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 less than or equal to S less than or equal to 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.