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We provide a streamlined construction of the Friedrichs extension of adensely-defined self-adjoint and semibounded operator $A$ on a Hilbert space$\mathcal{H}$, by means of a symmetric pair of operators. A \emph{symmetricpair} 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 ina certain sense. With the appropriate definitions of $\mathcal{H}_1$ and $J$ interms of $A$ and $\mathcal{H}$, we show that $(JJ^\star)^{-1}$ is theFriedrichs extension of $A$. Furthermore, we use related ideas (including thenotion of unbounded containment) to construct a generalization of theconstruction of the Krein extension of $A$ as laid out in a previous paper ofthe authors. These results are applied to the study of the graph Laplacian oninfinite networks, in relation to the Hilbert spaces $\ell^2(G)$ and$\mathcal{H}_{\mathcal E}$ (the energy space).

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