Let $A$ be a unital C\*-algebra. Let $x,y\in A$ be self adjoint elements in $A$, with $x$ being invertible. Can we say that the spectrum of $x^{-1}y$ is a subset of the real line? I know this is true if $x$ was also positive.



Regarding an element being self adjoint
The golden rule for conjectures in operator theory:


**Every ad-hoc conjecture is most likely false for $2 \times 2$-matrices.** :-)


So here's a $2 \times 2$-counterexample for the question:
Let $A = \mathbb{C}^{2 \times 2}$ and
$$
x = 
\begin{pmatrix}
 0 & 1 \\ 
 1 & 0
\end{pmatrix}
\quad \text{and} \quad 
y = 
\begin{pmatrix}
 4 & 3 \\ 
 3 & -2
\end{pmatrix}
$$.


