Analytic functions of finite valence, with applications to Toeplitz operators (Q1071128)

For f analytic on \({\mathcal U}=\{| z| <1\}\), the valence function is defined by \(\nu_ f(w)=card[f^{-1}\{w\}\cap {\mathcal U}]\). \textit{I. N. Baker}, \textit{J. A. Deddens}, and \textit{J. L. Ullman} [Duke Math. J 41, 739-745 (1974; Zbl 0296.30020)] proved that if f is (the restriction of) an entire function and k is the smallest non-zero value of \(\nu_ f\), then \(f(z)\equiv h(z^ k)\) for some entire function h. They asked whether an appropriate analogue holds for functions which are not entire, with the role of \(z^ k\) played by a k-fold Blaschke product. This paper constructs an example showing the answer is no. It goes on to study pairs f and g whose valence functions \(\nu_ f\) and \(\nu_ g\) are related. For instance, if \(\nu_ f\equiv \nu_ g<M<\infty\), do f and g arise from a common function by some type of composition ? Yes, if f and g are entire; no, in general. The paper answers some questions concerning commutants and similarity of analytic Toeplitz operators. It also provides a counterexample to a result of \textit{D. N. Clark} [Contributions to analysis and geometry, Suppl. Am. J. Math., 63-72 (1981; Zbl 0578.47016)] on similarity of rational and analytic Toeplitz operators. A suggestion is made as to what underlies the difficulty there.