Example 2

For $x$ to be hermitian, it is necessary and sufficient that Sp$_A(x)$ be contained in $\mathbf{R}$. In fact, let $f$ be the continuous mapping on Sp$_A(x)$ given by $f(z) =z-\overline{z}$. Then $x$ is hermitian if and only if $f(x) = 0$, that is, if $f$ is zero, that is, if Sp$_A(x)$ is contained in $\mathbf{R}$.