When is the restriction of a normal operator not normal?

Question

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)?

Answer 1

For arbitrary operators, it could be that $W$ is a $T$-invariant subspace, but not a $T^*$ invariant subspace. For example, take $$ T = \pmatrix{1&1\\0&2} $$ Note that $W = \operatorname{span}[(1,0)]$ is $T$-invariant, but not $T^*$ invariant.

On the other hand, if $W$ is both $T$-invariant and $T^*$-invariant, we will necessarily have $(T|_W)^* = (T^*)|_W$. It's easy to see that this should be the case by noting that we can define the adjoint by the property $$ \langle T|_W x, y \rangle = \langle x,(T|_W)^*y \rangle \quad \forall x,y \in W $$

Answer 2

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.