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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Miroslav Englis; Kunyu Guo; Genkai Zhang
Journal: Proc. Amer. Math. Soc. 137 (2009), 3669-3678.
MSC (2000): Primary 32A36; Secondary 47B35, 47B06
Posted: June 16, 2009
MathSciNet review: 2529873
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Abstract | References | Similar articles | Additional information

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.

References:

1.: J. Arazy, S. D. Fisher, and J. Peetre, Hankel operators on weighted Bergman spaces, Amer. J. Math. 110 (1988), no. 6, 989-1053. MR 970119 (90a:47067)
2.: J. Arazy, S. D. Fisher, S. Janson, and J. Peetre, An identity for reproducing kernels in a planar domain and Hilbert-Schmidt Hankel operators, J. Reine Angew. Math. 406 (1990), 179-199. MR 1048240 (91b:47050)
3.: -, Membership of Hankel operators on the ball in unitary ideals, J. London Math. Soc. (2) 43 (1991), no. 3, 485-508. MR 1113389 (93c:47030)
4.: P. Boggiatto and F. Nicola, Non-commutative residues for anisotropic pseudo-differential operators in $\mathbb{R}\sp n$ , J. Funct. Anal. 203 (2003), no. 2, 305-320. MR 2003350 (2004g:58036)
5.: M. Feldman and R. Rochberg, Singular value estimates for commutators and Hankel operators on the unit ball and the Heisenberg group, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 121-159. MR 1044786 (91e:47024)
6.: V. Guillemin, Toeplitz operators in dimensions, Integral Equations Operator Theory 7 (1984), no. 2, 145-205. MR 750217 (86i:58130)
7.: A. Connes, The action functional in noncommutative geometry, Comm. Math. Phys. 117 (1988), no. 4, 673-683. MR 953826 (91b:58246)
8.: A. Connes, Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994. MR 1303779 (95j:46063)
9.: J. W. Helton and R. E. Howe, Traces of commutators of integral operators, Acta Math. 135 (1975), no. 3-4, 271-305. MR 0438188 (55:11106)
10.: R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), no. 2, 188-254. MR 587908 (83b:35166)
11.: S. Janson, J. Peetre, and R. Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61-138. MR 1008445 (91a:47029)
12.: S. Janson, H. Upmeier, and R. Wallstén, Schatten-norm identities for Hankel operators, J. Funct. Anal. 119 (1994), no. 1, 210-216. MR 1255278 (95a:47021)
13.: J. Peetre, On the $S\sb 4$ -norm of a Hankel form, Rev. Mat. Iberoamericana 8 (1992), no. 1, 121-130. MR 1178451 (93f:47027)
14.: V. V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210 (2004e:47040)
15.: R. Ponge, Noncommutative residue invariants for CR and contact manifolds, J. Reine Angew. Math. 614 (2008), 117-151. MR 2376284 (2008j:58039)
16.: R. Rochberg, Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J. 31 (1982), no. 6, 913-925. MR 674875 (84d:47036)
17.: W. Rudin, Function theory in the unit ball of $\mathbb{C}^n$ , Springer-Verlag, New York-Berlin, 1980. MR 601594 (82i:32002)
18.: E. M. Stein, Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
19.: K. H. Zhu, Schatten class Hankel operators on the Bergman space of the unit ball, Amer. J. Math. 113 (1991), no. 1, 147-167. MR 1087805 (91m:47036)

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Additional Information:

Miroslav Englis
Affiliation: Mathematics Institute AS CR, Zitná 25, 11567 Prague 1, Czech Republic - and - Mathematics Institute, Silesian University, Na Rybnícku 1, 74601 Opava, Czech Republic
Email: englis@math.cas.cz

Kunyu Guo
Affiliation: Department of Mathematics, Fudan University, Shanghai 200433, People's Republic of China
Email: kyguo@fudan.edu.cn

Genkai Zhang
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, Göteborg, Sweden
Email: genkai@chalmers.se

DOI: 10.1090/S0002-9939-09-09331-9
PII: S 0002-9939(09)09331-9
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
Posted: June 16, 2009
Additional Notes: The research of the first author was supported by GA CR grant No. 201/06/128 and AV CR 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
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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