I need to calculate the numerical range of the operator $T:D(T)\subseteq L^2(0,1) \to L^2(0,1)$ defined by $$D(T):=\{ u \in H^1(0,1): u(1)=0 \}, \ Tu:=u', \ u \in D(T),$$ where $H^1(0,1)$ is the Sobolev space of functions in $L^2(0,1)$ whose first weak derivative belongs to $L^2(0,1)$.
My attempt
For each $u \in D(T)$ with $\|u \|=1$ we have that $$\langle Tu,u \rangle=\int_0^1u' \overline{u}dx=-|u(0)|^2 - \int_0^1u\overline{u'}dx.$$ If we add $\langle Tu,u \rangle$ on both sides of this equality, we get that $$2\langle Tu,u \rangle=-|u(0)|^2 + 2 i \int_0^1 \mbox{Im}(u' \overline{u})dx.$$ I think that the last one equality implies that the numerical range $W(T)$ of $T$ is given by $$W(T):=\{\langle Tu,u \rangle: u \in D(T), \|u \|=1 \}= \{z \in \mathbb{C} : \mbox{Re}(z) \leq 0 \}.$$
Is my argument right? Do you know another way to calculate $W(T)$?
Thank you.