The numerical ranges of automorphic composition operators
Let \(H^2\) be the Hardy space of functions holomorphic on the unit open disc \(\mathbb{U}\). Let \(\varphi\) be a holomorphic self-map of \(\mathbb{U}\). Let \(C_\varphi\) be the bounded composition operator induced by \(\varphi\), the numerical range of \(C_\varphi\) is denoted by \(W(C_\varphi)\). Some important results established in this paper include the following: Result 1: Suppose \(\varphi\) is a conformal automorphism of \(\mathbb{V}\) that is either parabolic or hyperbolic. Then \(W(C_\varphi)\) is a disc centered at the origin. Result 2: If \(\varphi\) is a canonical hyperbolic automorphism of \(\mathbb{U}\) with dilation parameter \(0< r< 1\), then \(W(C_\varphi)\) is the open disc of radius \(1/\sqrt{r}\) centered at the origin. Result 3: Suppose that \(\varphi\) is an elliptic automorphism of \(\mathbb{U}\) with rotation parameter \(\omega\). If \(\omega\) is not a root of unity, then \(\overline W(C_\varphi)\) is a disc centered at the origin.
parabolic, elliptic and hyperbolic automorphisms
scientific article; zbMATH DE number 1556947