Ideal-triangularizability of upward directed sets of positive operators

Abstract

‎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.