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Multiplication Operators on the Lipschitz Space of an Infinite Graph

José A. Issa-Barbará, Rubén A. Martínez-Avendaño

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.

Multiplication Operators on the Lipschitz Space of an Infinite Graph

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.
Paper Structure (9 sections, 24 theorems, 133 equations)

This paper contains 9 sections, 24 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. The Lipschitz Space of an Infinite Graph
  3. The little Lipschitz space of a graph
  4. Weak and strong convergence of functions
  5. Boundedness of the multiplication operators
  6. The spectrum of a bounded multiplication operator
  7. Compactness of the multiplication operators
  8. Estimates on the essential norm of a multiplication operator
  9. Isometries

Key Result

Proposition 2.3

$\| \cdot \|_{\Hat{\mathcal{L}}}^{a}$ is a norm in $\Hat{\mathcal{L}}$ for any $a \in G$.

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 42 more