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 PerronFrobenius 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2021
 arXiv:
 arXiv:2102.03772
 Bibcode:
 2021arXiv210203772G
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Spectral Theory;
 47B65;
 47A10
 EPrint:
 9 pages