TY - JOUR

T1 - Uniform continuity and quantization on bounded symmetric domains

AU - Bauer, Wolfram

AU - Hagger, Raffael

AU - Vasilevski, Nikolai

N1 - Funding Information: Received 23 December 2016; revised 10 July 2017; published online 11 August 2017. 2010 Mathematics Subject Classification 47B35 (primary), 81S10, 32M15 (secondary). The first and third authors acknowledge support through DFG (Deutsche Forschungsgemeinschaft), BA 3793/4-1. Publisher Copyright: © 2017 London Mathematical Society. Copyright: Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2017/10/2

Y1 - 2017/10/2

N2 - 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,gϵC(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.

AB - 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,gϵC(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.

UR - http://www.scopus.com/inward/record.url?scp=85030313049&partnerID=8YFLogxK

U2 - 10.1112/jlms.12069

DO - 10.1112/jlms.12069

M3 - Article

AN - SCOPUS:85030313049

VL - 96

SP - 345

EP - 366

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 2

ER -