Spectrum of product of self adjoint operators

Question

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $F$.

Let $T,S\in\mathcal{B}(F)$, be two self-adjoint operators. Why $$\sigma (TS)\subseteq\mathbb{R}?$$

Answer 1

Edit: You need additionally that one of the operators is positive, see the comment by Martin Argerami below.

This follows from combining the next two facts: $$ \sigma( T S ) \cup \{0\} = \sigma( S T ) \cup \{0\}, $$ this is sometimes called "Jacobson's lemma", and it can be proved by using, e.g., https://math.stackexchange.com/a/1928728/58577

The second fact is $$ \sigma( U ) = \overline{\sigma( U^\star) }.$$