Ideal properties of regular operators beween Banach lattices (Q1060400)

The main result of the paper is a positive answer to the question of \textit{C. D. Aliprantis} and \textit{O. Burkinshaw} [Trans. Am. Math. Soc. 274, 227-238 (1982; Zbl 0498.47013)]: Theorem 4.4. Let E and F be Banach lattices so that F has order continuous norm. Suppose \(T: E\to F\) is a positive Dunford-Pettis operator and \(0\leq S\leq T\). Then S is a Dunford-Pettis operator. The argument of this result hinges on a certain approximation theorem which has some other applications to similar problems.