Abstract/Summary

We consider Toeplitz operatorsTλfwith symbolfacting on the standard weighted Bergmanspaces over a bounded symmetric domain Ω⊂Cn.Hereλ>genus−1 is the weight parameter.The classical asymptotic relation for the semi-commutatorlimλ→∞∥∥∥TλfTλg−Tλfg∥∥∥λ=0,withf,g∈C(Bn),(∗)where Ω =Bndenotes the complex unit ball, is extended to larger classes of bounded andunbounded operator symbol-functions and to more general domains. We deal with operatorsymbols that generically are neither continuous inside Ω nor admit a continuous extension to theboundary. Letβdenote the Bergman metric distance function on Ω. We prove that(∗) remainstrue forfandgin the space UC(Ω) of allβ-uniformly continuous functions on Ω. Note that thisspace contains also unbounded functions. In case of the complex unit ball Ω =Bn⊂Cnwe showthat(∗) holds true for bounded symbols in VMO(Bn), where the vanishing oscillation insideBnismeasured with respect toβ.Atthesametime(∗) fails for generic bounded measurable symbols.We construct a corresponding counterexample using oscillating symbols that are continuousoutside of a single point in Ω