Rigidity of Volterra-type integral operators on Hardy spaces of the unit ball

TL;DR

This paper investigates the properties of Volterra-type integral operators on Hardy spaces of the unit ball, revealing their rigid behavior and equivalences among various singularity notions, with implications for their structure on these spaces.

Contribution

It establishes the equivalence of compactness, strict singularity, and $ ext{l}^p$-singularity for $J_b$ on $H^p$, and shows $J_b$ cannot fix an isomorphic copy of $ ext{l}^2$ when $p eq 2$.

Findings

Compactness, strict singularity, and $ ext{l}^p$-singularity are equivalent for $J_b$ on $H^p$.

$J_b$ cannot fix an isomorphic copy of $ ext{l}^2$ on $H^p$ when $p eq 2$.

The operator exhibits strong rigidity behavior on Hardy spaces of the unit ball.

Abstract

We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p < \infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $\ell^2$ when $p \ne 2.$