Reducing subspaces for analytic multipliers of the Bergman space

In Douglas et al. (2011) [4], some incisive results are obtained on the structure of the reducing subspaces for the multiplication operator Mϕ by a finite Blaschke product ϕ on the Bergman space on the unit disk. In particular, the linear dimension of the commutant, , is shown to equal the number of connected components of the Riemann surface, ϕ−1∘ϕ. Using techniques from Douglas et al. (2011) [4] and a uniformization result that expresses ϕ as a holomorphic covering map in a neighborhood of the boundary of the disk, we prove that Aϕ is commutative, and moreover, that the minimal reducing subspaces are pairwise orthogonal. Finally, an analytic/arithmetic description of the minimal reducing subspaces is also provided, along with the taxonomy of the possible structures of the reducing subspaces in case ϕ has eight zeros. These results have implications in both operator theory and the geometry of finite Blaschke products.