Invariant subspaces for operator semigroups with commutators of rank at most one

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997), 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladič, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998), 452-465].

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