Prove/Disprove: There doesn't exist $A\in M_n(\mathbb{C})$ s.t. $AA^*+I=\frac{3}{2}A$

Question

Prove/Disprove: There doesn't exist $A\in M_n(\mathbb{C})$ such that $AA^*+I=\frac{3}{2}A$

I think it is true, but struggling with the proof.

I tried to assume by contradiction there exists such $A$.

I found that such $A$ is invertible, and that $AA^*+I$ is self-adjoint, but couldn't work it out.

Any help is appreciated.

Answer 1

Hint : If $AA^*+I$ is self-adjoint, then so is $A$; this implies that A is diagonalizable with real eigenvalues, and also that $A^2+I=\frac{3}{2}A$.