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Journal of Operator Theory

Volume 69, Issue 1, Winter 2013  pp. 135-159.

Multiplication operators on the energy space

Authors Palle E.T. Jorgensen (1) and Erin P.J. Pearse (2)
Author institution: (1) University of Iowa, Iowa City, IA 52246-1419, U.S.A.
(2) California State Polytechnic University, San Luis Obispo, CA 93405-0403, U.S.A.

Summary:  We consider the multiplication operators on $\mathcal{H}_{\mathcal{E}}$ (the space of functions of finite energy supported on an infinite network), characterize them in terms of positive semidefinite functions. We show why they are typically not self-adjoint, and compute their adjoints in terms of a reproducing kernel. We also consider the bounded elements of $\mathcal{H}_{\mathcal{E}}$ and use the (possibly unbounded) multiplication operators corresponding to them to construct a boundary theory for the network. In the case when the only harmonic functions of finite energy are constant, we show that the corresponding Gel'fand space is the 1-point compactification of the underlying network.

Keywords:  multiplication operator, Dirichlet form, graph energy, discrete potential theory, graph Laplacian, weighted graph, spectral graph theory, resistance network, Gel'fand space, reproducing kernel Hilbert space

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