TY  - JOUR
ID  - Jorgensen2015
LB  - Jorgensen2015
AU  - Jorgensen, Palle
AU  - Tian, Feng
TI  - Frames and factorization of graph Laplacians
ST  - Frames and factorization of graph Laplacians
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2015
VL  - 35
IS  - 3
SP  - 293
EP  - 332
DO  - http://dx.doi.org/10.7494/OpMath.2015.35.3.293
UR  - http://dx.doi.org/10.7494/OpMath.2015.35.3.293
AB  - 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 \(l^{2}\left(V\right)\).
KW  - unbounded operators
KW  - deficiency-indices
KW  - Hilbert space
KW  - boundary values
KW  - weighted graph
KW  - reproducing kernel
KW  - Dirichlet form
KW  - graph Laplacian
KW  - resistance network
KW  - harmonic analysis
KW  - harmonic analysis, frame
KW  - Parseval frame
KW  - Friedrichs extension
KW  - reversible random walk
KW  - resistance distance
KW  - energy Hilbert space
SN  - 1232-9274
ER  -