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

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 $(JJ^\star)^{-1}$ 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).