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arXiv · 1603.08192

The Proper Dissipative Extensions of a Dual Pair

Christoph Fischbacher·Sergey Naboko·Ian Wood
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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.

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Christoph Fischbacher, Sergey Naboko, Ian Wood. 2016-03-27. The Proper Dissipative Extensions of a Dual Pair. https://arxiv.org/abs/1603.08192

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