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COMPACT TOEPLITZ OPERATORS WITH CONTINUOUS SYMBOLS

Published online by Cambridge University Press:  01 May 2009

TRIEU LE
TRIEU LE*
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada e-mail: t29le@math.uwaterloo.ca
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Abstract

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For any rotation-invariant positive regular Borel measure ν on the closed unit ball $\overline{\mathbb{B}}_n$ whose support contains the unit sphere $\mathbb{S}_n$, let L2a be the closure in L2 = L2($\overline{\mathbb{B}}_n, dν) of all analytic polynomials. For a bounded Borel function f on $\overline{\mathbb{B}}_n$, the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on $\overline{\mathbb{B}}_n$, then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

Keywords

Primary 47B35

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 257 - 261
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Lewis, A.Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973–1974), 433–439. MR 0322595 (48 #957)Google Scholar
2.Takahiko, Nakazi and Yoneda, Rikio, Compact Toeplitz operators with continuous symbols on weighted Bergman spaces, Glasgow Math. J. 42 (1) (2000), 31–35. MR 1739694 (2000i:47052)Google Scholar
3.Walter, Rudin, Function theory in the unit ball of Cn, vol. 241 (Springer-Verlag, New York, 1980). MR 601594 (82i:32002)Google Scholar
4.Kehe, Zhu, Spaces of holomorphic functions in the unit ball, vol. 226 (Springer-Verlag, New York, 2005). MR 2115155 (2006d:46035)Google Scholar