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Finite rank Toeplitz operators on the Bergman space


Author: Daniel H. Luecking
Journal: Proc. Amer. Math. Soc. 136 (2008), 1717-1723
MSC (2000): Primary 46E20
DOI: https://doi.org/10.1090/S0002-9939-07-09119-8
Published electronically: November 30, 2007
MathSciNet review: 2373601
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Abstract: Given a complex Borel measure $ \mu$ with compact support in the complex plane $ \mathbb{C}$ the sesquilinear form defined on analytic polynomials $ f$ and $ g$ by $ B_\mu(f,g) = \int f\bar g \,d\mu$, determines an operator $ T_\mu$ from the space of such polynomials $ \mathcal{P}$ to the space of linear functionals on $ \overline{\mathcal{P}}$. This operator is called the Toeplitz operator with symbol $ \mu$. We show that $ T_\mu$ has finite rank if and only if $ \mu$ is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.


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Additional Information

Daniel H. Luecking
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Ar- kansas 72701
Email: luecking@uark.edu

DOI: https://doi.org/10.1090/S0002-9939-07-09119-8
Keywords: Bergman space, Toeplitz operator
Received by editor(s): January 4, 2007
Received by editor(s) in revised form: February 21, 2007
Published electronically: November 30, 2007
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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