Article

# Reducing subspaces for analytic multipliers of the Bergman space

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.

0 0
·
0 Bookmarks
·
127 Views
• Source
• Source
##### Article: Multiplication operators on the Bergman space via analytic continuation
[hide abstract]
ABSTRACT: In this paper, using the group-like property of local inverses of a finite Blaschke product ϕ, we will show that the largest C⁎-algebra in the commutant of the multiplication operator Mϕ by ϕ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk. If the order of the Blaschke product ϕ is less than or equal to eight, then every C⁎-algebra contained in the commutant of Mϕ is abelian and hence the number of minimal reducing subspaces of Mϕ equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk.
• Source
##### Article: Multiplication operators on the Bergman space via the Hardy space of the bidisk
[hide abstract]
ABSTRACT: In this paper, we develop a machinery to study multiplication op-erators on the Bergman space via the Hardy space of the bidisk. We show that only a multiplication operator by a finite Blaschke product has a unique re-ducing subspace on which its restriction is unitarily equivalent to the Bergman shift. Using the machinery we study the structure of reducing subspaces uni-tary equivalence of a multiplication operator on the Bergman space. As a consequence, we completely classify reducing subspaces of the multiplication operator by a Blaschke product φ with order three on the Bergman space to solve a conjecture of Zhu [38] and obtain that the number of minimal reduc-ing subspaces of the multiplication operator equals the number of connected components of the Riemann surface of φ −1 • φ over D.
Journal Fur Die Reine Und Angewandte Mathematik - J REINE ANGEW MATH. 01/2009; 2009(628).