I am assuming $T$ is a compact operator and $\{T_j\}$ is a sequence of compact operators such that $\|T-T_j\| < \epsilon_j$ where $\epsilon_j$ is a quantity that goes to zero as $j \to \infty$.
It is easy to show that this implies $\|T^*T-T^*_jT_j\| \leq C\epsilon_j$ for some constant $C$. I am wondering if it is also true that $\|\sqrt{T^*T}-\sqrt{T^*_jT_j}\| \leq D\epsilon_j$ for some constant $D$? I have seen several proofs that $\sqrt{T^*_jT_j}$ converges to $T^*T$ in the operator norm, but I am wondering about the rate of this convergence.
I am pretty sure that just assuming $\|T^*T-T^*_jT_j\| \leq C\epsilon_j$ does not imply that $\|\sqrt{T^*T}-\sqrt{T^*_jT_j}\| \leq D\epsilon_j$. But I was wondering if $\|T-T_j\| < \epsilon_j$ did imply the result.
Any help or direction to a reference would be greatly appreciated. If the result is false, a counterexample would also be great.