Essential self-adjointness and the $L^2$-Liouville property

TL;DR

This paper explores the relationship between essential self-adjointness of operators and harmonic functions in the kernel of their adjoints, with applications to Laplacians on manifolds and graphs, and examines the role of Green's functions in this context.

Contribution

It establishes new connections between self-adjointness and the constancy of kernel functions, with specific insights into Laplacians on manifolds and graphs, and the properties of Green's functions.

Findings

Self-adjointness relates to the constancy of kernel functions.

Green's functions can produce non-constant square-integrable harmonic functions.

Applications to Laplacians on manifolds and graphs demonstrate these relationships.

Abstract

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green's function and when it gives a non-constant harmonic function which is square integrable.