Reducing Subspaces of Analytic Toeplitz Operators on the Bergman Space of the Annulus

Abstract

Let $\mathbb {A}_{r}$ be the annulus $\{z\mid \,|z|<r<1\}$ in the complex plane, $L_{a}^{2}(\mathbb {A}_{r})$ be the Bergman space on $\mathbb {A}_{r}$, B be a finite Blaschke product $B(z)=e^{i\theta }\prod \nolimits _{i=1}^{N}\frac{z-\alpha _{i}}{1-\overline{\alpha _{i}}z}$ with $|\alpha _{i}|<r$ for $1\le i\le N$. In this case, local inverses of B on $\mathbb {A}_{r}$ consist of a cyclic group with order N. It is shown that there is an one-to-one correspondence between a minimal reducing subspace of the Toeplitz operator $T_{B}$ on $L_{a}^{2}(\mathbb {A}_{r})$ and a character of the cyclic group, reducing subspaces of Toeplitz operators are studied from an algebraic point of view and Douglas and Kim’s result is generalized.