Lemma 4

Let E be a Hilbert space and let $u$ be a normal endomorphism of E. Let F be a closed subspace of E containing a total set of proper vectors of $u$. Then $F^{\circ}$ is stable under $u$ and the endomorphism $\widetilde{u}$ of $F^{\circ}$ deduced from $u$ is normal.

Since $u$ is normal, every proper vector of $u$ is also a proper vector of $u^*$ (EVT, V, p. 43, cor.). The assumption therefore implies that F is stable under $u$ and under $u^*$. By EVT, V, p. 41, prop. 4 (ii), we therefore have $u(F^{\circ})\subset F^{\circ}$ and $u^*(F^{\circ})\subset F^{\circ}$. It follows that the adjoint of $\widetilde{u}$ is the endomorphism of $F^{\circ}$ deduced from $u^*$. Since $u$ is normal, the endomorphism $\widetilde{u}$ is normal.