Unbounded graph-Laplacians in energy space, and their extensions

Abstract

Our purpose is to develop computational tools for determining spectra for operators associated with infinite weighted graphs. While there is a substantial literature concerning graph-Laplacians on infinite networks, much less developed is the distinction between the operator theory for the l2 space of the set V of vertices vs the case when the Hilbert space is defined by an energy form. A network is a triple (V,E,c) where V is a (typically countable infinite) set of vertices in a graph, with E denoting the set of edges. The function c is defined on E. It is given at the outset, symmetric and positive on E. We introduce a graph-Laplacian Δ, and an energy Hilbert space \(\mathcal{H}_{E}\) (both depending on c). While it is known that Δ is essentially selfadjoint on its natural domain in l2(V), its realization in \(\mathcal{H}_{E}\) is not. We give a characterization of the Friedrichs extension of the \(\mathcal{H}_{E}\)-Laplacian, and prove a formula for its computation. We obtain several corollaries regarding the diagonalization of infinite matrices. To every weighted finite-interaction countable infinite graph there is a naturally associated infinite banded matrix. With the use of the Friedrichs spectral resolution, we obtain a diagonalization formula for this family of infinite matrices. With examples we give concrete illustrations of both spectral types, and spectral multiplicities.