On the duality of DW-compact operators and DW-DP operators
Geraldo Botelho, Ariel Monção
TL;DR
The paper addresses duality properties between $DW$-compact and $DW$-DP operators on Banach lattices, establishing when the $DW$-compact property is preserved by adjoints and providing a full criterion for when the adjoint of a $DW$-compact operator remains $DW$-compact. It develops the core equivalences for $DW$-compact operators, derives sufficient and necessary conditions involving order continuity, discreteness, and the positive Schur property, and characterizes the adjoint behavior in terms of $E^*$ and $F^*$. For $DW$-$DP$ operators, it shows that order-continuous norms in the underlying spaces ensure desirable adjoint implications, while the lack of order continuity yields counterexamples, leading to a necessity result for the spaces. Overall, the work clarifies duality structures in these operator classes and identifies structural Banach lattice properties that govern adjoint behavior, with implications for when classical compactness dualities extend to $DW$-type concepts.
Abstract
We give a necessary condition and a sufficient condition on the Banach lattices E and F so that an operator from E to F is DW-compact whenever its adjoint is DW-compact. We do the same, with different conditions, for DW-DP operators. Moreover, we characterize the Banach lattices E and F for which the adjoint of every DW-compact operator from E to F is DW-compact.