Uniform continuity and quantization on bounded symmetric domains

Bauer, W., Hagger, R. and Vasilevski, N. (2017) Uniform continuity and quantization on bounded symmetric domains. Journal of the London Mathematical Society, 96 (2). pp. 345-366. ISSN 1469-7750 doi: 10.1112/jlms.12069

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

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/84022
Identification Number/DOI	10.1112/jlms.12069
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Wiley
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