If $T,S \in L(V)$ are positive operators, how can I show that $TS$ is self-adjoint?

Question

If we let $V$ be a finite dim. real/ complex inner product space, and $T \in L(V)$ and $S \in L(V)$ we let be positive operators, how can I prove that $TS$ is self-adjoint?

I tried to decompose $TS = T^{1/2}ST^{1/2} = T^{1/2}S^{1/2}T^{1/2}S^{1/2}$

but have not been able to prove this. Thank you!

Answer 1

It is not always true.

For example $$ A=\left(\begin{matrix} 3 & 1\\ 1 &3\end{matrix}\right)\quad\text{and}\quad B=\left(\begin{matrix} 1 & 0\\ 0 &3\end{matrix}\right). $$ Then $$ AB=\left(\begin{matrix} 3 & 3\\ 1 &9\end{matrix}\right), $$ which is non-symmetric. Positive operators NEED to be symmetric.