Multiplication operators on the energy space

TL;DR

This paper investigates multiplication operators on the energy space of functions on infinite networks, characterizing their boundedness, adjoints, and boundary representations, revealing complex interactions between function properties and network behavior.

Contribution

It provides a detailed analysis of multiplication operators on the energy space, including their boundedness criteria, adjoint formulas, and boundary structure, extending previous work with new operator-theoretic insights.

Findings

Multiplication operators are not Hermitian unless functions are constant.

Boundedness of multiplication operators relates to positive semidefinite functions and random walk behavior.

The boundary of the network can be described via a subalgebra of these operators, often embedding into the Gelfand space.

Abstract

This paper studies the "energy space" $\mathcal{H}_{\mathcal{E}}$ (the Hilbert space of functions of finite energy, aka the Dirichlet-finite functions) on an infinite network (weighted connected graph), from the point of view of the multiplication operators $M_f$ associated to functions $f$ on the network. We show that the multiplication operators $M_f$ are not Hermitian unless $f$ is constant, and compute the adjoint $M_f^\star$ in terms of a reproducing kernel for $\mathcal{H}_{\mathcal{E}}$. A characterization of the bounded multiplication operators is given in terms of positive semidefinite functions, and we give some conditions on $f$ which ensure $M_f$ is bounded. Examples show that it is not sufficient that $f$ be bounded or have finite energy. Conditions for the boundedness of $M_f$ are also expressed in terms of the behavior of the simple random walk on the network. We also consider the bounded elements of $\mathcal{H}_{\mathcal{E}}$ and the (possibly unbounded) multiplication operators corresponding to them. In a previous paper, the authors used functional integration to construct a type of boundary for infinite networks. The boundary is described here in terms of a certain subalgebra of these multiplication operators, and is shown to embed into the Gel'fand space of that subalgebra. In the case when the only harmonic functions of finite energy are constant, we show that the Gel'fand space is the 1-point compactification of the underlying network.