The nonproper dissipative extensions of a dual pair

Abstract:

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) self-adjoint. 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 sufficient 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.