Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian

Abstract

We compare the space of square-summable functions on an infinite graph (denoted ${\ell }^2(G)$) with the space of functions of finite energy (denoted ${\mathcal{H}}_{\mathcal{E}}$). There is a notion of inclusion that allows ${\ell }^2(G)$ to be embedded into ${\mathcal{H}}_{\mathcal{E}}$, but the required inclusion operator is unbounded in most interesting cases. These observations assist in the construction of the Krein extension of the Laplace operator on ${\mathcal{H}}_{\mathcal{E}}$. We investigate the Krein extension and compare it to the Friedrichs extension developed by the authors in a previous paper.

17 pages, 3 figures. arXiv admin note: text overlap with arXiv:1103.5792

References in corpus (1)

• Stochastic Completeness of Graphs