IUMJ

Title: How small can nonzero commutators be?

Authors: Janez Bernik and Heydar Radjavi

Issue: Volume 54 (2005), Issue 2, 309-320

Abstract:

If $\mathcal{G}$ is a group of complex matrices, can $AB-BA$ be "small," if nonzero, for all $A$, $B \in \mathcal{G}$? We provide answers to this question in several instances. If the spectral radius of $AB-BA$ is bounded above, when suitably normalized, by $\sqrt{3}$, then $\mathcal{G}$ is solvable and $\mathcal{G}'$ is triangularizable. If $\mathcal{G}$ is connected or compact, it is itself triangularizable (and thus abelian in the compact case). For these and some other cases then, the bound on the spectral radii of commutators is zero if it is less than $\sqrt{3}$, and this number is sharp. Extensions to the case of semigroups are given and limitations discussed.