# Reducing subspaces for analytic multipliers of the Bergman space

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Kai Wang, Apr 28, 2015 Available from:### Click to see the full-text of:

Article: Reducing subspaces for analytic multipliers of the Bergman space

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- "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. "

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**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

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**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 - [Show abstract] [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