Uniform continuity and quantization on bounded symmetric domains

We consider Toeplitz operators Tfλ with symbol f acting on the standard weighted Bergman spaces over a bounded symmetric domain Ω⊂Cn. Here λ>genus−1 is the weight parameter. The classical asymptotic relation for the semi‐commutator * limλ→∞TfλTgλ−Tfgλλ=0,withf,g∈C(Bn¯),where Ω=Bn denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol‐functions and to more general domains. We deal with operator symbols that generically are neither continuous inside Ω nor admit a continuous extension to the boundary. Let β denote the Bergman metric distance function on Ω. We prove that remains true for f and g in the space UC (Ω) of all β‐uniformly continuous functions on Ω. Note that this space contains also unbounded functions. In case of the complex unit ball Ω=Bn⊂Cn we show that holds true for bounded symbols in VMO (Bn), where the vanishing oscillation inside Bn is measured with respect to β. At the same time fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in Ω.