Dunford-Pettis operators on Banach lattices

Abstract:

Consider a Banach lattice $E$ and two positive operators $S,T:E \to E$ that satisfy $0 \leqslant S \leqslant T$. In $[{\mathbf {2,3}}]$ we examined the case when $T$ is a compact (or weakly compact) operator and studied what effect this had on an operator (such as $S$) dominated by $T$. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator $T$, to ensure that $S$ (or some power of $S$) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters. Theorem. Let $E$ be a Banach lattice, and let $S,T:E \to E$ be two positive operators such that $0 \leqslant S \leqslant T$. If $T$ is compact then (1) ${S^3}$ is a compact operator (although ${S^2}$ need not be compact); (2) ${S^2}$ is a Dunford-Pettis and weakly compact operator ( although $S$ need not be ); (3) $S$ is a weak Dunford-Pettis operator. In another direction, our techniques and results will be related to the lattice stracture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.