Reducing subspaces for analytic multipliers of the Bergman space

Journal of Functional Analysis (Impact Factor: 1.15). 10/2011; DOI: 10.1016/j.jfa.2012.06.008
Source: arXiv

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.

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