Graph Laplacians do not generate strongly continuous semigroups

TL;DR

This paper proves that graph Laplacians on infinite graphs do not generate strongly continuous semigroups when considering the product topology, highlighting a fundamental limitation in the analysis of such operators.

Contribution

It demonstrates that certain linear operators derived from graph Laplacians on infinite graphs cannot generate strongly continuous semigroups in the product topology, clarifying their analytical properties.

Findings

Graph Laplacians on infinite graphs do not generate strongly continuous semigroups.

Operators of the form α·Id + β·Δ_G do not produce such semigroups in the product topology.

The result clarifies limitations in the functional analysis of infinite graph Laplacians.

Abstract

We show that for graph Laplacians $\Delta_G$ on a connected locally finite simplicial undirected graph $G$ with countable infinite vertex set $V$ none of the operators $\alpha\,\mathrm{Id}+\beta\Delta_G, \alpha,\beta\in\mathbb{K},\beta \ne 0$, generate a strongly continuous semigroup on $\mathbb{K}^V$ when the latter is equipped with the product topology.