A note on the spectrum of irreducible operators and semigroups

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.