On positive strictly singular operators and domination

We study the domination problem by positive strictly singular % operators between Banach lattices. Precisely we show that if E and %F are two Banach lattices such that the norms on E' and F are %order continuous and E satisfies the subsequence splitting property, %and %0≤S≤ T : E → F are two positive operators, then T strictly %singular implies S strictly singular. The special case of %endomorphisms is also considered. Applications to the class of %strictly co-singular (or Pelczynski) operators are given too.