Exercise 16, § 5

a) Let $I =]0,1[$. The space $H^1(I)$ is contained in $\mathscr{C}(I)$.

b) The differential operator $u:f\mapsto -if'$ on $L^2(I)$ with domain $\mathscr{D}(I)$ is symmetric and closable. For every $f\in$ dom($u^*$), the derivative distribution $f'$ of $f$ belongs to $L^2(I)$ and one has $u^*(f) =if'$.

c) The deficiency spaces of $\overline{u}$ are of dimension 1; deduce an explicit description of all the self-adjoint extensions of $\overline{u}$.