Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces

This paper mainly concerns a tuple of multiplication operatorsdefined on the weighted and unweighted multi-variable Bergmanspaces, their joint reducing subspaces and the von Neumann algebragenerated by the orthogonal projections onto these subspaces. It isfound that the weights play an important role in the structures oflattices of joint reducing subspaces and of associated von Neumannalgebras. Also, a class of special weights is taken into account.Under a mild condition it is proved that if those multiplicationoperators are defined by the same symbols, then the correspondingvon Neumann algebras are $*$-isomorphic to the one defined on theunweighted Bergman space.