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

TL;DR

This paper investigates the spectral properties and long-term behavior of irreducible semigroups of Harris operators on Banach lattices, establishing conditions for trivial peripheral spectrum and convergence.

Contribution

It proves that the peripheral point spectrum is trivial under certain domination conditions and provides criteria for strong convergence of the semigroup and operator powers.

Findings

Peripheral point spectrum is trivial when dominated by a compact or kernel operator.

Point spectrum of powers of an operator intersects the unit circle at most at 1.

Provides conditions for strong convergence of the semigroup and its powers.

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.