Reducing Subspaces for $T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$ on Weighted Hardy Space over Bidisk 
Received:October 14, 2020 Revised:January 28, 2021 
Key Word:
reducing subspaces weighted Dirichlet space commutant algebra

Fund ProjectL:Supported by Fundamental Research Funds for the Central Universities (Grant No.201964007) and the National Natural Science Foundation of China (Grant Nos.11701537; 12071253). 

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Abstract: 
In this paper, we characterize the reducing subspaces for Toeplitz operator $T=M_{z^k}+M^*_{z^l}$, where $M_{z^k}$, $M_{z^l}$ are the multiplication operators on weighted Hardy space $\mathcal{H}_\omega^2(\mathbb{D}^2)$, $k=(k_1,k_2)$, $l=(l_1,l_2)$, $k\neq l$ and $k_i, l_i$ are positive integers for $i=1,2$. It is proved that the reducing subspace for $T$ generated by $z^m$ is minimal under proper assumptions on $\omega$. The Bergman space and weighted Dirichlet spaces $\mathcal{D}_\delta(\mathbb{D}^2)~(\delta>0)$ are weighted Hardy spaces which satisfy these assumptions. As an application, we describe the reducing subspaces for $T_{z^k+\bar{z}^l}$ on $\mathcal{D}_\delta(\mathbb{D}^2)~(\delta>0)$, which generalized the results on Bergman space over bidisk. 
Citation: 
DOI:10.3770/j.issn:20952651.2021.05.007 
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