Reducing subspaces of a class of multiplication operators
Bull. Korean Math. Soc. 2017 Vol. 54, No. 4, 1443-1455
Published online July 31, 2017
Bin Liu and Yanyue Shi
Ocean University of China, Ocean University of China
Abstract : 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 $M_{z^N}$.
Keywords : multiplication operator, reducing subspace, commutant algebra, unit ball
MSC numbers : Primary 47B35; Secondary 47C15
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