Details
| Original language | English |
|---|---|
| Pages (from-to) | 2617-2640 |
| Number of pages | 24 |
| Journal | Journal of functional analysis |
| Volume | 261 |
| Issue number | 9 |
| Publication status | Published - 22 Jul 2011 |
| Externally published | Yes |
Abstract
We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the "finite rank problem". We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role.
Keywords
- Commuting operators, Finite rank problem, Zero-products of Toeplitz operators
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
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In: Journal of functional analysis, Vol. 261, No. 9, 22.07.2011, p. 2617-2640.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space
AU - Bauer, Wolfram
AU - Le, Trieu
N1 - Funding Information: * Corresponding author. E-mail addresses: wbauer@uni-math.gwdg.de (W. Bauer), Trieu.Le2@utoledo.edu (T. Le). 1 The author has been supported by an “Emmy-Noether scholarship” of DFG (Deutsche Forschungsgemeinschaft). Copyright: Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/7/22
Y1 - 2011/7/22
N2 - We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the "finite rank problem". We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role.
AB - We study three different problems in the area of Toeplitz operators on the Segal-Bargmann space in Cn. Extending results obtained previously by the first author and Y.L. Lee, and by the second author, we first determine the commutant of a given Toeplitz operator with a radial symbol belonging to the class Sym>0(Cn) of symbols having certain growth at infinity. We then provide explicit examples of zero-products of non-trivial Toeplitz operators. These examples show the essential difference between Toeplitz operators on the Segal-Bargmann space and on the Bergman space over the unit ball. Finally, we discuss the "finite rank problem". We show that there are no non-trivial rank one Toeplitz operators Tf for f∈Sym>0(Cn). In all these problems, the growth at infinity of the symbols plays a crucial role.
KW - Commuting operators
KW - Finite rank problem
KW - Zero-products of Toeplitz operators
UR - http://www.scopus.com/inward/record.url?scp=80052085105&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2011.07.006
DO - 10.1016/j.jfa.2011.07.006
M3 - Article
AN - SCOPUS:80052085105
VL - 261
SP - 2617
EP - 2640
JO - Journal of functional analysis
JF - Journal of functional analysis
SN - 0022-1236
IS - 9
ER -