Some Generalizations on Positive Dunford–Pettis Operators

Abstract

We generalize some results established in [3] by giving new necessary and sufficient conditions for which the class of Dunford-Pettis operators satisfies the duality property. Mathematics Subject Classification (2000). Primary 46A40, 46B40; Secondary 46B42.

Positive almost Dunford–Pettis operators and their duality

“…Like the Dunford-Pettis operators, [2], there is no automatic duality result for the class of almost Dunford-Pettis operators. In fact, the identity operator I d L 1 [0,1] : L 1 [0, 1] → L 1 [0, 1] is almost Dunford-Pettis but its adjoint I d L ∞ [0,1] : L ∞ [0, 1] → L ∞ [0, 1] is not almost Dunford-Pettis.…”

About positive weak almost Dunford-Pettis operators and their duality

“…Recall from page 215 of [2] that a Banach lattice G has the Schur property if and only if G has positive Schur property and the lattice operations in G are weakly sequentially continuous. ( * )…”

Some characterizations of almost Dunford–Pettis operators and applications

“…On the other hand, as Dunford-Pettis operators [4], the subspace of almost Dunford-Pettis operators does not satisfy the duality property. This issue will be raised in a forthcoming paper [5].…”