The Proper Dissipative Extensions of a Dual Pair
  Let $A$ and $(-\widetilde{A})$ be dissipative operators on a Hilbert space $\mathcal{H}$ and let $(A,\widetilde{A})$ form a dual pair, i.e. $A\subset\widetilde{A}^*$, resp.\ $\widetilde{A}\subset A^*$. We present a method of determining the proper dissipative extensions $\widehat{A}$ of this dual pair, i.e. $A\subset \widehat{A}\subset\widetilde{A}^*$ provided that $\mathcal{D}(A)\cap\mathcal{D}(\widetilde{A})$ is dense in $\mathcal{H}$.   Applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed.   Finally, we investigate the stability of the numerical range of the different dissipative extensions. 
Christoph Fischbacher, Sergey Naboko and Ian Wood
math.FA