Frames and factorization of graph Laplacians

• Autorzy: Jorgensen P. , Tian F.

Identyfikatory

DOI

http://dx.doi.Org/10.7494/OpMath.2015.35.3.293

• Języki publikacji: EN

Abstrakty

EN

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 [formula] 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 [formula] we characterize the Priedrichs extension of the [formula]-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 [formula] where [formula] 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 [formula] consisting of dipoles. Now Δ is a well-defined semibounded Hermitian operator in both of the Hilbert [formula] and [formula]. It is known to automatically be essentially selfadjoint as an [formula]-operator, but generally not as an [formula] operator. Hence as an [formula] operator it has a Friedrichs extension. In this paper we offer two results for the Priedrichs extension: a characterization and a factorization. The latter is via [formula].

Słowa kluczowe

EN

unbounded operators; deficiency-indices; Hilbert space; boundary values; weighted graph; reproducing kernel; Dirichlet form; graph Laplacian; resistance network; harmonic analysis; frame; Parseval frame; Friedrichs extension; reversible random walk; resistance distance; energy Hilbert space

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2015
• Tom: Vol. 35, no. 3
• Strony: 293--332
• Opis fizyczny: Bibliogr. 35 poz., rys.

Twórcy

autor

Jorgensen P.

• The University of Iowa Department of Mathematics Iowa City, IA 52242-1419, USA

autor

Tian F.

• Wright State University Department of Mathematics Dayton, OH 45435, USA

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