Compact linear combinations of composition operators on Hardy spaces

TL;DR

This paper proves that the compactness of linear combinations of composition operators on Hardy spaces is independent of the space parameter p, resolving a conjecture about differences of such operators.

Contribution

It establishes that the compactness of linear combinations of composition operators on Hardy spaces does not depend on p, confirming a conjecture by Choe et al.

Findings

Compactness is independent of p for 0<p<∞.

Confirmed conjecture on compact differences of composition operators.

Provides a unified understanding of composition operator compactness across Hardy spaces.

Abstract

Let $\varphi_j$, $j=1,2, \dots, N$, be holomorphic self-maps of the unit disk $\mathbb{D}$ of $\mathbb{C}$. We prove that the compactness of a linear combination of the composition operators $C_{\varphi_j}: f\mapsto f\circ\varphi_j$ on the Hardy space $H^p(\mathbb{D})$ does not depend on $p$ for $0<p<\infty$. This answers a conjecture of Choe et al. about the compact differences $C_{\varphi_1} - C_{\varphi_2}$ on $H^p(\mathbb{D})$, $0<p<\infty$.