[Math] Trace of an operator

Question

Suppose $\rho$ is a positve trace-class operator and $x$ is positve bounded operator on a Hilbert space $\mathcal{H}$, I am unable to prove that trace of $\rho x$ is positive,
where trace($x$):= $\sum \limits _{i=0}^\infty \langle x\xi_i, \xi_i\rangle $, where {$\xi_i$} is an orthonormal basis. I don't know product of positive operators is again positive or not.

Answer 1

In general, $\rho x$ is not positive. But a simple, useful trick saves the day.

Let $x^{1/2}$ be the square root of $x$. Since $\operatorname{tr} (AB)=\operatorname{tr}(BA)$, it follows that $$ \operatorname{tr} (\rho x)=\operatorname{tr} (\rho x^{1/2}x^{1/2})=\operatorname{tr} (x^{1/2}\rho x^{1/2}) $$ Here $x^{1/2}\rho x^{1/2}$ is a positive operator, hence its trace is positive.