square root of positive operators

Question

It $T, S$ are positive operators, do we have that $\sqrt{TS}=\sqrt{T}\sqrt{S}$? Are there any basic rules that hold for square roots of positive numbers that don't hold for positive operators?

Answer 1

First of all, in general $TS$ is not even selfadjoint, so it is not obvious what $\sqrt{TS}$ means. More than that, I don't think there is a canonical way to give meaning to it.

In the same vein, $T^{1/2}S^{1/2}$ will usually not be positive.

Also, if $(TS)^{1/2}=T^{1/2}S^{1/2}$, this means that $TS=[(TS)^{1/2}]^2=T^{1/2}S^{1/2}T^{1/2}S^{1/2}$. In the case where $T$, $S$ are invertible, this only occurs if $TS=ST$.

To see an easy example of the problems at hand, let $$ T=\begin{bmatrix}1&0\\0&2\end{bmatrix},\ \ S=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}. $$ Then $T^{1/2}=\begin{bmatrix}1&0\\0&\sqrt2\end{bmatrix}$, and $S^{1/2}=S$. Then $$ T^{1/2}S^{1/2}=\begin{bmatrix}1/2&1/2\\ \sqrt2/2&\sqrt2/2\end{bmatrix}. $$ What would be the square root of this? In any case, $$ (T^{1/2}S^{1/2})^2=\begin{bmatrix}(1+\sqrt2)/4&(1+\sqrt2)/4\\ (2+\sqrt2)/4&(2+\sqrt2)/4\end{bmatrix}, $$ which is very different from $$ TS=\begin{bmatrix}1/2&1/2\\ 1&1\end{bmatrix} $$