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

Abstract

In a recent paper of the authors together with A. Aleman, it is shown that the Bloch space $\mathcal{B}$ in the unit disc has the following radicality property: if an analytic function $g$ satisfies that $g^n\in \mathcal{B}$, then $g^m\in \mathcal{B}$, for all $m\le n$. Since $\mathcal{B}$ coincides with the space $\mathcal{T}(A_{\alpha }^p)$ of analytic symbols $g$ such that the Volterra-type operator $T_{g}f(z)={\int }_0^{z}f(\zeta )g^{\prime }(\zeta )d\zeta$ 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 $\mathcal{T}(A_{\omega }^p)$ also has the radicality property, for any radial weight $\omega$. Unlike the classical case, the lack of a precise description of $\mathcal{T}(A_{\omega }^p)$ 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.