Frames and factorization of graph Laplacians

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 \(E\) 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 \(l^{2}\left(V\right)\).

Abstract licence: CC BY

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