Topological structure of the space of composition operators on $L^\infty$ of an unbounded, locally finite metric space

TL;DR

This paper investigates the topological properties of composition operators on bounded functions over unbounded, locally finite metric spaces, focusing on the operator and essential norm topologies, and characterizes when differences of such operators are compact.

Contribution

It provides a detailed analysis of the topological structure of composition operators on $L^$ spaces and characterizes the compactness of their differences, advancing understanding in operator theory.

Findings

Analyzed the topological space of composition operators in operator and essential norm topologies.

Characterized the compactness of differences of two composition operators.

Provided conditions for compactness in the context of unbounded, locally finite metric spaces.

Abstract

We study properties of the topological space of composition operators on the Banach algebra of bounded functions on an unbounded, locally finite metric space in the operator norm topology and essential norm topology. Moreover, we characterize the compactness of differences of two such composition operators.