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).
 Author Name Affiliation Bian REN School of Mathematic Sciences, Ocean University of China, Shandong 266100, P. R. China Yanyue SHI School of Mathematic Sciences, Ocean University of China, Shandong 266100, P. R. China
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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.