Symmetric Pairs and Self-Adjoint Extensions of Operators, with Applications to Energy Networks

摘要

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space , by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators and which are compatible in a certain sense. With the appropriate definitions of and J in terms of A and , we show that is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces and (the energy space).