Exercise 9, § 5

Let $u$ be a closed partial operator with dense domain on a Hilbert space E. The following assertions are equivalent:

a) The operator $u$ is normal ;

b) There exist self-adjoint operators $u_1$ and $u_2$ on E, which commute in the sense of the resolvent, and such that $u=u_1+iu_2$;

c) We have dom($u$) $=$ dom($u^*$) and $\|u(x)\|=\|u^*(x)\|$ for all $x\in$ dom($u$);

d) We have $uu^*=u^*u$.

If the resolvent set of $u$ is nonempty, these assertions are equivalent to

e) For every $\lambda \notin$ Sp($u$), the resolvent $R(u, \lambda )$ is normal.