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Toeplitz and Hankel operators and Dixmier traces on the unit ball of $ \mathbb{C}^n$

Authors: Miroslav Englis, Kunyu Guo and Genkai Zhang
Journal: Proc. Amer. Math. Soc. 137 (2009), 3669-3678
MSC (2000): Primary 32A36; Secondary 47B35, 47B06
Published electronically: June 16, 2009
MathSciNet review: 2529873
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Abstract: We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of  $ \mathbb{C}^d$. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.

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Additional Information

Miroslav Englis
Affiliation: Mathematics Institute AS ČR, Žitná 25, 11567 Prague 1, Czech Republic – and – Mathematics Institute, Silesian University, Na Rybníčku 1, 74601 Opava, Czech Republic

Kunyu Guo
Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden

Keywords: Schatten - von Neumann classes, Macaev classes, trace, Dixmier trace, Toeplitz operators, Hankel operators, pseudo-Toeplitz operators, pseudo-differential operators, boundary CR operators, invariant Banach spaces
Received by editor(s): February 19, 2007
Received by editor(s) in revised form: July 12, 2007
Published electronically: June 16, 2009
Additional Notes: The research of the first author was supported by GA ČR grant No. 201/06/128 and AV ČR Institutional Research Plan No. AV0Z10190503, of the second author by NSFC(10525106) and NKBRPC(2006CB805905), and of the third author by the Swedish Science Council (VR) and SIDA-Swedish Research Links
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.