Domination by Positive Strictly Singular Operators

The problem of domination for positive compact operators between Banach lattices was solved by Dodds and Fremlin in [ 6 ]: given a Banach lattice E with order continuous dual norm, an order continuous Banach lattice F and two positive operators 0 ⩽ S ⩽ T : E → F , the operator S is compact if T is. A similar problem has been considered in the class of weakly compact operators by Abramovich [ 1 ] and in a general form by Wickstead [ 26 ]. Precisely, Wickstead's result shows that the operator S is weakly compact if T is whenever one of the following two conditions holds: either E ′ is order continuous or F is order continuous. When it comes to Dunford–Pettis operators, Kalton and Saab [ 15 ] have proved that the operator S is Dunford–Pettis if T is, provided that the Banach lattice F is order continuous. On the other hand, Aliprantis and Burkinshaw settled the problem of domination for compact [ 2 ] and weakly compact [ 3 ] endomorphisms (that is, the case when E = F ). For example, they proved that if either the norm on E or the norm on E ′ is order continuous, then the compactness of T is inherited by the power operator S 2 . Also, they showed that, for E an arbitrary Banach lattice, T being compact always implies that S 3 is compact. More recently, Wickstead studied converses for the Dodds–Fremlin and Kalton–Saab theorems in [ 27 ] and for the Aliprantis–Burkinshaw theorems in [ 28 ].