Structural rigidity of generalised Volterra operators on $H^p$

TL;DR

This paper investigates the structural properties of generalized Volterra operators on Hardy spaces, revealing rigidity phenomena and their implications for the operators' behavior on infinite-dimensional subspaces.

Contribution

It establishes a rigidity property for non-compact Volterra operators on $H^p$, showing their subspace structure and $ ext{ell}^2$-singularity for $p eq 2$, which was previously unknown.

Findings

If $T_g$ is bounded below on an infinite-dimensional subspace, that subspace contains an $ ext{ell}^p$-like subspace.

Any Volterra operator $T_g$ on $H^p$ is $ ext{ell}^2$-singular for $p eq 2$.

The results hold for $g ext{ in } BMOA$ and extend understanding of operator structure on Hardy spaces.

Abstract

We show that the non-compact generalised analytic Volterra operators $T_g$, where $g \in \mathit{BMOA}$, have the following structural rigidity property on the Hardy spaces $H^p$ for $1 \le p < \infty$ and $p \neq 2$: if $T_g$ is bounded below on an infinite-dimensional subspace $M \subset H^p$, then $M$ contains a subspace linearly isomorphic to $\ell^p$. This implies in particular that any Volterra operator $T_g\colon H^p \to H^p$ is $\ell^2$-singular for $p \neq 2$.