The Proper Dissipative Extensions of a Dual Pair

Let $A$ and $(-\tilde{A})$ be dissipative operators on a Hilbert space $\mathcal{H}$ and let $(A,\tilde{A})$ form a dual pair, i.e. $A\subset {\tilde{A}}^{\ast }$, resp.\ $\tilde{A}\subset A^{\ast }$. We present a method of determining the proper dissipative extensions $\widehat{A}$ of this dual pair, i.e. $A\subset \widehat{A}\subset {\tilde{A}}^{\ast }$ provided that $\mathcal{D}(A)\cap \mathcal{D}(\tilde{A})$ 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.