Analytic functions of finite valence, with applications to Toeplitz operators
type: Item
instance of: scholarly article
title: Analytic functions of finite valence, with applications to Toeplitz operators
zbMATH Open document ID: 0585.30031
author: Kenneth Stephenson
published in: Michigan Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 3937496
zbMATH Keywords: Blaschke product
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003127
OpenAlex ID: W1966341053
DOI: 10.1307/MMJ/1029003127
description: scientific article; zbMATH DE number 3937496