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

In [1] 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(Aαp) of analytic symbols g such that the Volterra-type operator Tgf(z)=∫0zf(ζ)g′(ζ)dζ is bounded on the classical weighted Bergman space Aαp, the radicality property was used to study the composition of paraproducts Tg and Sgf=Tfg on Aαp. Motivated by this fact, we prove that T(Aωp) also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of T(Aωp) for a general radial weight, induces us to prove the radicality property for Aωp from precise norm-operator results for compositions of analytic paraproducts.