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

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