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Reducing subspaces for a class of multiplication operators on the Dirichlet space


Author: Liankuo Zhao
Journal: Proc. Amer. Math. Soc. 137 (2009), 3091-3097
MSC (2000): Primary 47A15, 46E22; Secondary 47S99.
DOI: https://doi.org/10.1090/S0002-9939-09-09859-1
Published electronically: March 11, 2009
MathSciNet review: 2506467
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Abstract: In this paper, we discuss reducing subspaces of multiplication operators $ M_\phi$ on the Dirichlet space $ \mathcal{D}$ defined by a Blaschke product $ \phi$ with two zeros $ a$, $ b$ in the unit disk $ \mathbb{D}$ and show that when $ a+b=0$, $ M_\phi$ has two proper ones; otherwise it has none. This is different from the cases of the Hardy space and the Bergman space.


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Additional Information

Liankuo Zhao
Affiliation: School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, People’s Republic of China
Email: lkzhao@sxnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-09-09859-1
Keywords: Reducing subspace, multiplication operator, Dirichlet space
Received by editor(s): June 25, 2008
Received by editor(s) in revised form: December 17, 2008
Published electronically: March 11, 2009
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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