Characterizations of strictly singular operators on Banach lattices

New characterizations of strictly singular operators between Banach lattices are given. It is proved that, for Banach lattices X and Y such that X has finite cotype and Y satisfies a lower 2-estimate, an operator T : X → Y is strictly singular if and only if it is disjointly strictly singular and ℓ2-singular. Moreover, if T is regular then the same equivalence holds provided that Y is just order continuous. Furthermore, it is shown that these results fail if the conditions on the lattices are relaxed.