Compactness of Toeplitz Operators on the Bergman Space

Abstract

Let φ\in L^{\infty}(\D). We study compactness criteria for \(T_φ\) on the Bergman space A^{2}(\D). Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function φ, however, its Berezin transform \tildeφ does not readily reveal the intrinsic properties of φ. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_φ$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.