| 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. |
