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

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

2009-03-31

arxiv.org/abs/0903.5412v1

Mathematics

Functional Analysis

10 pages; to appear in Journal of Functional Analysis

Scientific paper

10.1016/j.jfa.2009.03.010

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].

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