Let $H$ be a (separable complex) Hilbert space and let $A$ and $B$ be two densely-defined, maximally-defined linear operators on $H$ with domains $D(A)$ and $D(B)$ respectively. (By maximall-defined, I mean that $A$ and $B$ admit no extensions). Then, we can define the operator $A+B$ on $D(A)\cap D(B)$, however, in general, the operator $\left( D(A)\cap D(B),A+B\right)$ will not be maximally-defined. The question is: does this operator admit a unique maximal extension?
My conjecture is that the answer is no, but I would absolutely love for the answer to be yes.