TITLE: When is the restriction of a normal operator not normal?
QUESTION [6 upvotes]: I was proving the spectral theorem for normal operators on finite-dimensional complex vector spaces today during a test, when I arrived at the point in which

If $T\in\operatorname{End}(V)$ is normal, then if $W$ is a subspace of $V$ is $T$-invariant "surely" the restriction $T|_W\colon W\to W$ must be normal too.

But the professor said he could find a counterexample.
Now, the claim above was true in that case, but I'm still curious about that example.
Practically, he said that it can happen that the restriction of the adjoint $T^*$ may not coincide with the adjoint of $T$ in $W$, in other words
$$
(T|_W)^*\ne (T^*)|_W.
$$
When is it true (in finite-dimensional vector spaces)?

REPLY [3 votes]: It is always true for finite dimensional vector spaces. If $T \colon V \rightarrow V$ is a normal operator and $W$ is $T$-invariant, then $W$ is also $T^*$-invariant and hence $T|_W$ is normal (and $(T|_W)^* = T^*|_W$). You can find a proof in Axler's book "Linear Algebra Done Right", Chapter 7.