Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

Abstract

We consider the reducing subspaces of $M_{z^N}$ on $A_{\alpha }^2({\mathbb{D}}^k)$, where $k\ge 3$, $z^N=z_1^{N_1}\cdots z_k^{N_k}$, and $N_i\ne N_j$ for $i\ne j$. We prove that each reducing subspace of $M_{z^N}$ is a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases that $\alpha =0$ and $\alpha \in (-1,+\mathrm{\infty })\setminus \mathbb{Q}$, respectively. Finally, we give a complete description of minimal reducing subspaces of $M_{z^N}$ on $A_{\alpha }^2({\mathbb{D}}^3)$ with $\alpha >-1$.