demi Dunford--Pettis operator; Banach lattice; positive Schur property
almost demi Dunford--Pettis operator; Banach lattice; positive Schur property
Summary:
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.
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