TITLE: An inequality for positive operators
QUESTION [7 upvotes]: Let $S$ and $T$ be positive operators on a Hilbert space $\mathcal{H}$. Suppose that $S \le T$. Since the square root function is operator monotone, it follows that $S^{1/2} \le T^{1/2}$. Does the inequality
$$S^{1/2}RS^{1/2} \le T^{1/2}RT^{1/2}$$
hold for all positive operators $R$?

REPLY [2 votes]: Here's an easy example on dimension 2. Let
$$
S=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix},\ T=\begin{bmatrix} 1&0\\0&1\end{bmatrix}, \ R=\begin{bmatrix} 1&0\\0&0\end{bmatrix}. 
$$
Then $S^{1/2}=S$, $T^{1/2}=T$, and $S^{1/2}\leq T^{1/2}$. But 
$$
S^{1/2}RS^{1/2}=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix},\ \ T^{1/2}RT^{1/2}=\begin{bmatrix} 1&0\\0&0\end{bmatrix},
$$
which are not comparable.