On quasi-similarity and reducing subspaces of multiplication operator on the Fock space

Let F α 2 ( α > 0 ) denote the Fock space which consists of all entire functions f in L 2 ( C , d λ α ) . We prove that the multiplication operator M z n is quasi-similar to ⨁ 1 n M z on F α 2 . Then the reducing subspaces of M z n are characterized on F α 2 .