Polynomial hulls with convex fibers
Let Y be a compact set in \({\mathbb{C}}^ n\) and denote by \(\hat Y\) the polynomial hull of Y, i.e. the set of those points in \({\mathbb{C}}^ n\) at which the modulus of each polynomial on \({\mathbb{C}}^ n\) is dominated by its maximum over Y. Except in special cases it is not known how to calculate the points of \(\hat Y\), given Y. In this paper a class of sets Y in \({\mathbb{C}}^ 2\) is presented for which a satisfactory description of \(\hat Y\) is possible. For each \(\lambda\) in \({\mathbb{C}}\) one puts \(Y_{\lambda}=\{w|(\lambda,w)\in Y\}\) and \({\hat Y}_{\lambda}\) is defined similarly. \(\Gamma\) denotes the unit circle. The first result is the following: assume that \(Y\subset \{(\lambda,w)| \lambda \in \Gamma \}\) and that \(Y_{\lambda}\) is convex for each \(\lambda\in \Gamma\). Then \(\hat Y\) consists of Y together with the union of the graphs \(w=\phi(\lambda)\) of all bounded analytic functions on \(| \lambda |<1\) with \(\phi(\lambda)\in Y_{\lambda}\) for almost all \(\lambda\) in \(\Gamma\). In the particular case that each \(Y_{\lambda}\), \(\lambda\in \Gamma\), is a disk: \(Y_{\lambda}=\{w| | w- \alpha(\lambda)| \leq R(\lambda)\}\), where \(\alpha\) is a given complex-valued function continuous on \(\Gamma\) and \(R>0\) and \(\in C^ 2(\Gamma)\), the following is shown: Assuming that \(| \alpha(\lambda)| \leq R(\lambda)\), \(\lambda\in \Gamma\), and that there is a b, \(| b|<1\), such that \({\hat Y}_ b\) contains more than one point, there exist analytic functions A,B,C,D on \(| \lambda |<1\) such that \(\hat Y\cap \{| \lambda |<1\}=\{(\lambda,w)| |(A(\lambda)w+C(\lambda))(B(\lambda)w+D(\lambda))^{-1}| \leq 1,| \lambda |<1\}.\) A,B,C,D are calculated from \(\alpha\) and R. The method of proof makes key use of work of \textit{J. Garnett} [''Bounded analytic functions'' (1981; Zbl 0469.30024); Chapter 4].
scientific article; zbMATH DE number 3855484