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

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

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.**P. Ahern and Ž. Čučković,*A theorem of Brown-Halmos type for Bergman space Toeplitz operators*, J. Funct. Anal.**187**(2001), 200-210. MR**1867348 (2002h:47040)****2.**A. Brown and P.R. Halmos,*Algebraic properties of Toeplitz operators*, J. Reine Angew. Math.**213**(1964), 89-102. MR**0160136 (28:3350)****3.**B.R. Choe and Y.J. Lee,*Commuting Toeplitz operators on the harmonic Bergman spaces*, Michigan Math. J.**46**(1999), 163-174. MR**1682896 (2000a:47054)****4.**Ž. Čučković and N.V. Rao,*Mellin transform, monomial symbols, and commuting Toeplitz operators*, J. Funct. Anal.**154**(1998), 195-214. MR**1616532 (99f:47033)****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.**I. Louhichi, E. Strouse and L. Zakariasy,*Products of Toeplitz operators on the Bergman space*, Integral Equations Operator Theory**54**(2006), 525-539. MR**2222982 (2007a:47033)****7.**I. Louhichi and L. Zakariasy,*On Toeplitz operators with quasihomogeneous symbols*, Arch. Math.**85**(2005), 248-257. MR**2172383 (2006e:47061)****8.**R. Remmert,*Classical Topics in Complex Function Theory*. Graduate Texts in Methematics, 172, Springer, New York, 1998. MR**1483074 (98g:30002)****9.**L. Zakariasy,*The rank of Hankel operators on harmonic Bergman spaces*, Proc. Amer. Math. Soc.**131**(2003), 1177-1180. MR**1948109 (2003k:47038)****10.**Z.H. Zhou and X.T. Dong,*Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball*, Integral Equations Operator Theory**64**(2009), 137-154. MR**2501175****11.**N. Zorboska,*The Berezin transform and radial operators*, Proc. Amer. Math. Soc.**131**(2003), 793-800. MR**1937440 (2003h:47064)**

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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:
https://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.