Commutators of small rank and reducibility of operator semigroups

Abstract:

It is easy to see that if $\mathcal {G}$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\mathcal {G}$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $\mathcal {S}$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $\mathcal {S}$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $\mathcal {G}_1\oplus \mathcal {G}_2$, where $\mathcal {G}_1$ is at most $3\times 3$ and $\mathcal {G}_2$ is abelian.