Skip to main content

On the Theory of Dissipative Extensions

Fischbacher, Christoph Stefan (2017) On the Theory of Dissipative Extensions. Doctor of Philosophy (PhD) thesis, University of Kent,. (KAR id:61093)

Language: English
Download (1MB) Preview
[thumbnail of 132Thesis Corrected C. Fischbacher.pdf]
This file may not be suitable for users of assistive technology.
Request an accessible format


We consider the problem of constructing dissipative extensions of given dissipative

Firstly, we discuss the dissipative extensions of symmetric operators and give a suffcient condition for when these extensions are completely non-selfadjoint. Moreover,

Then, we consider operators A and \widetilde{A} that form a dual pair, which means that A\subset \widetilde{A}^*, respectively \widetilde{A}\subset A^* Assuming that A and (-\widetilde{A}) are dissipative, we present a method of determining the proper dissipative extensions \widehat{A} of this dual pair, i.e. we determine all dissipative operators \widehat{A} such that A\subset \subset\widehat{A}\subset\widetilde{A}^* provided that D(A)\cap D(\widetilde{A}) is dense in H. We discuss applications to symmetric operators, symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators. Also, we investigate the stability of the numerical ranges of the various proper dissipative extensions of the dual pair (A,\widetilde{A}).

dissipative. If there exists a dissipative operator (-\widetilde{A}) such that A and \widetilde{A} form a dual

the case, we propose a construction of a dual pair (A_0,\widetilde{A}_0), where A_0\subset A and \widetilde{A}_0\subset\widetilde{A} such that A_K is a proper extension of (A_0,\widetilde{A}_0).

selfadjoint and sectorial extensions of positive symmetric operators as described by Birman, Krein, Vishik and Grubb.

We finish with a discussion of non-proper extensions: Given a dual pair (A,\widetilde{A}) that satisfies certain assumptions, we construct all dissipative extensions of A that have domain contained in D(\widetilde{A}^*). Applying this result, we recover Crandall and Phillip's description

a strictly positive closable quadratic form, we find a criterion for an arbitrary extension of A to be dissipative.

Item Type: Thesis (Doctor of Philosophy (PhD))
Thesis advisor: Wood, Ian
Uncontrolled keywords: Operator theory, extension theory, dissipative operators, differential operators
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
Depositing User: Users 1 not found.
Date Deposited: 30 Mar 2017 11:00 UTC
Last Modified: 16 Feb 2021 13:44 UTC
Resource URI: (The current URI for this page, for reference purposes)
  • Depositors only (login required):