Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On operators which commute with analytic Toeplitz operators modulo the finite rank operators


Authors: Kunyu Guo and Kai Wang
Journal: Proc. Amer. Math. Soc. 134 (2006), 2571-2576
MSC (2000): Primary 47B35, 47B20
Published electronically: February 17, 2006
MathSciNet review: 2213734
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that an operator $ S$ on the Hardy space $ H^2({\mathbb{D}}^n)$ (or $ H^2({\mathbb{B}}_n)$) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if $ S=T_g+F$. Here $ F$ is a finite rank operator, and in the case $ n=1$, $ g$ is a sum of a rational function and a bounded analytic function, and in the case $ n\geq 2$, $ g$ is a bounded analytic function.


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  • [CG] Xiaoman Chen and Kunyu Guo, Analytic Hilbert modules, Chapman & Hall/CRC Research Notes in Mathematics, vol. 433, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1988884
  • [Da] Kenneth R. Davidson, On operators commuting with Toeplitz operators modulo the compact operators, J. Functional Analysis 24 (1977), no. 3, 291–302. MR 0454715
  • [Gu1] Caixing Gu, On operators commuting with Toeplitz operators modulo the finite rank operators, J. Funct. Anal. 215 (2004), no. 1, 178–205. MR 2085114, 10.1016/j.jfa.2004.01.011
  • [Gu2] Caixing Gu, Some algebraic properties of Toeplitz and Hankel operators on polydisk, Arch. Math. (Basel) 80 (2003), no. 4, 393–405. MR 1982839
  • [GuZ] Caixing Gu and Dechao Zheng, The semi-commutator of Toeplitz operators on the bidisc, J. Operator Theory 38 (1997), no. 1, 173–193. MR 1462020
  • [Guo] Kun Yu Guo and Shun Hua Sun, The essential commutant of the analytic Toeplitz algebra and some problems related to it, Acta Math. Sinica (Chin. Ser.) 39 (1996), no. 3, 300–313 (Chinese, with English and Chinese summaries). MR 1413350
  • [Kr] Steven G. Krantz, Function theory of several complex variables, John Wiley & Sons, Inc., New York, 1982. Pure and Applied Mathematics; A Wiley-Interscience Publication. MR 635928
  • [R1] Walter Rudin, New constructions of functions holomorphic in the unit ball of 𝐶ⁿ, CBMS Regional Conference Series in Mathematics, vol. 63, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 840468
  • [R2] W. Rudin, Function theory in polydisks, New York, 1969.
  • [R3] Walter Rudin, Function theory in the unit ball of 𝐶ⁿ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
  • [P] S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699

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Additional Information

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

Kai Wang
Affiliation: School of Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China
Email: 031018009@fudan.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08259-1
Keywords: Hardy space, Toeplitz operator, finite rank operator
Received by editor(s): December 13, 2004
Received by editor(s) in revised form: March 18, 2005
Published electronically: February 17, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.