Multiplication operators on the Bergman space via analytic continuation

Douglas, Ronald G.; Sun, Shunhua; Zheng, Dechao

Multiplication operators on the Bergman space via analytic continuation

Authors

Douglas, Ronald G.

Sun, Shunhua

Zheng, Dechao

Description

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.