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Complex Analysis and Operator Theory

, Volume 13, Issue 8, pp 4195–4206 | Cite as

Reducing Subspaces of Analytic Toeplitz Operators on the Bergman Space of the Annulus

  • Anjian XuEmail author
Article
  • 115 Downloads

Abstract

Let \(\mathbb {A}_{r}\) be the annulus \(\{z\mid \,|z|<r<1\}\) in the complex plane, \(L_{a}^{2}(\mathbb {A}_{r})\) be the Bergman space on \(\mathbb {A}_{r}\), B be a finite Blaschke product \(B(z)=e^{i\theta }\prod \nolimits _{i=1}^{N}\frac{z-\alpha _{i}}{1-\overline{\alpha _{i}}z}\) with \(|\alpha _{i}|<r\) for \(1\le i\le N\). In this case, local inverses of B on \(\mathbb {A}_{r}\) consist of a cyclic group with order N. It is shown that there is an one-to-one correspondence between a minimal reducing subspace of the Toeplitz operator \(T_{B}\) on \(L_{a}^{2}(\mathbb {A}_{r})\) and a character of the cyclic group, reducing subspaces of Toeplitz operators are studied from an algebraic point of view and Douglas and Kim’s result is generalized.

Keywords

Toeplitz operator Bergman space Blaschke product Annulus 

Mathematics Subject Classification

Primary 47B35 46B32 Secondary 05A38 15A15 

Notes

Acknowledgements

Thanks for the referee’s important suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing University of TechnologyChongqingChina

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