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

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 \(p \ne 2.\)