On the connected component of compact composition operators on the Hardy space

TL;DR

This paper demonstrates the existence of non-compact composition operators within the same connected component as compact ones on the Hardy space, answering a longstanding question and providing new bounds using Aleksandrov-Clark measures.

Contribution

It shows non-compact composition operators are connected to compact ones on the Hardy space, and improves bounds on their essential norms using Aleksandrov-Clark measures.

Findings

Existence of non-compact composition operators in the same connected component as compact ones.

An improved lower bound for the essential norm of differences of composition operators.

Application of Aleksandrov-Clark measures as a key analytical tool.

Abstract

We show that there exist non-compact composition operators in the connected component of the compact ones on the classical Hardy space $H^2$ on the unit disc. This answers a question posed by Shapiro and Sundberg in 1990. We also establish an improved version of a theorem of MacCluer, giving a lower bound for the essential norm of a difference of composition operators in terms of the angular derivatives of their symbols. As a main tool we use Aleksandrov-Clark measures.