  Given a positive, irreducible and bounded C_0-semigroup on a Banach lattice
with order continuous norm, we prove that the peripheral point spectrum of its
generator is trivial whenever one of its operators dominates a non-trivial
compact or kernel operator. For a discrete semigroup, i.e. for powers of a
single operator T, we show that the point spectrum of some power T^k intersects
the unit circle at most in 1. As a consequence, we obtain a sufficient
condition for strong convergence of the C_0-semigroup and for a subsequence of
the powers of T, respectively.
