Thomson-type 定理 for the multiplication operators on the Bergman space of the annulus

Abstract

Thomson's theorem implies that on the Bergman space over the unit disk if h is holomorphic on the closed unit disk, then there is a finite Blaschke product B such that h can be written as a function of B, and the commutant of the multiplication operator M h by h equals that of MB. This is essentially generalized to the Bergman space over an annulus under a mild condition. It is also seen that the situation is complicated compared with the classical Bergman space over the unit disk. We also consider the associated reducing subspaces of the concerned multiplication operators.