Compact Differences of Composition Operators on Holomorphic Function Spaces in the Unit Ball

TL;DR

This paper establishes a lower bound for the essential norm difference of composition operators on certain holomorphic function spaces in the unit ball, providing a criterion for their compactness and answering a longstanding question.

Contribution

It introduces a lower bound for the essential norm difference of composition operators and characterizes their compactness on holomorphic function spaces in the unit ball.

Findings

Lower bound for the essential norm difference of composition operators

Necessary and sufficient condition for compactness of operator differences

Resolution of a question posed by MacCluer and Weir in 2005

Abstract

We find a lower bound for the essential norm of the difference of two composition operators acting on $H^2(B_N)$ or $A^2_s(B_N)$ ($s>-1$). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.