The Non-Proper Dissipative Extensions of a Dual Pair

Abstract

We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\ge 0$ 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 $\mathcal{D}((S-iV{)}^{\ast })$ 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.

20 pages, Extension of Lemma 5.7 and Corollary 5.9, additional example in Remark 5.8 and more references added