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 $g^n\in B$, then $g^m\in B$, for all $m\le n$. Since $B$ coincides with the space $T\left(A_{\alpha }^p\right)$ of analytic symbols g such that the Volterra-type operator $T_{g}f\left(z\right)={\int }_0^{z}f\left(\zeta \right)g^{\prime }\left(\zeta \right)d\zeta$ is bounded on the classical weighted Bergman space $A_{\alpha }^p$, the radicality property was used to study the composition of paraproducts $T_g$ and $S_{g}f=T_{f}g$ on $A_{\alpha }^p$. Motivated by this fact, we prove that $T\left(A_{\omega }^p\right)$ also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of $T\left(A_{\omega }^p\right)$ for a general radial weight, induces us to prove the radicality property for $A_{\omega }^p$ from precise norm-operator results for compositions of analytic paraproducts.