## Banach Journal of Mathematical Analysis

### Reducing subspaces for a class of nonanalytic Toeplitz operators

#### Abstract

In this paper, we give a uniform characterization for the reducing subspaces for $T_{\varphi}$ with the symbol $\varphi(z)=z^{k}+\bar{z}^{l}$ ($k,l\in\mathbb{Z}_{+}^{2}$) on the Bergman spaces over the bidisk, including the known cases that $\varphi(z_{1},z_{2})=z_{1}^{N}z_{2}^{M}$ and $\varphi(z_{1},z_{2})=z_{1}^{N}+\overline{z}_{2}^{M}$ with $N,M\in\mathbb{Z}_{+}$. Meanwhile, the reducing subspaces for $T_{z^{N}+\overline{z}^{M}}$ ($N,M\in \mathbb{Z}_{+}$) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra $\mathcal{V}^{*}(\varphi)$.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 456-480.

Dates
Accepted: 29 July 2017
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.bjma/1513674118

Digital Object Identifier
doi:10.1215/17358787-2017-0035

Mathematical Reviews number (MathSciNet)
MR3779723

Zentralblatt MATH identifier
06873510

#### Citation

Deng, Jia; Lu, Yufeng; Shi, Yanyue; Hu, Yinyin. Reducing subspaces for a class of nonanalytic Toeplitz operators. Banach J. Math. Anal. 12 (2018), no. 2, 456--480. doi:10.1215/17358787-2017-0035. https://projecteuclid.org/euclid.bjma/1513674118

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