Analytic Toeplitz operators with automorphic symbol

Abstract

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