Q. Let V be a $n-$dimensional space over $\mathbb{C}$ and $T$ a normal operator. Prove that all T-invariant space is $T^{*}$-invariant.
A. If $T$ is normal over $\mathbb{C}$ then $V$ admit a basis of eigenvectors of $T$ such that $V$ is a direct sum of $E_{i}$, $i=1,...,k$ where $E_{i}$ is the eigenspace of $\lambda_{i}$. $E_{i}$ is $T-$invariant so if i prove that $E_{i}$ is $T^{*}-$invariant the result follow.
is that correct?