Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces

Abstract

Abstract This paper gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].