Strict singularity of a Volterra-type integral operator on $H^p$

We prove that a Volterra-type integral operator $T_gf(z) = \int_0^z f(\zeta)g’(\zeta)d\zeta, , z \in \mathbb D,$definedonHardyspaces$H^p, , 1 \le p < \infty,$fixesanisomorphiccopyof$\ell^p,$iftheoperator$T_g$ is not compact. In particular, this shows that the strict singularity of the operator $T_g$ coincides with the compactness of the operator $T_g$ on spaces $H^p\mathrm{.}$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator $T_g$ on $H^1$.