Operators whose adjoints and second adjoints are almost Dunford-Pettis

TL;DR

This paper characterizes Banach lattices with the positive Schur property via second adjoints of operators being almost Dunford-Pettis, and extends results on when adjoints and second adjoints of certain operators are almost Dunford-Pettis.

Contribution

It provides new characterizations of Banach lattices and extends conditions under which operator adjoints are almost Dunford-Pettis.

Findings

Characterization of Banach lattices with positive Schur property.

Conditions under which adjoints of operators are almost Dunford-Pettis.

Identification of classes of operators with almost Dunford-Pettis second adjoints.

Abstract

First we characterize the Banach lattices E whose biduals have the positive Schur property by means of second adjoints of operators on E being almost Dunford-Pettis. Next we extend some known results concerning conditions on the Banach lattices E and F under which the adjoint T* and the second adjoint T** of any positive almost Dunford-Pettis operator T from E to F are almost Dunford-Pettis. Finally, we prove when T* and T** are almost Dunford-Pettis for any (non necessarily almost Dunford-Pettis) T that is either bounded, regular, order bounded or weakly compact.