Define the operator $A:H^1(0,1)\to L^2(0,1)$ as $Au=u'$. I want to show that if we let the domain of $A$ to be $D(A)=\{u\in H^1(0,1): u(1)=0\}$, then $A$ generates a $C^0$-semigroup on $L^2(0,1)$.
I want to use the Hille-Yosida theorem. First, $D(A)$ is dense in $L^2(0,1)$. Then for a positive number $\lambda$ and $u\in L^2(0,1)$, I got $$ ((\lambda I-A)^{-1}u) (t)= e^{\lambda t}\int_t^1e^{-\lambda x}u(x)dx,$$ and by Minkowski's and Holder's inequality, $$ \|(\lambda I-A)^{-1}u\|_{L^2(0,1)} \leq \frac{\sqrt{2\lambda+e^{-2\lambda}-1}}{2\lambda}\|u\|_{L^2(0,1)}. $$ This implies $\|(\lambda I-A)^{-1}\|\leq \frac{M}{\sqrt{\lambda}}$ for some constant $M$ and Hille-Yosida theorem cannot be used. Can anyone give me a hint?