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 HE 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 HE we characterize the Friedrichs extension of the HE -graph Laplacian. We consider infinite connected network-graphs G = (V;E), V for vertices, and E for edges. To every conductance function c on the edges E of G, there is an associated pair (HE ; Δ) where HE in an energy Hilbert space, and Δ(= Δc ) 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 HE consisting of dipoles. Now Δ is a well-defined semibounded Hermitian operator in both of the Hilbert l2 (V) and HE . It is known to automatically be essentially selfadjoint as an l2 (V)-operator, but generally not as an HE operator. Hence as an HE 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 l2 (V).