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. $Asubsetwidetilde{A}^*$, resp. $widetilde{A}subset A^*$. We present a method of determining the proper dissipative extensions $widehat{A}$ of this dual pair, i.e. $Asubset widehat{A}subsetwidetilde{A}^*$ provided that $mathcal{D}(A)capmathcal{D}(widetilde{A})$ 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.