Closure of an operator I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ 
$$A_0= f^{(4)}-f^{(6)}$$
$$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(1)=f^{(5)}(1)=0\big\}$$

The closure of the domain in $L^2$ is simply $L^2$: Obviously it holds $C_0^\infty(0,1)\subset D(A_0)$. The set of smooth function is dense in $L^2(0,1)$, hence its closure is $L^2(0,1)$. This implies that the closure of $D(A_0)$ is $L^2(0,1)$ as well.
