T. Laffey showed (Linear and Multilinear Algebra 6 (1978), 269 305) that a semigroup of matrices is triangularizable if the ranks of all the commutators of elements of the semigroup are at most 1. Our main theorem is an extension of this result to semigroups of algebraic operators on a Banach space. We also obtain a related theorem for a pair [A, B] of arbitrary bounded operators satisfying rank (AB&BA)=1 and several related conditions. In addition, it is shown that a semigroup of algebraically unipotent operators of bounded degree is triangularizable. Academic Press

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