The Proper Dissipative Extensions of a Dual Pair

TL;DR

This paper develops a method to identify proper dissipative extensions of dual pairs of dissipative operators on Hilbert spaces, with applications to symmetric and differential operators, and studies their numerical range stability.

Contribution

It introduces a systematic approach for finding dissipative extensions of dual pairs, expanding understanding of their structure and stability in various operator classes.

Findings

Method for determining dissipative extensions of dual pairs.

Applications to symmetric and perturbed operators.

Analysis of numerical range stability of extensions.

Abstract

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.