A quantitative version of the commutator theorem for zero trace matrices

Abstract

Let $A$ be a $m\times m$ complex matrix with zero trace and let \e>0. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and \|B\|\|C\|\le K_\e m^\e\|A\| where K_\e depends only on \e. Moreover, the matrix $B$ can be taken to be normal