Multiplication Operators on the Lipschitz Space of an Infinite Graph

TL;DR

This paper studies the Lipschitz space on infinite graphs, proving it is a Banach space, and characterizes multiplication operators including their boundedness, compactness, spectra, and isometric conditions.

Contribution

It introduces the Banach space structure of Lipschitz functions on infinite graphs and provides a detailed analysis of multiplication operators on these spaces.

Findings

Lipschitz space is a Banach space with a natural norm.

Characterization of bounded and compact multiplication operators.

Spectral properties and isometric conditions for these operators.

Abstract

The Lipschitz space of an infinite (locally-finite) graph is defined as the set of functions on the vertices of the graph such that the differences of the values between adjacent vertices remain bounded. In this paper we prove that this set is a Banach space when endowed with its natural norm, and we define the little Lipschitz space as the subspace where these differences tend to zero. We consider the multiplication operators on these spaces and characterize their boundedness, compactness and the spectra. We also obtain estimates of the norm and essential norm, and we characterize when these operators are isometric.