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

We show that the non-compact generalised analytic Volterra operators $T_g$, where $g\in BMOA$, have the following structural rigidity property on the Hardy spaces $H^p$ for $1\le p<\mathrm{\infty }$ and $p\mathrm{\ne }2$: if $T_g$ is bounded below on an infinite-dimensional subspace $M\subset H^p$, then $M$ contains a subspace linearly isomorphic to ${\mathrm{\ell }}^p$. This implies in particular that any Volterra operator $T_g\text{\hspace{0.17em}⁣}:H^p\to H^p$ is ${\mathrm{\ell }}^2$-singular for $p \neq 2$.