Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2017; 54(4): 1443-1455

Online first article May 25, 2017      Printed July 31, 2017

https://doi.org/10.4134/BKMS.b160618

Copyright © The Korean Mathematical Society.

Reducing subspaces of a class of multiplication operators

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