Proposition 11

Let $u$ be a symmetric partial operator on E, and let $\lambda \in \mathbf{C}$. If $u+\lambda 1_E$ and $u+\lambda 1_E$ are surjective, then $u$ is self-adjoint.

It is enough to prove that dom($u^*$)$\subset$ dom($u$). Let $x\in$ dom($u^*$). By hypothesis there exists $y\in$ dom($u$) such that $u(y) +\lambda y=u^*(x) +\lambda x$. Let us prove that $y=x$. For every $z\in$ dom($u$), we have

since $u$ is symmetric. Since the operator $u+\lambda 1_E$ is surjective, we indeed have $y=x$, hence $x\in$ dom($u$).

We shall see later (cf. prop. 17 of IV, p. 248) that if $\lambda \in \mathbf{C}-\mathbf{R}$, then the converse is true.