Let $\displaystyle \mathcal{D} = \{f \in L^{2}(0,\infty)$; for every c>0, f is absolutetly continuous on [0,c], f(0)=0, and $\displaystyle f' \in L^2(0,\infty)\}$. Define Af=if' for $\displaystyle f \in \mathcal{D}$. Show that A is a densely defined closed operator and find dom A*. Show that A is symmetric with deficiency indices n_+=0 and n_-=1.

Let $\displaystyle \mathcal{E} = \{f \in L^{2}(-\infty,0)$; for every c<0, f is absolutetly continuous on [c,0], f(0)=0, and $\displaystyle f' \in L^2(-\infty,0)\}$. Define Af=if' for $\displaystyle f \in \mathcal{E}$. Show that A is a densely defined closed operator and find dom A*. Show that A is symmetric with deficiency indices n_+=1 and n_-=0.

Use the above two exercies to prove the following:

If k,l are any nonnegative integers or $\displaystyle \infty$, show that there is a closed symmetric operator A with n_+=k and n_-=l.

Your help is much appreciated