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 $\Vert B\Vert \Vert C{\Vert }_2\le (\log m+O(1){)}^{1/2}\Vert A{\Vert }_2$ where $\Vert D\Vert$ is the norm of $D$ as an operator on ${\ell }_2^m$ and $\Vert D{\Vert }_2$ 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 $A=[B,C]$, $\Vert B\Vert \Vert C{\Vert }_2\ge |\log m-O(1){|}^{1/2}\Vert A{\Vert }_2$ for some absolute constant $c>0$.

References in corpus (1)

• A quantitative version of the commutator theorem for zero trace matrices