, Volume 17, Issue 3, pp 875-898
Date: 02 Nov 2012

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

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Abstract

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.