Function Spaces and Their Operators
A Conference in Honor of Richard Rochberg on the Occasion of his 65th Birthday

Dechao Zheng from Vanderbilt University; Multiplication operators on the Bergman space, analytic continuation and Riemann surfaces; May 29, 2008.

Abstract: In this talk, I will present the recent joint work with R. Douglas and S. Sun on multiplication operators by finite Blaschke products on the Bergman space of the unit disk.Using the group-like property of local inverses of a a finite Blaschke product $\phi$, we show that any von Neumann algebra contained in the commutant of the multiplication operator by $\phi$ on the Bergman space is abelian and finite dimensional. We show that the number of nontrivial minimal reducing subspaces of $M_{\phi}$ equals the number of connected components of the Riemann surface of $\phi^{-1}\circ\phi $ over the unit disk and any two distinct minimal reducing subspaces of $M_{\phi}$ are not unitarily equivalent.