Reducing subspaces for analytic multipliers of the Bergman space

Journal of Functional Analysis (Impact Factor: 1.32). 10/2011; 263(6). DOI: 10.1016/j.jfa.2012.06.008
Source: arXiv


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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Available from: Kai Wang, Apr 28, 2015
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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.
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    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. © 2012 Science China Press and Springer-Verlag Berlin Heidelberg.
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    ABSTRACT: In this paper, we mainly study geometric constructions of thin Blaschke products $B$ and reducing subspace problem of multiplication operators induced by such symbols $B$ on the Bergman space. Considering such multiplication operators $M_B$, we present a representation of those operators commuting with both $M_B$ and $M_B^*$. It is shown that for "most" thin Blaschke products $B$, $M_B$ is irreducible, i.e. $M_B$ has no nontrivial reducing subspace; and such a thin Blaschke product $B$ is constructed. As an application of the methods, it is proved that for "most" finite Blaschke products $\phi$, $M_\phi$ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for when $M_B$ defined by a thin Blaschke product $B$ has a nontrivial reducing subspace.
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