Details
| Original language | English |
|---|---|
| Pages (from-to) | 3531-3551 |
| Number of pages | 21 |
| Journal | Journal of functional analysis |
| Volume | 274 |
| Issue number | 12 |
| Early online date | 4 Jan 2018 |
| Publication status | Published - 15 Jun 2018 |
Abstract
For Toeplitz operators Tf (t) acting on the weighted Fock space Ht 2, we consider the semi-commutator Tf (t)Tg (t)−Tfg (t), where t>0 is a certain weight parameter that may be interpreted as Planck's constant ħ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit (⁎)limt→0‖Tf (t)Tg (t)−Tfg (t)‖t. It is well-known that ‖Tf (t)Tg (t)−Tfg (t)‖t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f,g∈BUC(Cn) by Bauer and Coburn. We now further generalize (⁎) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO∩L∞ of bounded functions having vanishing mean oscillation on Cn. Our approach is based on the algebraic identity Tf (t)Tg (t)−Tfg (t)=−(Hf¯ (t))⁎Hg (t), where Hg (t) denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (⁎) to vanish. For g we only have to impose limsupt→0‖Hg (t)‖t<∞ e.g. g∈L∞(Cn). We prove that the set of all symbols f∈L∞(Cn) with the property that limt→0‖Tf (t)Tg (t)−Tfg (t)‖t=limt→0‖Tg (t)Tf (t)−Tgf (t)‖t=0 for all g∈L∞(Cn) coincides with VMO∩L∞. Additionally, we show that limt→0‖Tf (t)‖t=‖f‖∞ holds for all f∈L∞(Cn). Finally, we present new examples, including bounded smooth functions, where (⁎) does not vanish.
Keywords
- Heat transform, Semi-classical limit, Semi-commutator, Vanishing mean oscillation
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
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In: Journal of functional analysis, Vol. 274, No. 12, 15.06.2018, p. 3531-3551.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Toeplitz quantization on Fock space
AU - Bauer, W.
AU - Coburn, L. A.
AU - Hagger, R.
PY - 2018/6/15
Y1 - 2018/6/15
N2 - For Toeplitz operators Tf (t) acting on the weighted Fock space Ht 2, we consider the semi-commutator Tf (t)Tg (t)−Tfg (t), where t>0 is a certain weight parameter that may be interpreted as Planck's constant ħ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit (⁎)limt→0‖Tf (t)Tg (t)−Tfg (t)‖t. It is well-known that ‖Tf (t)Tg (t)−Tfg (t)‖t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f,g∈BUC(Cn) by Bauer and Coburn. We now further generalize (⁎) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO∩L∞ of bounded functions having vanishing mean oscillation on Cn. Our approach is based on the algebraic identity Tf (t)Tg (t)−Tfg (t)=−(Hf¯ (t))⁎Hg (t), where Hg (t) denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (⁎) to vanish. For g we only have to impose limsupt→0‖Hg (t)‖t<∞ e.g. g∈L∞(Cn). We prove that the set of all symbols f∈L∞(Cn) with the property that limt→0‖Tf (t)Tg (t)−Tfg (t)‖t=limt→0‖Tg (t)Tf (t)−Tgf (t)‖t=0 for all g∈L∞(Cn) coincides with VMO∩L∞. Additionally, we show that limt→0‖Tf (t)‖t=‖f‖∞ holds for all f∈L∞(Cn). Finally, we present new examples, including bounded smooth functions, where (⁎) does not vanish.
AB - For Toeplitz operators Tf (t) acting on the weighted Fock space Ht 2, we consider the semi-commutator Tf (t)Tg (t)−Tfg (t), where t>0 is a certain weight parameter that may be interpreted as Planck's constant ħ in Rieffel's deformation quantization. In particular, we are interested in the semi-classical limit (⁎)limt→0‖Tf (t)Tg (t)−Tfg (t)‖t. It is well-known that ‖Tf (t)Tg (t)−Tfg (t)‖t tends to 0 under certain smoothness assumptions imposed on f and g. This result was recently extended to f,g∈BUC(Cn) by Bauer and Coburn. We now further generalize (⁎) to (not necessarily bounded) uniformly continuous functions and symbols in the algebra VMO∩L∞ of bounded functions having vanishing mean oscillation on Cn. Our approach is based on the algebraic identity Tf (t)Tg (t)−Tfg (t)=−(Hf¯ (t))⁎Hg (t), where Hg (t) denotes the Hankel operator corresponding to the symbol g, and norm estimates in terms of the (weighted) heat transform. As a consequence, only f (or likewise only g) has to be contained in one of the above classes for (⁎) to vanish. For g we only have to impose limsupt→0‖Hg (t)‖t<∞ e.g. g∈L∞(Cn). We prove that the set of all symbols f∈L∞(Cn) with the property that limt→0‖Tf (t)Tg (t)−Tfg (t)‖t=limt→0‖Tg (t)Tf (t)−Tgf (t)‖t=0 for all g∈L∞(Cn) coincides with VMO∩L∞. Additionally, we show that limt→0‖Tf (t)‖t=‖f‖∞ holds for all f∈L∞(Cn). Finally, we present new examples, including bounded smooth functions, where (⁎) does not vanish.
KW - Heat transform
KW - Semi-classical limit
KW - Semi-commutator
KW - Vanishing mean oscillation
UR - http://www.scopus.com/inward/record.url?scp=85040543330&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1704.05652
DO - 10.48550/arXiv.1704.05652
M3 - Article
AN - SCOPUS:85040543330
VL - 274
SP - 3531
EP - 3551
JO - Journal of functional analysis
JF - Journal of functional analysis
SN - 0022-1236
IS - 12
ER -