Suppose that $G$ is an $n\times n$ Gaussian random matrix of i.i.d. entries $N(0,1/n)$ and $D$ is an $n\times n$ deterministic diagonal elements. I'd like to know if there have been results on the singular values of the matrix product $GD$.

There is classical result on the operator norm of $GD$, which says that $$ \|GD\|_{op} \approx \|D\|_{op} \pm \frac{\|D\|_F}{\sqrt{n}}, $$ where $\|D\|_F$ is the Frobenius norm of $D$.

I'd like to know if there are similar results on other singular values, or on the trace norm $\|GD\|_\ast$, preferably a lower bound.