Reducing Subspaces Of Multiplication Operators Induced By A Class Of Multivariate Polynomials

In recent years,the study of reduced subspace of multiplication operators on the the space of general analytic functions has been an important subject of great concern and some remarkable achievements have been made.These results also realize the combination of analysis,algebra,and group theory.In this paper,the weighted sequence space is taken as the main space research object,and the reduced subspace problem of multiplication operators induced by a class of multivariate multinomial is mainly discussed.The main contents are as follows.(1)In this paper,the concept of stability in the theory of graded modules established in reference[40]is extended to define double stability,and the relation between minimal and double stability is given.On this basis,it is proved that the graded module induced by Toeplitz operator Tz+w-has double stability structure.(2)The kernel method is established in the theory of graded modules,and some basic properties of the method are discussed.As its application,the condition of unitary equivalence of multiplication operators Mz in different weighted sequence spaces is first described.Secondly,a special weighted sequence space is defined,that is,proportion sequence space,on which the von Neumann algebra generated by multiplication operators Mz+w is obtained.(3)The concept of super-shift dilation operators is given,and it is proved that a class of weighted shift operators in weighted sequence space is minimal super-shift dilation,and give two cases in which the multiplication operator Mz can super-shift dilation.