Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, we have no reason to believe that this operator will be closed, which begs the question, is it closable?
I hope I'm not being an idiot again. . . Any ideas?