On the radicality property for spaces of symbols of bounded Volterra operators

In a recent paper of the authors together with A. Aleman, it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that gⁿ B, then gᵐ B, for all m n. Since B coincides with the space T (Aᵖ_) of analytic symbols g such that the Volterra-type operator Tgf (z) = ₀ᶻ f () g' () \, d is bounded on the classical weighted Bergman space Aᵖ_, the radicality property was used to study the composition of paraproducts Tg and Sgf=Tfg on Aᵖ_. Motivated by this fact, we prove that T (Aᵖ_) also has the radicality property, for any radial weight. Unlike the classical case, the lack of a precise description of T (Aᵖ_) for a general radial weight, induces us to prove the radicality property for Aᵖ_ from precise norm-operator results for compositions of analytic paraproducts.