Proposition 16

Let $u$ be a closed operator with dense domain on a complex Hilbert space E. The spectrum of $u^*$ is the image of the spectrum of $u$ under complex conjugation and, for every element $\lambda$ of the resolvent set of $u$, one has $R(u, \lambda )^*= R(u^*, \lambda )$. In particular, if $u$ is self-adjoint, the endomorphism $R(u, \lambda )$ is normal for every $\lambda \notin$ Sp($u$).

Let $\lambda \in \mathbf{C}-$ Sp($u$) be an element of the resolvent set of $u$. One has $u=\lambda 1_E-R(u, \lambda )^{-1}$, hence

(IV, p. 236 and prop. 9 of IV, p. 239). Applying remark 5, we deduce that $\overline{\lambda}\in \mathbf{C}-$ Sp($u^*$) and that $R(u, \lambda )^*= R(u^*, \lambda )$. Consequently, the spectrum of $u^*$ is contained in the image of Sp($u$) under complex conjugation. We obtain equality by applying this property to $u^*$, since $u^{**}=u$ (cor. to prop. 8 of IV, p. 237). The last assertion then results from formula (9), p. 245.