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

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 $p \neq 2$.