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There is probably an elementary proof, but that's an immediate consequence of the continuous functional calculus :

$\frac{A-A^*}{2i} \frac{A-A^*}{2i} = f (A) A)$

where $f$ is the imaginary part function on $\mathbb{C}$. And when you apply a continuous function $f$ to a normal operator $A$ you have : $Spec(f(A)) = f(Spec(A))$ (you can see that by restricting to the abelian sub-algebra sub-$C^*$-algebra generated by $A$).

1

There is probably an elementary proof, but that's an immediate consequence of the continuous functional calculus : $\frac{A-A^*}{2i} = f ( A)$ where $f$ is the imaginary part function on $\mathbb{C}$. And when you apply a continuous function $f$ to a normal operator $A$ you have : $Spec(f(A)) = f(Spec(A))$ (you can see that by restricting to the abelian sub-algebra generated by $A$).