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The Proper Dissipative Extensions of a Dual Pair

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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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Authors and Affiliations

  1. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NF, UK

    Christoph Fischbacher & Ian Wood

  2. Department of Math. Physics, Institute of Physics, St. Petersburg State University, 1 Ulianovskaia, St. Petergof, St. Petersburg, 198504, Russia

    Sergey Naboko

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  1. Christoph Fischbacher
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  2. Sergey Naboko
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  3. Ian Wood
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Correspondence to Christoph Fischbacher.

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Fischbacher, C., Naboko, S. & Wood, I. The Proper Dissipative Extensions of a Dual Pair. Integr. Equ. Oper. Theory 85, 573–599 (2016). https://doi.org/10.1007/s00020-016-2310-5

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