Journal of Operator Theory
Volume 69, Issue 1, Winter 2013 pp. 135-159.
Multiplication operators on the energy spaceAuthors: 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 HE (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 HE 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.
DOI: http://dx.doi.org/10.7900/jot.2010jul20.1886
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
Contents Full-Text PDF