Frames and Factorization of Graph Laplacians

Abstract

Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space ${\mathcal{H}}_E$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in ${\mathcal{H}}_E$ we characterize the Friedrichs extension of the ${\mathcal{H}}_E$-graph Laplacian. We consider infinite connected network-graphs $G=\left(V,E\right)$, $V$ for vertices, and \emph{E} for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair $\left({\mathcal{H}}_E,\Delta \right)$ where ${\mathcal{H}}_E$ in an energy Hilbert space, and $\Delta \left(={\Delta }_c\right)$ is the $c$-Graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in ${\mathcal{H}}_E$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^2\left(V\right)$ and ${\mathcal{H}}_E$. It is known to automatically be essentially selfadjoint as an $l^2\left(V\right)$-operator, but generally not as an ${\mathcal{H}}_E$ operator. Hence as an ${\mathcal{H}}_E$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^2\left(V\right)$.

39 pages, 12 figures

References in corpus (1)

• Essential selfadjointness of the graph-Laplacian