ABSTRACT. We give a characterization of those cr-Dedekind complete Banach lattices for which every continuous linear operator T: E -> co is a difference of two positive linear operators from E into cq .

The paper contains several characterizations of Banach lattices E with the dual positive Schur property (i.e., 0 f n− −−−− → 0 implies f n → 0) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford-Pettis property.

We give a characterization of those cr-Dedekind complete Banach lattices for which every continuous linear operator T: E -> co is a difference of two positive linear operators from E into cq .

The purpose of this note is to characterize those Banach lattices (£, ||-||) which have the property: an operator T: E-> c 0 is a Dunford-Pettis operator if and only if T is regular (*) (i.e., T is the difference of two positive operators). Our characterization generalizes a theorem recently proved by Holub [6] The proof of our Theorem will be preceded by some lemmas. LEMMA 1. Let (E, ||||) denote a a-Dedekind complete Banach lattice. The following statements are equivalent: (i) Every Dunford-Pettis operator T :E-*c 0 is regular. (ii) The norm \\-\\ is order continuous. Proof. (i)=>(ii). If |||| were not order continuous then {E, \\.\\) would contain a positively complemented closed Riesz subspace order and topologically isomorphic to C°. Let P:E^>€" be a positive projection and let T:f°^>c 0 be a weakly compact operator which is non-compact. The space F° has the Dunford-Pettis property, and so T is a Dunford-Pettis operator. Therefore the composition T°P:E-*c 0 is a Dunford-Pettis operator too. The assumption implies that T °P = T x -T 2 , where T t : £-» c 0 (i = 1, 2) are positive. If y e C, then Ty = T°Py = T x y -T 2 y, i.e., T = 5, -5 2 , where 5, denotes the restriction of 7J to /T. Operators 5,, as positive operators, are compact because they map the unit ball (which is an order interval), into an interval in c 0 , i.e., into a compact set. Hence T is a compact operator and we have a contradiction.(ii) ^> (i) Order bounded subsets of E are weakly compact and so they are mapped by a Dunford-Pettis operator into norm compact subsets which are order bounded in c 0 . Thus every Dunford-Pettis operator from E into c 0 is regular.

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