Multiplication operators on the Bergman space of bounded domains in C^d

Abstract

In this paper we study multiplication operators on Bergman spaces of high dimensional bounded domains and those von Neumann algebras induced by them via the geometry of domains and function theory of their symbols. In particular, using local inverses and $L^2_a$-removability, we show that for a holomorphic proper map $\Phi =(\phi_1, \phi_2, \cdots , \phi_d)$ on a bounded domain $\Omega$ in $\mathbb{C}^{d}$, the dimension of the von Neumann algebra $\mathcal{V}^*(\Phi ,\Omega) $ consisting of bounded operators on the Bergman space $L_a^2(\Omega)$, which commute with both $ M_{\phi_j}$ and its adjoint $M_{\phi_j}^*$ for each $j$, equals the number of components of the complex manifold $\mathcal{S}_{\Phi }= \{(z,w)\in \Omega^2: \Phi (z)=\Phi (w),\, z\not\in \Phi ^{-1}(\Phi (Z))\},$ where $Z$ is the zero variety of the Jacobian $J\Phi $ of $ \Phi.$ This extends the main result in \cite{DSZ} in high dimensional complex domains. Moreover we show that the von Neumann algebra $\mathcal{V}^*(\Phi ,\Omega) $ may not be abelian in general although Douglas, Putinar and Wang \cite{DPW} showed that $\mathcal{V}^*(\Phi ,\mathbb{D})$ for the unit disk $\mathbb{D}$ is abelian.