Components of linear-fractional composition operators.
Let \(\phi\) denote a holomorphic function on the open unit disc \(U\) in the complex plane with \(\phi(U) \subset U\). The composition operator \(C_{\phi}(f):=f \circ \phi\) is bounded in the Hardy space \(H^2\). Denote by \(\text{comp} (H^2)\) the collection of all composition operators on \(H^2\) endowed with the metric induced by the operator norm. \textit{J. H. Shapiro} and \textit{C. Sundberg} [Pac. J. Math. 145, 117--121 (1990; Zbl 0732.30027)], improving work by Berkson, studied isolated points in \(\text{comp} (H^2)\) and showed, e.g., that only extreme points of \(H^{\infty}(U)\) can induce isolated composition operators. They also conjectured that if two composition operators \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(C_{\phi} - C_{\psi}\) is compact. Two analytic self-maps \(\phi\) and \(\psi\) are said to have the same first order boundary data if, whenever \(\phi\) or \(\psi\) has a finite angular derivative at \(\zeta\) in the boundary of \(U\), then \(\phi\) and \(\psi\) have the same radial limit and the same angular derivative at \(\zeta\). \textit{B. D. MacCluer} [Integral Equations Oper. Theory 12, 725--738 (1989; Zbl 0685.47027)] showed that if \(C_{\phi}\) and \(C_{\psi}\) belong to the same component of \(\text{comp} (H^2)\), then \(\phi\) and \(\psi\) have the same first order boundary data. The author proves the converse, if both \(\phi\) and \(\psi\) are linear fractional, and even that \(C_{\phi}\) and \(C_{\psi}\) can be joined by a continuous path in \(\text{comp} (H^2)\). In the final section, the author proves that, for analytic self-maps \(\phi\) and \(\psi\) which extend to be \(C^2\) on the closed unit disc, a necessary condition for \(C_{\phi}- C_{\psi}\) to be compact is that \(\phi\) and \(\psi\) have the same second order boundary data, thus obtaining an example disproving the conjecture of Shapiro and Sundberg. He also shows that the component of a non-automorphic, linear fractional self-map \(\phi\) on \(U\) always contains composition operators whose symbols are not linear fractional. In a remark, the author points out that \textit{J.~Moorhouse} and \textit{C.~Toews} [Contemp. Math. 321, 207-213 (2003; Zbl 1052.47018)] were the first to disprove the conjecture of Shapiro and Sundberg.
scientific article; zbMATH DE number 1915745
scientific article