The Hilbert Schmidt version of the commutator theorem for zero trace matrices

Abstract

Let $A$ be a $m\times m$ complex matrix with zero trace. Then there are $m\times m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\|\|C\|_2\le (\log m+O(1))^{1/2}\|A\|_2$ where $\|D\|$ is the norm of $D$ as an operator on $\ell_2^m$ and $\|D\|_2$ is the Hilbert--Schmidt norm of $D$. Moreover, the matrix $B$ can be taken to be normal. Conversely there is a zero trace $m\times m$ matrix $A$ such that whenever $A=[B,C]$, $\|B\|\|C\|_2\ge |\log m-O(1)|^{1/2}\|A\|_2$ for some absolute constant $c>0$.