Irreducible semigroups of positive operators on Banach lattices.

The classical Perron–Frobenius theory asserts that an irreducible matrixhas cyclic peripheral spectrum and its spectral radiusis 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 ifis a commutative irreducible semigroup of positive operators oncontaining a compact operatorthen there exist positive disjoint vectorsinsuch that every operator inacts as a positive scalar multiple of a permutation on. Compactness ofmay be replaced with the assumption thatis peripherally Riesz, i.e. the peripheral spectrum ofis separated from the rest of the spectrum and the corresponding spectral subspaceis finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator, we show thatis a cyclic permutation on,, and iffor someinandthenfor someand. We also extend results of Drnovšek and Levin about peripheral spectra of irreducible operators. [ABSTRACT FROM AUTHOR]