Suppose $A$ in $\mathbb{R}^{p \times p}$ is a strictly positive defined symmetric matrix. Now suppose there exists a weakly consistent estimator for the matrix $A$, say $B_n$, which satisfies $\| B_n - A\|_{\infty} \rightarrow_p 0$.
Does any know the properties of $$A^{1 /2} B_n^{1 / 2 } - B_n^{1 / 2} A^{1 / 2}.$$ Thanks so much!
PS: the property I would like to obtain is something like $$\frac{\|A^{1 /2} B_n^{1 / 2 } - B_n^{1 / 2} A^{1 / 2}\|}{\|A - B_n\|} \, \rightarrow_p 0$$ under some proper norm $\|\cdot\|$ as $n$ goes to infinity. Does this possible?
Let $C:=A^{1/2}$ and $D:=B^{1/2}$. Then it is not necessarily the case that $$\frac{\|CD-DC\|}{\|C^2-D^2\|}\to0$$ as $D\to C$.
Here is a counterexample. Let $C:=\begin{pmatrix}2&0\\0&1\end{pmatrix}$, $D:=\begin{pmatrix}2&\epsilon\\\epsilon&1\end{pmatrix}$, with $\epsilon\to0$, and $A=C^2$, $B=D^2$. Then $$\|CD-DC\|_F=\left\|\begin{pmatrix}0&\epsilon\\-\epsilon&0\end{pmatrix}\right\|_F=\sqrt2|\epsilon|$$ $$\|A-B\|_F=\|C^2-D^2\|_F=\left\|\begin{pmatrix}\epsilon^2 & 3 \epsilon \\ 3 \epsilon &\epsilon^2\end{pmatrix}\right\|_F=\sqrt2|\epsilon|\sqrt{9+\epsilon^2}$$ Hence the ratio $\frac{\|CD-DC\|_F}{\|A-B\|_F}\to\frac{1}{3}$ as $\epsilon\to0$.
N.B. All norms on matrices are equivalent, so it does not matter what specific norm one picks.