Multiplication Operators Defined By A Class Of Polynomials

Recently,multiplication operators acting on analytic function spaces,especially on Bergman space,is an important subject that has received considerable attention.Not only it provides a way to understand connections of complex analysis and von Neumann algebras, but also gives a new idea to the solutions of some open problems,such as isomorphism problem in free group factor and the Invariant Subspace Problem,by obtaining equivalent forms of these problems.This paper starts with an introduction to the research background and development process of multiplication operator theory,and then summary the results about the property and structure of the reducing spaces of multiplication operators acting on Bergman spaces and the commutant of the von Neumann algebras generated by it.In this paper,we consider those multiplication operators Mp de=fined by a special class of polynomials p(z,w)=zk+wl on La2(D2).This paper also deals with reducing subspaces of Mp,the von Neumann algebra W*(p)generated by Mp,and its commutant V*(p)(?)W*(p)’. The structure of V*(p)is completely determined,along with those reducing subspaces of Mp.