Commutants of analytic Toeplitz operators on the Bergman space
HTML articles powered by AMS MathViewer
- by Sheldon Axler, Željko Čučković and N. V. Rao PDF
- Proc. Amer. Math. Soc. 128 (2000), 1951-1953 Request permission
Abstract:
In this note we show that if two Toeplitz operators on a Bergman space commute and the symbol of one of them is analytic and nonconstant, then the other one is also analytic.References
- W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
- Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
- 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
- E. M. Čirka, Approximation by holomorphic functions on smooth manifolds in $\textbf {C}^{n}$, Mat. Sb. (N.S.) 78 (120) (1969), 101–123 (Russian). MR 0239121
- 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
- Željko Čučković, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277–285. MR 1251902
- eljko C̆u ković and N. V. Rao, Mellin transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal. 154 (1998), no. 1, 195–214. MR 1616532, DOI 10.1006/jfan.1997.3204
- Alexander J. Izzo, Uniform algebras generated by holomorphic and pluriharmonic functions, Trans. Amer. Math. Soc. 339 (1993), no. 2, 835–847. MR 1139494, DOI 10.1090/S0002-9947-1993-1139494-6
- 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
- Sheldon Axler
- Affiliation: Department of Mathematics, San Francisco State University, San Francisco, California 94132
- MR Author ID: 201457
- ORCID: 0000-0003-1733-6080
- Email: axler@sfsu.edu
- Željko Čučković
- Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606
- MR Author ID: 294593
- Email: zcuckovi@math.utoledo.edu
- N. V. Rao
- Email: rnagise@math.utoledo.edu
- Received by editor(s): August 8, 1998
- Published electronically: October 29, 1999
- Additional Notes: The first author was partially supported by the National Science Foundation
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1951-1953
- MSC (2000): Primary 47B35
- DOI: https://doi.org/10.1090/S0002-9939-99-05436-2
- MathSciNet review: 1694299