Details
| Original language | English |
|---|---|
| Pages (from-to) | 57-78 |
| Number of pages | 22 |
| Journal | Journal of functional analysis |
| Volume | 259 |
| Issue number | 1 |
| Publication status | Published - 9 Apr 2010 |
| Externally published | Yes |
Abstract
In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1<p<∞ whenever g̃(s)∈C0(C{double-struck}n) (vanishes at infinity) or g̃(s)∈Lp(Cn,dv), respectively, for some s with 0<s<14, where g̃(s) is the heat transform of g on C{double-struck}n. Moreover, we show that compactness of Tg implies that g̃(s) is in C0(Cn) for all s>14 and use this to show that, for g∈BMO1(C{double-struck}n), we have g̃(s) is in C0(C{double-struck}n) for some s>0 only if g̃(s) is in C0(Cn) for all s>0. This " backwards heat flow" result seems to be unknown for g∈BMO1 and even g∈L∞. Finally, we show that our compactness and vanishing " backwards heat flow" results hold in the context of the weighted Bergman space La2(B{double-struck}n,dvα), where the "heat flow" g̃(s) is replaced by the Berezin transform Bα(g) on La2(B{double-struck}n,dvα) for α>-1.
Keywords
- Berezin transform, Berezin-Toeplitz operator, Compact operators, Segal-Bargmann space
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
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In: Journal of functional analysis, Vol. 259, No. 1, 09.04.2010, p. 57-78.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Heat flow, BMO, and the compactness of Toeplitz operators
AU - Bauer, W.
AU - Coburn, L. A.
AU - Isralowitz, J.
N1 - Funding Information: * Corresponding author. E-mail addresses: wolfram.bauer@uni-greifswald.de (W. Bauer), lcoburn@buffalo.edu (L.A. Coburn), jbi2@buffalo.edu (J. Isralowitz). 1 Supported by an Emmy-Noether grant of DFG (Deutsche Forschungsgemeinschaft). Copyright: Copyright 2010 Elsevier B.V., All rights reserved.
PY - 2010/4/9
Y1 - 2010/4/9
N2 - In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1(s)∈C0(C{double-struck}n) (vanishes at infinity) or g̃(s)∈Lp(Cn,dv), respectively, for some s with 0n. Moreover, we show that compactness of Tg implies that g̃(s) is in C0(Cn) for all s>14 and use this to show that, for g∈BMO1(C{double-struck}n), we have g̃(s) is in C0(C{double-struck}n) for some s>0 only if g̃(s) is in C0(Cn) for all s>0. This " backwards heat flow" result seems to be unknown for g∈BMO1 and even g∈L∞. Finally, we show that our compactness and vanishing " backwards heat flow" results hold in the context of the weighted Bergman space La2(B{double-struck}n,dvα), where the "heat flow" g̃(s) is replaced by the Berezin transform Bα(g) on La2(B{double-struck}n,dvα) for α>-1.
AB - In this paper, it is shown that the Berezin-Toeplitz operator Tg is compact or in the Schatten class Sp of the Segal-Bargmann space for 1(s)∈C0(C{double-struck}n) (vanishes at infinity) or g̃(s)∈Lp(Cn,dv), respectively, for some s with 0n. Moreover, we show that compactness of Tg implies that g̃(s) is in C0(Cn) for all s>14 and use this to show that, for g∈BMO1(C{double-struck}n), we have g̃(s) is in C0(C{double-struck}n) for some s>0 only if g̃(s) is in C0(Cn) for all s>0. This " backwards heat flow" result seems to be unknown for g∈BMO1 and even g∈L∞. Finally, we show that our compactness and vanishing " backwards heat flow" results hold in the context of the weighted Bergman space La2(B{double-struck}n,dvα), where the "heat flow" g̃(s) is replaced by the Berezin transform Bα(g) on La2(B{double-struck}n,dvα) for α>-1.
KW - Berezin transform
KW - Berezin-Toeplitz operator
KW - Compact operators
KW - Segal-Bargmann space
UR - http://www.scopus.com/inward/record.url?scp=77951939234&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2010.03.016
DO - 10.1016/j.jfa.2010.03.016
M3 - Article
AN - SCOPUS:77951939234
VL - 259
SP - 57
EP - 78
JO - Journal of functional analysis
JF - Journal of functional analysis
SN - 0022-1236
IS - 1
ER -