On Quasi-similarity And Reducing Subspaces Of Multiplication Operator On The Fock Space

The research of the commutant and reducing subspaces of operators is an interesting problem in operator theory. From the information of the commutant, people study the similar equivalence and unitary equivalence of operators. Toeplitz operator is a class of concrete operator. There were some results on the similarity and reducing subspaces problem on Toeplitz operators on the Hardy space and the Bergman space. In this paper, we will investigate the situation on the Fock space.Let F2α(α＞0)denote the Fock space which consists of all entire functions f in L2(C, dλα). In this paper, first we prove that multiplication operator Mzn is quasi-similar to (?) Mz on the Fock space. Similarly, we obtain that multiplication operator M(z-a)n, M(z+a)n and M(a-z)n are also quasi-similar to (?) Mz. Then we characterize the reducing subspaces of Mzn, by directly analyzing the commutant of operator, it is shown that Mzn has exactly2n reducing subspaces and n minimal reducing subspaces on the Fock space.The main structure of this paper is:In the first section, we introduce the concept of the Fock space, quasi-similarity and reducing subspaces. In addition, we list Stoltz’s theorem which is needed in the process.In the second section, first we prove that multiplication operator Mzn is quasi-similar to0Mz on the Fock space. Similarly, we obtain that multiplication operator M(z-a)n, M(z-a)n and M(a-z)n are also quasi-similar to (?) MzIn the third section, with the help of projection operators, we characterize the reducing subspaces of Mzn, by directly analyzing the commutant of operator, it is shown that Mzn has exactly2n reducing subspaces and n minimal reducing subspaces on the Fock space.