The spectrum of an R-homomorphism

Abstract

AbstractLet E be a real Banach space ordered by a closed, normal and generating cone. Suppose also that the order induced on E has the Riesz decomposition property. It is shown that if T:E → E is a positive linear operator with the property that y, z, a ∈ E with a ≧ Ty, Tz implies there is x ∈ E with x ≧ y, z and a ≧ Tx then the approximate point spectrum and spectrum of T are cyclic subsets of the complex plane. That is, if α = |α|γ lies in one of these sets then so does |α|γk for all integers k.