Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks

Abstract

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator $A$ on a Hilbert space $\mathcal{H}$, by means of a symmetric pair of operators. A \emph{symmetric pair} is comprised of densely defined operators $J:{\mathcal{H}}_1\to {\mathcal{H}}_2$ and $K:{\mathcal{H}}_2\to {\mathcal{H}}_1$ which are compatible in a certain sense. With the appropriate definitions of ${\mathcal{H}}_1$ and $J$ in terms of $A$ and $\mathcal{H}$, we show that $(J{J}^{\star }{)}^{-1}$ 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 ${\ell }^2(G)$ and ${\mathcal{H}}_{\mathcal{E}}$ (the energy space).

arXiv comment11 pages, 1 figure. arXiv admin note: text overlap with arXiv:1504.01332, arXiv:1103.5792. To appear: Complex Analysis and Operator Theory