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

Abstract

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

18 pages