Corollary 2

Let E be a real Hilbert space. Let $u$ be a closed symmetric operator on E. The partial operator $u_{(\mathbf{C})}$ on $E_{(\mathbf{C})}$ is symmetric and closed, and admits at least one self-adjoint extension.

The partial operator $u_{(\mathbf{C})}$ is closed and symmetric by prop. 2 of IV, p. 228 and lemma 3 of IV, p. 239. The $\mathbf{R}$-linear isomorphism $j$ of $E_{(\mathbf{C})}$ onto $E_{(\mathbf{C})}$ such that $j(z\otimes x) =\overline{z}\otimes x$ for every $z\in \mathbf{C}$ and every $x\in E$ induces an isomorphism of Hilbert spaces of Ker($u^*_{(\mathbf{C})}-i$) onto Ker($u^*_{(\mathbf{C})}+i$), and one can apply corollary 1.