Self-adjoint extensions of network Laplacians and applications to resistance metrics

Let $(G,c)$ be an infinite network, and let $\mathcal{E}$ be the canonical energy form. Let $\Delta_2$ be the Laplace operator with dense domain in $\ell^2(G)$ and let $\Delta_{\mathcal{E}}$ be the Laplace operator with dense domain in the Hilbert space $\mathcal{H}_\mathcal{E}$ of finite energy functions on $G$. It is known that $\Delta_2$ is essentially self-adjoint, but that $\Delta_{\mathcal{E}}$ is \emph{not}. In this paper, we characterize the Friedrichs extension of $\Delta_{\mathcal{E}}$ in terms of $\Delta_2$ and show that the spectral measures of the two operators are mutually absolutely continuous with Radon-Nikodym derivative $\lambda$ (the spectral parameter), in the complement of $\lambda=0$. We also give applications to the effective resistance on $(G,c)$. For transient networks, the Dirac measure at $\lambda = 0$ contributes to the spectral resolution of the Friedrichs extension of $\Delta_{\mathcal{E}}$ but not to that of the self-adjoint $\ell^2$ Laplacian.