Ideal-triangularizability of Upward Directed Sets of Positive Operators

In this paper we consider the question when an upward directed set of positive ideal-triangularizable operators on a Banach lattice is (simultaneously) ideal-triangularizable. We prove that a majorized upward directed set of ideal-triangularizable positive operators, which are compact or abstract integral operators is ideal-triangularizable. We also prove that a finite subset of an additive semigroup of positive power compact quasinilpotent operators is ideal-triangularizable. Moreover, we prove that an additive semigroup of positive power compact quasinilpotent operators of bounded compactness index is ideal-triangularizable.