Fischbacher, Christoph Stefan (2017) On the Theory of Dissipative Extensions. Doctor of Philosophy (PhD) thesis, University of Kent,. (KAR id:61093)
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Abstract
We consider the problem of constructing dissipative extensions of given dissipative
operators.
Firstly, we discuss the dissipative extensions of symmetric operators and give a suffcient condition for when these extensions are completely nonselfadjoint. Moreover,
given a closed and densely defined operator A, we construct its closed extensions which we parametrize by suitable subspaces of D(A^*).
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}).
Assuming that zero is in the field of regularity of a given dissipative operator A, we then construct its Kreinvon Neumann extension A_K, which we show to be maximally
dissipative. If there exists a dissipative operator (\widetilde{A}) such that A and \widetilde{A} form a dual
pair, we discuss when A_K is a proper extension of the dual pair (A,\widetilde{A}) and if this is not
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).
After this, we consider dual pairs (A, \widetilde{A}) of sectorial operators and construct proper sectorial extensions that satisfy certain conditions on their numerical range. We apply this result to positive symmetric operators, where we recover the theory of nonnegative
selfadjoint and sectorial extensions of positive symmetric operators as described by Birman, Krein, Vishik and Grubb.
Moreover, for the case of proper extensions of a dual pair (A_0,\widetilde{A}_0)of sectorial operators, we develop a theory along the lines of the BirmanKreinVishik theory and define an order in the imaginary parts of the various proper dissipative extensions of (A,\widetilde{A}).
We finish with a discussion of nonproper 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
of all dissipative extensions of a symmetric operator perturbed by a bounded dissipative operator. Lastly, given a dissipative operator A whose imaginary part induces
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:  https://kar.kent.ac.uk/id/eprint/61093 (The current URI for this page, for reference purposes) 
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