On the Theory of Dissipative Extensions

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 non-selfadjoint. 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 Krein-von 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 non-negative

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 Birman-Krein-Vishik 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 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

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.