Exercise 24, § 4

Let E be a complex Hilbert space and $u$ a symmetric partial operator on E. Let $F = E\oplus E$ and let $v$ be the partial operator on F with domain dom($u^*$)$\oplus$ dom($\overline{u}$) such that $v(x, y) = (\overline{u}(y), u^*(x))$.

a) The partial operator $v$ is self-adjoint.

b) Let $\widetilde{E}$ be the image of the linear mapping $\delta :x\mapsto (x, x)$. The reduction of $v$ to $\widetilde{E}$ is the partial operator with domain $\delta$(dom($u$)) such that $(x, x)\mapsto (u(x), u(x))$.