Consider, on $\Omega=(0,1),$ the unbounded operator $A=\Delta, \,$ (the Laplacean operator) with domain $D(A)=\{y\in W^{2,p}(0,1)/ y'(0)=0 \;\&\; y'(1)=0\}$.
I am wondering if the operator $A$ is the infinitesimal generator of a compact semigroup on $L^p(0,1)$ for $p>1$?