2014/04/05 by Palle Jorgensen, Feng Tian, Jorgensen, Palle +1
Mathematics · #05C50 #05C75 #22E70 #31A15 #31C20 #42C15 #46N30 #46N50 #58J65 #65R10 #81S25 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47L60 #Secondary 46N20 #math.FA #msc:05C50 #msc:05C75 #msc:22E70 #msc:31A15 #msc:31C20 #msc:42C15 #msc:46N20 #msc:46N30 #msc:46N50 #msc:47L60 #msc:58J65 #msc:65R10 #msc:81S25
paper · pdf · doi:10.48550/arxiv.1404.1424
39 pages, 12 figures
arxiv created 2014/04/05 · arxiv updated 2014/04/08
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 \mathscrHE 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 \mathscrHE we characterize the Friedrichs extension of the \mathscrHE-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 (\mathscrHE,Δ) where \mathscrHE 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 \mathscrHE consisting of dipoles. Now Δ is a well-defined semibounded Hermitian operator in both of the Hilbert l2(V) and \mathscrHE. It is known to automatically be essentially selfadjoint as an l2(V)-operator, but generally not as an \mathscrHE operator. Hence as an \mathscrHE 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).