Strict singularity of Volterra type operators on Hardy spaces

In this paper, we first characterize the boundedness and compactness of Volterra type operator $S_gf(z) = \int_0^z f’(\zeta)g(\zeta)d\zeta, \ z \in $\mathbb{D}$,$definedonHardyspaces$H^p, , 0< p <\infty$\mathrm{.}Thespectrumof$S_g$isalsoobtained\mathrm{.}Thenweprovethat$S_g$fixesanisomorphiccopyof$\ell^p$andanisomorphiccopyof$\ell^2$iftheoperator$S_g$ is not compact on $H^p(1\le p<\mathrm{\infty })$. In particular, this implies that the strict singularity of the operator $S_g$ coincides with the compactness of the operator $S_g$ on $H^p$. At last, we post an open question for further study.