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

Abstract

We prove that the Volterra-type integral operator \[ T_gf(z) = \int _0^z f(\zeta )g’(\zeta )d\zeta , \quad z \in \mathbb {D},\] defined on the Hardy spaces $H^p$ fixes an isomorphic copy of $\ell ^p$ if it is not compact. In particular, the strict singularity of $T_g$ coincides with its compactness on spaces $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of $T_g$ on $H^1$. Moreover, a non-compact $T_g$ acting on the space $BMOA$ fixes an isomorphic copy of $c_0.$