Commutators of small rank and reducibility of operator semigroups

TL;DR

This paper investigates the structure of operator semigroups with commutators of small rank, concluding that such conditions imply the reducibility of the semigroup and a specific block-diagonal form for unitary groups.

Contribution

It establishes that semigroups with commutators of rank at most two are reducible and characterizes the structure of unitary groups under this condition.

Findings

Semigroups with rank ≤ 2 commutators are reducible.

Unitary groups under the hypothesis are contained in a block-diagonal form with specific properties.

The structure of such groups includes a small 3x3 component and an abelian component.

Abstract

It is easy to see that if $\cG$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\cG$ 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 $\cS$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $\cS$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $\cG_1\oplus\cG_2$ where $\cG_1$ is at most $3\times 3$ and $\cG_2$ is abelian.