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

TL;DR

This paper explores the relationship between square-summable functions and finite energy functions on infinite graphs, focusing on the unbounded inclusion operator and the construction of the Krein extension of the energy Laplacian.

Contribution

It introduces the unbounded inclusion of $\, ext{l}^2(G)$ into $\, ext{H}_ ext{E}$ and constructs the Krein extension of the energy Laplacian, comparing it with the Friedrichs extension.

Findings

The inclusion operator from $\, ext{l}^2(G)$ to $\, ext{H}_ ext{E}$ is generally unbounded.

The Krein extension of the energy Laplacian is explicitly constructed.

Comparison between Krein and Friedrichs extensions provides insights into their spectral properties.

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.