双圆盘加权Hardy空间上$T_{z_1^{k_1}z_2^{k_2}+\bar{z}_1^{l_1}\bar{z}_2^{l_2}}$的约化子空间

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

中文关键词:  约化子空间  加权 Dirichlet 空间  换位代数

英文关键词:reducing subspaces  weighted Dirichlet space  commutant algebra

中文摘要:

      本文中, 我们主要刻画了Toeplitz算子$T=M_{z^k}+M^*_{z^l}$的约化子空间, 其中 $k_i, l_i$ ($i=1,2$) 均是正整数, $k=(k_1,k_2), l=(l_1,l_2)$ 且 $k\neq l$, $M_{z^k}$, $M_{z^l}$ 是双圆盘加权Hardy空间$\mathcal{H}_\omega^2(\mathbb{D}^2)$上的乘法算子. 对权系数 $\omega$ 适当限制, 我们证明了由 $z^m$ 生成的 $T$ 的约化子空间均是极小的. 特别地, Bergman 空间和加权 Dirichlet 空间 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 均是满足该限制条件的加权Hardy空间. 作为应用, 我们刻画了 $\mathcal{D}_\delta(\mathbb{D}^2)(\delta>0)$ 上 Toeplitz 算子 $T_{z^k+\bar{z}^l}$ 的约化子空间, 该结论是对双圆盘Bergman 空间上相关结论的推广.

英文摘要:

      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.