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.