Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Commuting Toeplitz operators on the polydisk


Authors: Boo Rim Choe, Hyungwoon Koo and Young Joo Lee
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 1727-1749
MSC (2000): Primary 47B35; Secondary 32A36
DOI: https://doi.org/10.1090/S0002-9947-03-03430-5
Published electronically: December 9, 2003
MathSciNet review: 2031039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.


References [Enhancements On Off] (What's this?)

  • 1. S. Axler and Z. Cuckovic, Commuting Toeplitz Operators with Harmonic Symbols, Integr. Equat. Oper. Th. 14 (1991), 1-11. MR 92f:47018
  • 2. S. Axler and Z. Cuckovic, and N. V. Rao, Commutants of Analytic Toeplitz Operators on the Bergman Space, Proc. Amer. Math. Soc. 128 (2000), 1951-1953. MR 2000m:47035
  • 3. B. R. Choe and Y. J. Lee, Pluriharmonic Symbols of Commuting Toeplitz Operators, Illinois J. Math. 37 (1993), 424-436. MR 94i:47041
  • 4. B. R. Choe and Y. J. Lee, Pluriharmonic Symbols of Essentially Commuting Toeplitz Operators, Illinois J. Math. 42 (1998), 280-293. MR 99g:47051
  • 5. B. R. Choe and Y. J. Lee, Commuting Toeplitz Operators on the Harmonic Bergman Space, Michigan Math. J. 46 (1999), 163-174. MR 2000a:47054
  • 6. Z. Cuckovic, Commuting Toeplitz Operators on the Bergman Space of an Annulus, Michigan Math. J. 43 (1996), 355-365. MR 97k:47026
  • 7. S. G. Krantz, Function Theory of Several Complex Variables, John Wiley & Sons, New York, 1982. MR 84c:32001
  • 8. J. Lee, An Invariant Mean Value Property in the Polydisc, Illinois J. Math. 42 (1998), 406-419. MR 99k:31003
  • 9. Y. J. Lee Pluriharmonic Symbols of Commuting Toeplitz Type Operators on the Weighted Bergman Spaces, Canadian Math. Bull. 41 (1998), 129-136. MR 99b:47035
  • 10. Y. J. Lee and K. Zhu, Some Differential and Integral Equations with Applications to Toeplitz Operators, Integr. Equat. Oper. Th. 44 (2002), 466-479.
  • 11. W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, 1969. MR 41:501
  • 12. W. Rudin, Functional Analysis, McGraw-Hill, 1973. MR 51:1315
  • 13. K. Stroethoff, Essentially Commuting Toeplitz Operators with Harmonic Symbols, Canadian J. Math. 45 (1993), 1080-1093. MR 94h:47046
  • 14. S. Sun and D. Zheng, Toeplitz Operators on the Polydisk, Proc. Amer. Math. Soc. 124 (1996), 3351-3356. MR 97a:47038
  • 15. D. Zheng, Hankel Operators and Toeplitz Operators on the Bergman Space, J. Functional Anal. 83 (1989), 98-120. MR 91b:47057
  • 16. D. Zheng, Semi-commutators of Toeplitz Operators on the Bergman Space, Integr. Equat. Oper. Th. 25 (1996), 347-372. MR 97e:47039
  • 17. D. Zheng, Commuting Toeplitz Operators with Pluriharmonic Symbols, Trans. Amer. Math. Soc. 350 (1998), 1595-1618. MR 98i:47027
  • 18. K. Zhu, Operator Theory in Function Spaces, Marcel Dekker. New York and Basel, 1990. MR 92c:47031

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47B35, 32A36

Retrieve articles in all journals with MSC (2000): 47B35, 32A36


Additional Information

Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Korea
Email: choebr@math.korea.ac.kr

Hyungwoon Koo
Affiliation: Department of Mathematics, Korea University, Seoul 136-701, Korea
Email: koohw@math.korea.ac.kr

Young Joo Lee
Affiliation: Department of Mathematics, Mokpo National University, Chonnam 534-729, Korea
Email: yjlee@mokpo.ac.kr

DOI: https://doi.org/10.1090/S0002-9947-03-03430-5
Keywords: Toeplitz operator, $n$-harmonic function, Bergman space
Received by editor(s): December 13, 2001
Published electronically: December 9, 2003
Additional Notes: This work was supported by the Korea Research Foundation Grant (KRF-2000-DP0014)
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society