On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators

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.