  We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$
and $V\geq 0$ are assumed to be symmetric but neither of them needs to be
(essentially) selfadjoint. After a brief discussion of the relation of the
operators $S\pm iV $to dual pairs with the so called common core property, we
present a necessary and suffcient condition for any extension of $A$ with
domain contained in $\mathcal{D}((S-iV)^*)$ 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.
