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Domination by positive disjointly strictly singular operators

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E --> E is dominated by a disjointly strictly singular operator, then S-2 is disjointly strictly singular.