I want to prove: given a bounded normal operator $T$ on a Hilbert space $H$. If $H_1$ is a $T$-invariant subspace, then the orthogonal complement $W_1$ of $H_1$ is also $T$-invariant.
I can prove the case when $H$ is finite-dimensional. If $T$ can be written as a matrix $$\begin{pmatrix} M & A\\ 0 & B \end{pmatrix},$$ then by $T^*T=TT^*$,we have $MM^*+AA^*=MM^*$. Take the trace, then we have A must be 0. However, I'm not sure how to translate the language when $H$ is infinite-dimensional where trace might not be well-defined.