I have been attempting the exercise:
Let $A_k: \mathcal{D}\left(A_k\right) \subset L^2([0,1]) \rightarrow L^2([0,1])$ be $A_k u=i u^{\prime}$ with domains $$ \begin{aligned} & \mathcal{D}\left(A_1\right)=H^1((0,1)), \\ & \mathcal{D}\left(A_2\right)=\left\{u \in H^1((0,1)) \mid u(0)=u(1)\right\}, \\ & \mathcal{D}\left(A_3\right)=\left\{u \in H^1((0,1)) \mid u(0)=0\right\} . \end{aligned} $$ Show that (a) the spectrum of $A_1$ is $\mathbb{C}$, (b) the spectrum of $A_2$ is the set $\{2 n \pi \mid n \in \mathbb{Z}\}$, and (c) the spectrum of $A_3$ is empty.
For (a), we have that the solution of $iu^\prime =\lambda u$ is $u = \exp(i\lambda x)$, which is in $H^1(0,1)$, so $A_1-\lambda I$ has a solution. Similarly for $A_2$ in (b), using the identical differential equation I have shown that the point spectrum includes the set $\{2 n \pi \}_{n \in \mathbb{Z}}$. However, I am uncertain how I can show that no other point lies in the point spectrum, nor the continuous or residual spectrum. I am also lost on (c). Is there an easy way of proving (b) and (c)?
Edit: I discovered this link: Spectrum of the derivative operator has a similar question that is answered. However, the answer uses that the range of a relevant operator is strictly dense, which I do not know how to prove, or (partially) constructs a function with compact support that is very difficult to come up with.