Determine the range of ${\rm i}\frac{\rm d}{{\rm d}t}$ on $\left\{f\in L^2([0,1]):f\text{ is absolutely continuous and }f(0)=f(1)=0\right\}$

Question

Let $\operatorname{AC}(0,1)$ denote the space of absolutely continuous functions $[0,1]\to\mathbb C$, $$Af:={\rm i}f'\;\;\;\text{for }f\in\mathcal D(A):=\left\{f\in L^2([0,1];\mathbb C):f\in\operatorname{AC}(0,1)\text{ and }f(0)=f(1)=0\right\}.$$

How can we determine the range $\mathcal R(A)$ of $A$?

I've read that $$\mathcal R(A)=\left\{g\in L^2([0,1];\mathbb C):\int_0^1g=0\right\},$$ but I absolutely don't understand why this equality holds. (Maybe we need to redefine $\mathcal D(A)$ such that $f'\in L^2((0,1))$ for all $f\in\mathcal D(A)$.) It the description of the range is totally wrong, how can we determine it instead?

Answer 1

The description of the range is not correct. If $f(x)=(1-x)\sqrt x$ then it is easy to see that $if' \notin L^{2}$ even though $f$ is AC and vanishes at $0$ and $1$.