How does taking real part of a matrix affect the Frobenius norm of its powers?

Question

Let $A \in C^{n \times n}$, and $\|A\|_2 \leq 1$. Does it hold that

$$\Vert \Re(A)^k \Vert_F \leq C_n\Vert \Re(A^k) \Vert_F ?$$

Answer 1

This is already wrong for $n=1$. Take $x$ real, $A=(x^2+ix)$. Then $\Re(A^2)=(x^4-x^2)$ and $(\Re A)^2=(x^4)$. For $|x|$ small, you immediately see that your proposed inequality cannot hold.