Abstract
R. Drnov�ek, D. Kokol-Bukov�ek, L. Livshits, G. MacDonald, M. Omladic, and H. Radjavi constructed an irreducible set of positive nilpotent operators on which is closed under multiplication, addition and multiplication by positive real scalars with the property that any finite subset is ideal-triangularizable. In this paper we prove the following: every algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable whenever the nilpotency index of its operators is bounded; every finite subset of an algebra of nilpotent operators which is generated by a set of positive operators on a Banach lattice is ideal-triangularizable.
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