© 2018 Amerian Mathematial Soiety. We consider dissipative operators A of the form A = S + iV, where both S and V ≥ 0 are assumed to be symmetric, but neither of them needs to be (essentially) self-adjoint. After a brief discussion of the relation of the operators S ± iV to dual pairs with the so-called common core property, we present a necessary and sufficient condition for any extension of A with domain contained in D((S − iV) ∗ ) to be dissipative. We will discuss several special situations in which this condition can be expressed in a particularly nice form—accessible to direct computations. Examples involving ordinary differential operators are given.