Analytic Toeplitz operators with automorphic symbol

Let $R$ denote the annulus $\{ z:1/2 < |z| < 1\}$ and let $\pi$ be a holomorphic universal covering map from the unit disk onto $R$. It is shown that if $\pi$ is a function of an inner function $\omega$, that is, if $\pi (z) = \pi (\omega (z))$, then $\omega$ is a linear fractional transformation. However, the analytic Toeplitz operator ${T_\pi }$ has nontrivial reducing subspaces. These facts answer in the negative a question raised by Nordgren [10]. Let $\phi$ be the function $\phi (z) = \pi (z) - 3/4$ and let $\phi = \chi F$ be the inner-outer factorization of $\phi$. An operator $C$ is produced which commutes with ${T_\phi }$ but does not commute with ${T_\chi }$ nor with ${T_F}$. This answers in the negative a question raised by Deddens and Wong [7]. The functions $\pi$ and $\phi$ are both automorphic under the group of covering transformations for $\pi$ and hence may be viewed as functions on the annulus $R$. This point of view is critical in these examples.