Frames and Factorization of Graph Laplacians

TL;DR

This paper constructs a canonical Parseval frame in the energy Hilbert space of infinite networks, enabling explicit analysis of the graph Laplacian's Friedrichs extension and its factorization.

Contribution

It introduces a new Parseval frame in the energy space of networks and characterizes the Friedrichs extension of the graph Laplacian using this frame.

Findings

Explicit formulas for the Parseval frame in energy Hilbert space.

Characterization of the Friedrichs extension of the graph Laplacian.

Factorization of the Friedrichs extension via $l^{2}(V)$.

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 $\mathscr{H}_{E}$ 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 $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-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 $\left(\mathscr{H}_{E},\Delta\right)$ where $\mathscr{H}_{E}$ 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 $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ 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 $l^{2}\left(V\right)$.