Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces

Abstract

Let $ω$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{ω,u}\) is compact on \(A_ω^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_ω^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{ω,u}$. Even in the unweighted Bergman space \(A^2\), our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on \(\widehat{\mathcal D}\)-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic \(\widehat{\mathcal D}\)-weight \(ω\) and bounded symbols \(u,v\) such that the product \(T_{ω,v}T_{ω,u}\) is noncompact, whereas its Berezin transform vanishes at the boundary.