Compact Toeplitz Operators and the Geometry of Fixed-Radius Averages

We characterize compact Toeplitz operators with bounded complex-valued symbols on the Bergman space of a bounded strongly pseudoconvex domain with $C^2$ boundary. For any fixed $R>0$, the operator $T_f$ is compact if and only if the Euclidean-volume averages of $f$ over Bergman geodesic balls of radius $R$ tend to zero at the boundary. This answers Zhu’s fixed-radius question affirmatively on the unit ball. On the polydisc ${\mathbb{D}}^n$, $n\ge 2$, averages over products of Bergman discs of fixed radii characterize compactness, whereas averages over genuine Bergman geodesic balls need not. For every sufficiently small fixed radius, we construct a bounded real-valued symbol, smooth in the domain, whose geodesic-ball averages tend to zero along every sequence approaching the boundary, while its Berezin transform converges to a positive value along one such sequence. Boundary scaling reduces the strongly pseudoconvex case to a uniqueness problem for fixed-radius ball averages on complex hyperbolic space. We prove the required uniqueness theorem under a sharp decay condition expressed in terms of a Busemann height function.