Products of Toeplitz operators on the harmonic Bergman space

Authors:
Xing-Tang Dong and Ze-Hua Zhou

Journal:
Proc. Amer. Math. Soc. **138** (2010), 1765-1773

MSC (2010):
Primary 47B35; Secondary 47B38

Published electronically:
December 16, 2009

MathSciNet review:
2587461

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we first discuss some basic results concerning Toeplitz operators with quasihomogeneous symbols (i.e., symbols being of the form , where is a radial function) on the harmonic Bergman space. Then we determine when the product of two Toeplitz operators with quasihomogeneous symbols is a Toeplitz operator.

**1.**Patrick Ahern and Željko Čučković,*A theorem of Brown-Halmos type for Bergman space Toeplitz operators*, J. Funct. Anal.**187**(2001), no. 1, 200–210. MR**1867348**, 10.1006/jfan.2001.3811**2.**Arlen Brown and P. R. Halmos,*Algebraic properties of Toeplitz operators*, J. Reine Angew. Math.**213**(1963/1964), 89–102. MR**0160136****3.**Boo Rim Choe and Young Joo Lee,*Commuting Toeplitz operators on the harmonic Bergman space*, Michigan Math. J.**46**(1999), no. 1, 163–174. MR**1682896**, 10.1307/mmj/1030132367**4.**Zeljko Cucković and N. V. Rao,*Mellin transform, monomial symbols, and commuting Toeplitz operators*, J. Funct. Anal.**154**(1998), no. 1, 195–214. MR**1616532**, 10.1006/jfan.1997.3204**5.**X.T. Dong and Z.H. Zhou,*Algebraic properties of Toeplitz operators with separately quasi- homogeneous symbols on the Bergman space of the unit ball*, J. Operator Theory, to appear.**6.**Issam Louhichi, Elizabeth Strouse, and Lova Zakariasy,*Products of Toeplitz operators on the Bergman space*, Integral Equations Operator Theory**54**(2006), no. 4, 525–539. MR**2222982**, 10.1007/s00020-005-1369-1**7.**Issam Louhichi and Lova Zakariasy,*On Toeplitz operators with quasihomogeneous symbols*, Arch. Math. (Basel)**85**(2005), no. 3, 248–257. MR**2172383**, 10.1007/s00013-005-1198-0**8.**Reinhold Remmert,*Classical topics in complex function theory*, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998. Translated from the German by Leslie Kay. MR**1483074****9.**Lova Zakariasy,*The rank of Hankel operators on harmonic Bergman spaces*, Proc. Amer. Math. Soc.**131**(2003), no. 4, 1177–1180 (electronic). MR**1948109**, 10.1090/S0002-9939-02-06638-8**10.**Ze-Hua Zhou and Xing-Tang Dong,*Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball*, Integral Equations Operator Theory**64**(2009), no. 1, 137–154. MR**2501175**, 10.1007/s00020-009-1677-y**11.**Nina Zorboska,*The Berezin transform and radial operators*, Proc. Amer. Math. Soc.**131**(2003), no. 3, 793–800 (electronic). MR**1937440**, 10.1090/S0002-9939-02-06691-1

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Additional Information

**Xing-Tang Dong**

Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China

Email:
dongxingtang@163.com

**Ze-Hua Zhou**

Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China

Email:
zehuazhou2003@yahoo.com.cn

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-10204-6

Keywords:
Toeplitz operators,
harmonic Bergman space,
quasihomogeneous symbols

Received by editor(s):
June 30, 2009

Received by editor(s) in revised form:
September 7, 2009

Published electronically:
December 16, 2009

Additional Notes:
The second author was supported in part by the National Natural Science Foundation of China (Grant Nos. 10971153, 10671141).

Communicated by:
Nigel J. Kalton

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.