## On the weak compactness of weak$$^*$$ Dunford-Pettis operators on Banach lattices.(English)Zbl 1380.46012

Summary: We characterize Banach lattices on which each positive weak$$^*$$ Dunford-Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $$F$$ is a Banach lattice with order continuous norm, then each positive weak$$^*$$ Dunford-Pettis operator $$T : E\to F$$ is weakly compact if, and only if, the norm of $$E'$$ is order continuous or $$F$$ is reflexive. On the other hand, when the Banach lattice $$F$$ is Dedekind $$\sigma$$-complete, we show that every positive weak$$^*$$ Dunford-Pettis operator $$T: E\to F$$ is M-weakly compact if, and only if, the norms of $$E'$$ and $$F$$ are order continuous or $$E$$ is finite-dimensional.

### MSC:

 46B42 Banach lattices 47B60 Linear operators on ordered spaces 47B65 Positive linear operators and order-bounded operators
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