Good evening! Given the operator $A$ acting on $L^2(0,1)$ with $Au = u'$, $\mathcal{D}(A) = \{u \in H^1(0,1):u(1)=0\}$, I am trying to show that $A$'s spectrum is empty, which is stated as easy to show in the text. However, I am not entirely sure how to prove this, and I'm not entirely sure it is true due to this other (slightly confusing) Stack question: Spectrum of the derivative operator: What's wrong in my argument?. Any hints would be appreciated.
Thanks!