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

Abstract

We prove that a Volterra-type integral operator $T_{g}f(z)={\int }_0^{z}f(\zeta )g^{\prime }(\zeta )d\zeta ,z\in \mathbb{D},$ defined on Hardy spaces $H^p,1\le p<\infty ,$ fixes an isomorphic copy of ${\ell }^p,$ if the operator $T_g$ 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 $H^p.$ As a consequence, we obtain a new proof for the equivalence of the compactness and the weak compactness of the operator $T_g$ on $H^1$.

14 pages, 1 figure