Some properties of the class of positive Dunford--Pettis operators (Q1018187)

The authors establish necessary and sufficient conditions for positive Dunford--Pettis operators to be M-weakly compact (respectively, L-weakly compact). They first prove that each Dunford--Pettis operator \(T\) from a Banach lattice \(E\) into a Banach space \(F\) is M-weakly compact if and only if the norm dual \(E'\) of \(E\) is order continuous or \(F=\{0\}\). Next, they establish an analogue of the Riesz theorem by proving that the closed unit ball \(B_{E}\) of a Banach lattice \(E\) is L-weakly compact if and only if \(E\) is finite-dimensional. From this, they deduce some interesting results.