Frames and factorization of graph Laplacians

Palle Jorgensen; Feng Tian

doi:10.7494/OpMath.2015.35.3.293

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Frames and factorization of graph Laplacians

Journal article

Open access

Peer reviewed

Frames and factorization of graph Laplacians

Palle Jorgensen and Feng Tian

Opuscula Mathematica, Vol.35(3), pp.293-332

04/04/2014

DOI: 10.7494/OpMath.2015.35.3.293

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https://doi.org/10.7494/OpMath.2015.35.3.293View

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

Applied mathematics

boundary values

deficiency

Dirichlet form

energy Hilbert space

frame

Friedrichs extension

Functional Analysis

graph Laplacian

harmonic analysis

Hilbert space

indices

Mathematics

om walk

Parseval frame

Primary 47L60

Quantitative methods

reproducing kernel

resistance distance

resistance network

reversible r

Secondary 46N20

unbounded operators

weighted graph

Details

Title: Subtitle: Frames and factorization of graph Laplacians
Creators: Palle Jorgensen
Feng Tian - Wright State University
Resource Type: Journal article
Publication Details: Opuscula Mathematica, Vol.35(3), pp.293-332
DOI: 10.7494/OpMath.2015.35.3.293
ISSN: 1232-9274
eISSN: 2300-6919
Publisher: AGH Univeristy of Science and Technology Press
Language: English
Date published: 04/04/2014
Academic Unit: Mathematics
Record Identifier: 9984241048302771

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