A note on the spectrum of irreducible operators and semigroups

2021/02/07 by Glück, Jochen
#47A10 #47B65 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2102.03772

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 ℓ¹ whose peripheral spectrum equals U. We also give a similar construction for the C₀-semigroup case.

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