Colloquium: Tuesday, March 15, 2 pm. Speaker: Gideon Schechtman (Weizmann) . Title: “A quantitative version of the commutator theorem for zero trace matrices”.
As is well known, a complex $m \times m$ matrix $A$ is a
commutator (i.e., there are matrices $B$ and $C$ of the same dimensions as
$A$ such that $A=[B,C]=BC-CB$) if and only if $A$ has zero trace. If
$\|\cdot\|$ is the operator norm from $\ell_2^m$ to itself
and $|\cdot|$ any ideal norm on $m\times m$ matrices then clearly for any
$A,B,C$ as above $|A|\le 2\|B\||C|$.
Does the converse hold? That is, if $A$ has zero trace are there $m\times
m$ matrices $B$ and $C$ such that $A=[B,C]$ and $\|B\||C|\le K|A|$ for some
absolute constant $K$? If not, what is the behavior of the best $K$ as a
function of $m$?
The talk will concentrate on two recent results on this problem. The first
is a couple of years old result of Johnson, Ozawa and myself which gives
some partial answers to this problem for the most interesting case of
$|\cdot|=\|\cdot\|$. The second is a more recent result of Angel and myself
which solves the problem for $|\cdot|=$ the Hilbert–Schmidt norm.