if $AA^*$ = $A + A^*$ then $A$ is normal

Question

I have been trying to prove the following:

if $AA^*$ = $A + A^*$ then $A$ is normal (meaning : $AA^* = A^*A$).

I have literally tried everything possible; I can't come up with proof.

Any ideas?

Answer 1

Observe that $$(A-I)(A^*-I)=AA^*-A-A^*+I=I$$ This means that $A-I$ and $A^*-I$ are inverses of each other. It follows that $$I=(A^*-I)(A-I)=A^*A-A-A^*+I$$ The desired conclusion follows.

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A more general result can be shown to be true. Proceeding along the same lines, one can show that if $AB=A+B$, then $AB=BA$.