I was wondering if anyone knows about an inequality for the Hilbert-Schmidt (H-S) norm of the type
$|Tr (Bg)|\leq Const\cdot||B||\cdot function(||g||_{2})$ for a bounded operator $B$ and a H-S operator $g$ on a infinite dimensional space.
The H-S norm is defined by $||\cdot||_{2}^{2}:=Tr[B^{*}B]$ and $||\cdot||$ is the operator norm.
There is already such an inequality for trace-class operators: $|Tr (Bg)|\leq||B||\cdot ||g||_{1}$
Such an inequality cannot exist. Take $H=\ell^2(\mathbb N)$, $B=I$, and $g$ the diagonal operator with diagonal $\{1/n\}_n$. Then $\text{Tr}(Bg)=\infty$. So $function(\|g\|_2)=function(\pi/\sqrt6)=\infty$. As we can do the same with scalar multiplies of $g$, your function has to be infinity at every point.