Let V be an inner product space with finite dimension. Suppose that T and U are two positive linear operators. I need an example to show that TU is not necessarily a positive operator.
A linear operator T on an inner product space with finite dimension is positive if T is Hermitian and for every $\displaystyle \alpha \neq 0, (T\alpha|\alpha) >0$ (i.e. inner product of $\displaystyle \alpha$ and $\displaystyle T\alpha$ is positive).