Commutants of analytic Toeplitz operators

on the Bergman space

Authors:
Sheldon Axler, Zeljko Cuckovic and N. V. Rao

Journal:
Proc. Amer. Math. Soc. **128** (2000), 1951-1953

MSC (2000):
Primary 47B35

DOI:
https://doi.org/10.1090/S0002-9939-99-05436-2

Published electronically:
October 29, 1999

MathSciNet review:
1694299

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Sheldon Axler and \v{Z}eljko \v{C}u\v{c}kovi\'{c},*Commuting Toeplitz Operators with harmonic symbols*, Integral Equations Operator Theory**14**(1991), 1-12. MR**92f:4018****2.**Sheldon Axler and Allen Shields,*Algebras generated by analytic and harmonic functions*, Indiana Univ. Math. J.**36**(1987), 631-638. MR**88h:46102****3.**Christopher J. Bishop,*Approximating continuous functions by holomorphic and harmonic functions*, Trans. Amer. Math. Soc.**311**(1989), 781-811. MR**89j:30051****4.**Arlen Brown and P. R. Halmos,*Algebraic properties of Toeplitz operators*, J. Reine Angew. Math.**213**(1964), 89-102. MR**28:3350****5.**E. M. \v{C}irka,*Approximation by holomorphic functions on smooth manifolds in*, Mat. Sb.**78**(1969), 101-123. MR**39:480****6.**Carl C. Cowen,*The commutant of an analytic Toeplitz operator*, Trans. Amer. Math. Soc.**239**(1978), 1-31. MR**58:2420****7.**\v{Z}eljko \v{C}u\v{c}kovi\'{c},*Commutants of Toeplitz operators on the Bergman space*, Pacific J. Math.**162**(1994), 277-285. MR**94j:47041****8.**\v{Z}eljko \v{C}u\v{c}kovi\'{c} and N. V. Rao,*Mellin Transform, monomial symbols and commuting Toeplitz operators*, J. Funct. Anal.**154**(1998), 195-214. MR**99f:47033****9.**Alexander J. Izzo,*Uniform algebras generated by holomorphic and pluriharmonic functions*, Trans. Amer. Math. Soc.**339**(1993), 835-847. MR**93m:46060****10.**James E. Thomson,*The commutant of a class of analytic Toeplitz operators*, Amer. J. Math.**99**(1977), 522-529. MR**57:1181****11.**James Thomson,*The commutant of a class of analytic Toeplitz operators*II, Indiana Univ. Math. J.**25**(1976), 793-800. MR**54:5891**

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Additional Information

**Sheldon Axler**

Affiliation:
Department of Mathematics, San Francisco State University, San Francisco, California 94132

Email:
axler@sfsu.edu

**Zeljko Cuckovic**

Affiliation:
Department of Mathematics, University of Toledo, Toledo, Ohio 43606

Email:
zcuckovi@math.utoledo.edu

**N. V. Rao**

Email:
rnagise@math.utoledo.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05436-2

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

Article copyright:
© Copyright 2000
American Mathematical Society