Invariant subspaces for semigroups of algebraic operators

T. Laffey showed (Linear and Multilinear Algebra 6 (1978), 269 305) that a semigroup of matrices is triangularizable if the ranks of all the commutato rs of elements of the semigroup are at most 1. Our main theorem is an exten sion 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 op erators satisfying rank (AB - BA) = 1 and several related conditions. In ad dition, it is shown that a semigroup of algebraically unipotent operators o f bounded degree is triangularizable. (C) 1998 Academic Press.