When is zero in the numerical range of a composition operator?
Let \(H^2=\{f(z)=\sum_{n=0}^\infty \widehat f(n)z^n:\sum_{n=0}^\infty| \widehat f(n)| ^2<\infty\} \) be the usual Hardy space on the unit disk \(D\). With respect to the inner product \(\langle f,g\rangle =\sum_{n=0}^\infty\widehat f(n)\overline{\widehat g(n)}\), the space \(H^2\) becomes a Hilbert space. The numerical range of an operator \(T\) on a Hilbert space \(H\) is the set \(W(T)=\{\langle Tf,f\rangle : f\in H\), \(\| f\| = 1\}\). In this paper, the authors study the numerical range of the composition operators \(T=C_\phi: f\mapsto f\circ \phi\) on \(H^2\), where \(\phi\) is a holomorphic selfmap of the unit disk \(D\). Previous results on this subject have been obtained by \textit{V. Matache} [Linear Algebra Appl. 331, 61--74 (2001; Zbl 0990.47022)]. In the paper under review, the authors determine all those \(\phi\), fixing a point \(p\) in \(D\), for which \(0\notin W(C_\phi)\). In fact, these are exactly those functions for which there is an \(r\in {}]0,1[\) such that \(| p| \leq \sqrt r\) and \(\phi(z)=\alpha_p(r\alpha_p(z))\), where \(\alpha_p(z)=(p-z)/(1-\overline p z)\). Several results on when \(0\) is an interior point of \(W(C_\phi)\) are obtained, too. For example, if \(\phi\) is a noninjective, nonconstant map of \(D\) into \(D\), then \(0\in W(C_\phi)^o\). For compact composition operators, the authors obtain a complete description of the corner points of the closure of \(W(C_\phi)\). Section 6 is devoted to the study of those composition operators on \(H^2\) that are sectorial, meaning that the numerical range is contained in a sector of the form \(\{z:| \arg z| <\theta\} \) for some positive acute angle \(\theta\). The writing in this paper is a pleasure to read, and this publication serves as a model of mathematical exposition.
scientific article; zbMATH DE number 1890560
scientific article