The Hardy class of geometric models and the essential spectral radius of composition operators
Let \(\phi\) be an analytic map of the unit disk \(\mathbb{D}\) into itself with \(\phi (0)=0\) and \(\phi '(0)\neq 0\). Then there exists an analytic map \(\sigma\) (the Kœnigs function of \(\phi\)) defined on \(\mathbb{D}\) such that \(\sigma \circ \phi = \lambda \sigma\), where \(\lambda = \phi '(0)\). Also, the map \(\phi\) induces a bounded composition operator on the Hardy space \(H^2(\mathbb{D} )\), given by the formula \(C_{\phi}(f)= f\circ \phi\). If \(\mathcal K\) is the subspace of all compact operators on \(H^2(\mathbb{D} )\), then the spectral radius of the projection of \(C_{\phi}\) in \(H^2(\mathbb{D} )/\mathcal K \) is called the essential spectral radius of \(C_{\phi}\) and denoted by \(r_2(C_{\phi})\). Recently, P.Bourdon and J.Shapiro found a lower bound for the essential spectral radius of \(C_{\phi}\) in terms of \(\lambda\) and \(h(\sigma)=\sup \{p>0: \sigma\in H^p (\mathbb{D}) \}\). The author shows, in the case when \(\phi\) is one-to-one, that the lower bound mentioned before is also an upper bound and obtains for \(r_2(C_{\phi})\) an exact formula \(r_2(C_{\phi})=|\lambda|^{h(\sigma)/2}\). Moreover, he produces a new characterization of the Hardy number \(h(\sigma)\) of \(\sigma\) in terms of the geometry of the image set \(G=\sigma(\mathbb{D})\).
scientific article; zbMATH DE number 997637