On a problem of Axler, Cuckovic and Rao
HTML articles powered by AMS MathViewer
- by Guangfu Cao PDF
- Proc. Amer. Math. Soc. 136 (2008), 931-935 Request permission
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).References
- 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, DOI 10.1090/S0002-9939-99-05436-2
- Christopher J. Bishop, Approximating continuous functions by holomorphic and harmonic functions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 781–811. MR 961619, DOI 10.1090/S0002-9947-1989-0961619-2
- Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
- Carl C. Cowen, The commutant of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239 (1978), 1–31. MR 482347, DOI 10.1090/S0002-9947-1978-0482347-9
- Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
- 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, DOI 10.1016/S0924-6509(09)70302-2
- Sun Hua Sun and De Chao Zheng, Toeplitz operators on the polydisk, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3351–3356. MR 1328380, DOI 10.1090/S0002-9939-96-03425-9
- James E. Thomson, The commutant of a class of analytic Toeplitz operators, Amer. J. Math. 99 (1977), no. 3, 522–529. MR 461196, DOI 10.2307/2373929
- James Thomson, The commutant of a class of analytic Toeplitz operators. II, Indiana Univ. Math. J. 25 (1976), no. 8, 793–800. MR 417843, DOI 10.1512/iumj.1976.25.25063
Additional Information
- Guangfu Cao
- Affiliation: Department of Mathematics, Guangzhou University, Guangzhou 510006, People’s Republic of China
- MR Author ID: 262473
- Email: guangfucao@163.com
- Received by editor(s): August 17, 2006
- Received by editor(s) in revised form: October 31, 2006
- Published electronically: November 23, 2007
- Additional Notes: Supported by National Natural Science Foundation of China
- Communicated by: Joseph A. Ball
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 931-935
- MSC (2000): Primary 47B35
- DOI: https://doi.org/10.1090/S0002-9939-07-08987-3
- MathSciNet review: 2361866