Abstract :
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 ST −T S is at most 1 for all {S,T} ⊂ 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. Omladiˇc, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups
of algebraic operators, J. Funct. Anal. 160 (1998) 452–465].
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