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On a problem of Axler, Cuckovic and Rao
Author:
Guangfu Cao
Journal:
Proc. Amer. Math. Soc. 136 (2008), 931-935
MSC (2000):
Primary 47B35
Posted:
November 23, 2007
MathSciNet review:
2361866
Full-text PDF
Abstract |
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Additional Information
Abstract: In this note we show that if two Toeplitz operators on a Bergman space of the (Levi) pseudoconvex domain commute and the symbol of one of them is analytic and non-constant, then the other one is also analytic. This gives an affirmative answer of a problem of S. Axler, Z. Cuckovic and N. V. Rao (1999).
- 1.
Sheldon
Axler, Željko
Čučković, and N.
V. Rao, Commutants of analytic Toeplitz
operators on the Bergman space, Proc. Amer.
Math. Soc. 128 (2000), no. 7, 1951–1953. MR 1694299
(2000m:47035), http://dx.doi.org/10.1090/S0002-9939-99-05436-2
- 2.
Christopher
J. Bishop, Approximating continuous functions by
holomorphic and harmonic functions, Trans.
Amer. Math. Soc. 311 (1989), no. 2, 781–811. MR 961619
(89j:30051), http://dx.doi.org/10.1090/S0002-9947-1989-0961619-2
- 3.
Arlen
Brown and P.
R. Halmos, Algebraic properties of Toeplitz operators, J.
Reine Angew. Math. 213 (1963/1964), 89–102. MR 0160136
(28 #3350)
- 4.
Carl
C. Cowen, The commutant of an analytic Toeplitz
operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. MR 0482347
(58 #2420), http://dx.doi.org/10.1090/S0002-9947-1978-0482347-9
- 5.
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
(84c:32001)
- 6.
R.
Narasimhan, Analysis on real and complex manifolds,
North-Holland Mathematical Library, vol. 35, North-Holland Publishing
Co., Amsterdam, 1985. Reprint of the 1973 edition. MR 832683
(87c:58002)
- 7.
Sun
Hua Sun and De
Chao Zheng, Toeplitz operators on the
polydisk, Proc. Amer. Math. Soc.
124 (1996), no. 11, 3351–3356. MR 1328380
(97a:47038), http://dx.doi.org/10.1090/S0002-9939-96-03425-9
- 8.
James
E. Thomson, The commutant of a class of analytic Toeplitz
operators, Amer. J. Math. 99 (1977), no. 3,
522–529. MR 0461196
(57 #1181)
- 9.
James
Thomson, The commutant of a class of analytic Toeplitz operators.
II, Indiana Univ. Math. J. 25 (1976), no. 8,
793–800. MR 0417843
(54 #5891)
- 1.
- Sheldon Axler, Zeljko Cuckovic and N. V. Rao, Commutants of analytic Teoplitz operators on the Bergman space, Proc. Amer. Math. Soc., 128 (1999) 1951-1953. MR 1694299 (2000m:47035)
- 2.
- Christopher J. Bishop, Approximating continuous functions by holomorphic and harmonic functions, Trans. Amer. Math. Soc., 311 (1989) 781-811. MR 961619 (89j:30051)
- 3.
- A. Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213 (1964) 89-102. MR 0160136 (28:3350)
- 4.
- Carl C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc., 239 (1978) 1-31. MR 0482347 (58:2420)
- 5.
- Steven G. Krantz, Function theory of several complex variables, New York, Chichester, Brisbane, Toronto, Singapore 1982. MR 635928 (84c:32001)
- 6.
- R. Narasimhan, Analysis on real and complex manifolds, North-Holland, Inc. 1973. MR 832683 (87c:58002)
- 7.
- S. Sun and D. Zheng, Toeplitz operators on the polydisk, Proc. Amer. Math. Soc., 124 (1996) 3351-3356. MR 1328380 (97a:47038)
- 8.
- James E. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math., 99 (1977) 522-529 MR 0461196 (57:1181)
- 9.
- James E. Thomson, The commutant of a class of analytic Toeplitz operators II, Indiana Univ. Math. J., 25 (1976) 793-800. MR 0417843 (54:5891)
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Additional Information
Guangfu Cao
Affiliation:
Department of Mathematics, Guangzhou University, Guangzhou 510006, People’s Republic of China
Email:
guangfucao@163.com
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08987-3
PII:
S 0002-9939(07)08987-3
Keywords:
Toeplitz operator,
commutant of an operator
Received by editor(s):
August 17, 2006
Received by editor(s) in revised form:
October 31, 2006
Posted:
November 23, 2007
Additional Notes:
Supported by National Natural Science Foundation of China
Communicated by:
Joseph A. Ball
Article copyright:
© Copyright 2007 American Mathematical Society
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