Omission of hypotheses in adjoint equivalence corollaries

Determine whether the assumption that the Banach space Y has the sequential (p,q)-Dunford–Pettis property can be omitted from Corollary 3.6, and whether the assumption that the closed unit ball B_{Y^*} is weak-star sequentially compact can be omitted from Corollary 3.8.

Background

The paper establishes equivalences between norm attainment, the weakly p-singular maximizing property, and corresponding properties for adjoint operators under several hypotheses. Corollary 3.6 assumes that B_X is relatively weakly p-precompact, that B_{Y^*} is relatively weakly q-precompact, and that Y has the sequential (p,q)-Dunford–Pettis property. Corollary 3.8 gives a related equivalence under relative weak p-precompactness of B_X, the Dunford–Pettis-star property of order p for Y, and weak-star sequential compactness of B_{Y^*}.

The authors explicitly leave unresolved whether the sequential (p,q)-Dunford–Pettis assumption in the first corollary and the weak-star sequential compactness assumption in the second are genuinely necessary. They note that a counterexample to either omission would provide a negative answer to Question 5.12 of García-Lirola and Petitjean.