1) What is the domain of the adjoint $A^\ast$ of the differential operator $Af = i \frac{d}{dx}$ with $D(A) = \mathcal C^\infty_c (0,\infty)$?
2) I want to compute the deficiency indices of $A$. By definition we have $d_+(A) = \dim \ker (A^\ast - i I) = \dim \ker (T^\ast - \lambda I)$ for $Im \lambda > 0 $ and $d_-(A) = \dim \ker (A^\ast + i I) = \dim \ker (T^\ast - \lambda I)$ for $Im \lambda < 0$. Now, the exponentials $e^{i \lambda x}$ are the possible candidates for eigenfunctions of $A^\ast$ and they only are square-integrable if $Im \lambda > 0$, so i would say $d_+(A) = 1$ while $D_-(A) = 0$ and hence the operator has no self-adjoint extension. Is this correct?