Exercise 12, § 1

If $E$ is a complex Hilbert space and $u$ an endomorphism compact normal of E, prove that $u$ possesses a cyclic vector (cf. def. 3 of IV, p. 191) if and only if the following conditions are satisfied:

(i) The kernel of $u$ is reduced to 0 ;

(ii) The spectral multiplicity of every nonzero eigenvalue of $u$ is equal to 1.