  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.
Drnov\v{s}ek, D. Kokol-Bukov\v{s}ek, T. Laffey, M. Omladi\v{c}, H. Radjavi, P.
Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct.
Anal. 160 (1998), 452-465].
