A Note on Spectral Sublinearity for Collections of Positive Compact Operators on Banach Lattices

Abstract

Abstract In this paper we consider the question when a triangularizable semigroup S of positive compact ideal-triangularizable operators on an order continuous Banach lattice X is ideal-triangularizable. We prove that triangularizability always implies ideal-triangularizability iff X contains at most one atom. Under this condition we connect ideal-triangularizability of S with spectral properties of S. Surprisingly, S is idealtriangularizable iff the spectral radius is subadditive on S iff the spectral radius is submultiplicative on S. We also consider a pair of positive compact operators A and T with the property that A has a T-stable spectrum.