Exercise 11, § 4

We identify the space $\mathscr{D}(]0,1[)$ with a subspace of $L^2([0,1])$. Let $u$ be the partial operator on $L^2([0,1])$ with domain $\mathscr{D}(]0,1[)$ such that $u(f) =i^{-1}f'$ for every function $f\in \mathscr{D}(]0,1[)$.

a) The partial operator $u$ is closable and $\overline{u}$ is symmetric.

b) The eigenspaces of $\overline{u}$ corresponding to $i$ and to $-i$ are not orthogonal.