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 24, 2021
Key Word: Reducing subspaces   Weighted Dirichlet space   Commutant algebra.
Fund ProjectL:Fundamental Research Funds for the Central Universities;the National Natural Science Foundation of China
 Author Name Affiliation Address Yanyue Shi School of Mathematical Sciences in Ocean University of China 青岛市崂山区松岭路238号中国海洋大学数学科学学院 Bian Ren School of Mathematical Sciences in Ocean University of China 青岛市崂山区松岭路238号中国海洋大学数学科学学院
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In this paper, we characterize the reducing subspaces for Toeplitz operator $T_{z^k+\bar{z}^l}=M_{z^k}+M^*_{z^l}$, where $M_{z^k}$, $M_{z^l}$ are the multiplication operators on weighted Hardy space $H_\omega^2(\mathbb{D}^2)$, $k=(k_1,k_2), l=(l_1,l_2)$, $k\neq l$ and $k_i, l_i$ are positive integers for $i=1,2$. It is proved that the reducing subspace generated by $z^m$ is minimal under proper assumptions. 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 of $T_{z^k+\bar{z}^l}$ on $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$, which generalized the results on Bergman space over bidisk.