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)?