Any pair of deficiency indices can occur

Let ; for every c>0, f is absolutetly continuous on [0,c], f(0)=0, and . Define Af=if' for . 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 ; for every c<0, f is absolutetly continuous on [c,0], f(0)=0, and . Define Af=if' for . 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 , show that there is a closed symmetric operator A with n_+=k and n_-=l.

Your help is much appreciated