Exercise 6, § 1

Let E be a complex Hilbert space.

(a) Let $u$ be a compact normal endomorphism of E. Every closed subspace of E invariant under $u$ is invariant under $u^*$.

(b) If E is of infinite dimension, there exists a normal endomorphism $u$ of E and a closed subspace $F\subset E$ such that:

(i) One has $u(F)\subset F$;

(ii) The space F is not invariant under $u^*$;

(iii) The space $F^{\circ}$ is not invariant under $u$;

(iv) The endomorphism of F induced by $u$ by passing to subspaces is not normal.