Compact Toeplitz operators on radial weighted Bergman spaces

Abstract

Let $\omega$ be a radial $\hat{\mathcal{D}}$-weight and $u$ be a bounded function on the unit disk $\mathbb{D}$. We prove that the Toeplitz operator $T_{\omega ,u}$ is compact on the weighted Bergman space $A_{\omega }^2$ if and only if its Berezin transform satisfies the vanishing condition ${\lim }_{|z|\to 1^{-}}{\tilde{u}}_{\omega }(z)=0.$ Our approach is based on a polynomial frame for $A_{\omega }^2$ and a detailed localization analysis of the resulting infinite matrix representation of $T_{\omega ,u}$. The argument is new even in the unweighted Bergman space $A^2$ which does not depend on the classical translation operators.

36 pages