Article

# Reducing subspaces for analytic multipliers of the Bergman space

(Impact Factor: 1.15). 10/2011; 263(6). 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.

1 Follower
·
211 Views
• Source
• "Representing T B as a bundle shift allows us to recover most of the results in [9],[10] except for two key ones: the fact that (W * (T B )) ′ is abelian and its linear dimension. A more careful analysis of the covering group associated to the Riemann surface {(z 1 , z 2 ) : B(z 1 ) = B(z 2 )} for B will be required for that. "
##### Article: Generalized Bundle Shift with Application to Multiplication operator on the Bergman space
[Hide abstract]
ABSTRACT: Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts that the number of the minimal reducing subspaces equals the number of connected components of the Riemann surface $B(z)=B(w)$ and the algebra of commutant of $T_{B}$ is commutative, are not proved. Moreover, the role of the Riemann surface is not made clear also.
• ##### Article: Reducibility and Unitarily Equivalence for a Class of Analytic Multipliers on the Dirichlet Space
[Hide abstract]
ABSTRACT: In this paper, we first prove that if $\phi$ is a finite Blaschke product with $N=2,3$ zeros, then $M_\phi$ is reducible on the Dirichlet space if and only if $\phi$ is equivalent to $z^N$ . Also, we prove that $M_\phi$ is unitary equivalent to Dirichlet shift of multiplicity $N$ if and only if $\phi =\lambda z^N$ for some unimodular constant $\lambda$ .
Complex Analysis and Operator Theory 01/2013; 7(6). DOI:10.1007/s11785-012-0267-1 · 0.52 Impact Factor
• Source
##### Article: von Neumann algebras generated by multiplication operators on the weighted Bergman space: a function-theory view into operator theory
[Hide abstract]
ABSTRACT: Recently, a class of Type II factors has been constructed, arising from holomorphic coverings of bounded planar domains. Those operators in Type II factors act on the Bergman space. In this paper, we develop new techniques to generalize those results to the case of the weighted Bergman spaces. In addition, a class of group-like von Neumann algebras are constructed, which are shown to be *-isomorphic to the group von Neumann algebras.
Science China Mathematics 04/2013; 56(4). DOI:10.1007/s11425-012-4440-9 · 0.71 Impact Factor