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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that an operator on the Hardy space (or ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if . Here is a finite rank operator, and in the case , is a sum of a rational function and a bounded analytic function, and in the case , is a bounded analytic function.

**[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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B35,
47B20

Retrieve articles in all journals with MSC (2000): 47B35, 47B20

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.