A note on the spectrum of irreducible operators and semigroups

TL;DR

This paper demonstrates that the peripheral spectrum of irreducible operators or semigroups can be more complex than previously thought, by constructing examples with prescribed spectral properties on the unit circle.

Contribution

It shows that the classical Perron-Frobenius spectral result does not extend to the entire peripheral spectrum, providing explicit counterexamples.

Findings

Constructed irreducible stochastic operators with prescribed peripheral spectrum.

Extended the construction to $C_0$-semigroups.

Counterexamples to classical spectral assertions.

Abstract

Let $T$ denote a positive operator with spectral radius $1$ on, say, an $L^p$-space. A classical result in infinite dimensional Perron--Frobenius theory says that, if $T$ is irreducible and power bounded, then its peripheral point spectrum is either empty or a subgroup of the unit circle. In this note we show that the analogous assertion for the entire peripheral spectrum fails. More precisely, for every finite union $U$ of finite subgroups of the unit circle we construct an irreducible stochastic operator on $\ell^1$ whose peripheral spectrum equals $U$. We also give a similar construction for the $C_0$-semigroup case.