Reducing subspaces for analytic multipliers of the Bergman space

TL;DR

This paper characterizes the minimal reducing subspaces of Bergman space multipliers induced by finite Blaschke products, linking their structure to the topology of the associated Riemann surface and providing explicit descriptions.

Contribution

It proves that minimal reducing subspaces are orthogonal and correspond to connected components of the Riemann surface, extending previous open problems.

Findings

Number of minimal reducing subspaces equals the number of connected components of the Riemann surface.

The double commutant of the multiplier algebra is abelian with dimension equal to that number.

Provides an explicit arithmetic description of these subspaces and classifies cases for degree eight.

Abstract

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