Reducing subspaces of a class of multiplication operators

Let $M_{z^N}$($N\in \mathbb{Z}_+^d$) be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n: n\in \mathbb{Z}_+^d\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that $d=2$, we find all the minimal reducing subspaces of $M_{z^N}(N=(N_1,N_2), N_1\neq N_2)$ on weighted Bergman space $A_\alpha^2(\mathbb{B}_2)(\alpha>-1)$ and Hardy space $H^2(\mathbb{B}_2)$, and characterize the structure of $\mathcal{V}^*(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$. KCI Citation Count: 0