Symmetric pairs and self-adjoint extensions of operators, with applications to energy networks

TL;DR

This paper introduces a new approach using symmetric pairs to construct self-adjoint extensions of operators, with applications to energy networks and graph Laplacians, providing streamlined methods for Friedrichs and Krein extensions.

Contribution

It presents a novel symmetric pair framework for constructing Friedrichs and Krein extensions of self-adjoint operators, simplifying existing methods and applying them to energy networks.

Findings

Constructed Friedrichs extension via symmetric pairs.

Generalized Krein extension construction.

Applied results to graph Laplacian on infinite networks.

Abstract

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator $A$ on a Hilbert space $\mathcal{H}$, by means of a symmetric pair of operators. A \emph{symmetric pair} is comprised of densely defined operators $J: \mathcal{H}_1 \to \mathcal{H}_2$ and $K: \mathcal{H}_2 \to \mathcal{H}_1$ which are compatible in a certain sense. With the appropriate definitions of $\mathcal{H}_1$ and $J$ in terms of $A$ and $\mathcal{H}$, we show that $(JJ^\star)^{-1}$ is the Friedrichs extension of $A$. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of $A$ as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces $\ell^2(G)$ and $\mathcal{H}_{\mathcal E}$ (the energy space).