Irreducible semigroups of positive operators on Banach lattices

Abstract

The classical Perron–Frobenius theory asserts that an irreducible matrix has cyclic peripheral spectrum and its spectral radius is an eigenvalue corresponding to a positive eigenvector. This was extended by Radjavi and Rosenthal to semigroups of matrices and of compact operators on -spaces. We extend this approach to operators on an arbitrary Banach lattice . We prove, in particular, that if is a commutative irreducible semigroup of positive operators on containing a compact operator then there exist positive disjoint vectors in such that every operator in acts as a positive scalar multiple of a permutation on . Compactness of may be replaced with the assumption that is peripherally Riesz, i.e. the peripheral spectrum of is separated from the rest of the spectrum and the corresponding spectral subspace is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator , we show that is a cyclic permutation on , , and if for some in and then for some and . We also extend results of Drnovšek and Levin about peripheral spectra of irreducible operators.