Reducing subspaces for analytic multipliers of the Bergman space

Abstract

We answer affirmatively the problem left open in \cite{DSZ,GSZZ} and prove that for a finite Blaschke product $\varphi$, the minimal reducing subspaces of the Bergman space multiplier $M_{\varphi }$ are pairwise orthogonal and their number is equal to the number $q$ of connected components of the Riemann surface of ${\varphi }^{-1}\circ \varphi$. In particular, the double commutant $\{M_{\varphi },M_{\varphi }^{\ast }{\}}^{\prime }$ is abelian of dimension $q$. An analytic/arithmetic description of the minimal reducing subspaces of $M_{\varphi }$ is also provided, along with a list of all possible cases in degree of $\varphi$ equal to eight.