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
$0 \leqslant S \leqslant T : E \longrightarrow F$
, 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
$E^{\prime}$
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
$E^{\prime}$
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].