A note on the spectrum of irreducible operators and semigroups

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 is irreducible and power bounded, then its peripheral point spectrum either empty or subgroup of the unit circle. In this note we show that analogous assertion for entire fails. More precisely, every finite union $U$ subgroups circle construct stochastic on $\ell^1$ whose equals $U$. We also give similar construction $C_0$-semigroup case.