Multiplication operators on the Bergman space via analytic continuation

2009/01/23 by Douglas, Ronald G., Sun, Shunhua, Zheng, Dechao
#46E20 #47B35 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0901.3787

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∘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 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∘ϕ over the unit disk.

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