Details
| Original language | English |
|---|---|
| Pages (from-to) | 345-366 |
| Number of pages | 22 |
| Journal | Journal of the London Mathematical Society |
| Volume | 96 |
| Issue number | 2 |
| Publication status | Published - 2 Oct 2017 |
Abstract
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.
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Journal of the London Mathematical Society, Vol. 96, No. 2, 02.10.2017, p. 345-366.
Research output: Contribution to journal › Article › Research › peer review
}
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 -