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arXiv · 0901.3787

Multiplication operators on the Bergman space via analytic continuation

Ronald G. Douglas·Shunhua Sun·Dechao Zheng
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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 $M_ϕ$ by $ϕ$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ 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 $M_ϕ$ is abelian and hence the number of minimal reducing subspaces of $M_ϕ$ equals the number of connected components of the Riemann surface of $ϕ^{-1}\circϕ$ over the unit disk.

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Ronald G. Douglas, Shunhua Sun, Dechao Zheng. 2009-01-23. Multiplication operators on the Bergman space via analytic continuation. https://arxiv.org/abs/0901.3787

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