Suppose that $A \in \mathbb{R}^{m,n}$ and $B \in \mathbb{R}^{n,m}$ and $A$ has a bounded 2-norm. We know that $AB$ is a positive semi-definite matrix with $\|AB\|_2 \leq 1$. Further, assume that $C \in \mathbb{R}^{n,n}$ and $\|C\|_2 \leq 1$. Can we compute a bound on the following matrix?
$$\|ACB\|_2 \leq ?$$
I couldn't apply basic properties for matrix norm e.g. $\|ACB\| \leq \|A\|\|C\|\|B\|$.
I can see that we have $\|ACB\|_2 \leq 1$ when $n=1$ because in that case $C$ is scalar and we have $ACB = CAB \Rightarrow \|ACB\|_2 = |C|\|AB\|_2 \leq 1.$
I can also use a lower bound assuming that $A$ is non-singular. For example, writing $\|AB\|_2 \geq \|B\|_2 \sigma_\min (A)$ yields
$$\|B\|_2 \sigma_\min(A) \leq 1 \Rightarrow \|B\|_2 \leq \frac{1}{\sigma_\min(A)},$$
and we get
$$\|ACB\|_2 \leq \|A\|_2\|C\|_2\|B\|_2 \leq \frac{\sigma_\max(A)}{\sigma_\min(A)} = \kappa(A).$$
Is there a sharper lower bound that I can use?