The conditions in Lemma IC is in fact necessary and sufficient.
But the three conditions have to be checked simultaneously.
This is because if
one finds an X1> 0 and a Y1> 0 satisfying conditions (i) and (ii) but not
condition (iii), this does not imply that there is no admissible H¥controller since there might be other X1> 0 and Y1> 0 that satisfy all three conditions.