Extended eigenvalues of composition operators
Let \(H\) be a separable Hilbert space and \(A\) be a nonzero bounded linear operator on \(H\). A \(\lambda \in \mathbb{C}\) is called an extended eigenvalue of \(A\) if there is a nonzero bounded linear operator \(X\) on \(H\) such that \[ AX = \lambda XA. \] The authors describe the set \(Ext(C_\varphi)\) of extended eigenvalues of the composition operator \(C_\varphi\) acting on the Hardy space \(H^2(\mathbb{D})\) in the case when \(\varphi\) is a linear fractional map of the unit disk \(\mathbb{D}\) into itself. In particular, they prove that if \(\varphi\) is a parabolic or hyperbolic automorphism of \(\mathbb{D}\), then \[ Ext(C_\varphi) = \partial \mathbb{D}. \] If \(\varphi\) is an elliptic automorphism of \(\mathbb{D}\) conjugated to the map \(z \rightarrow \omega z, z \in \mathbb{D}, |\omega|=1\), then \[ Ext(C_\varphi) = \{\omega^n : n \in \mathbb{Z}\}. \]
scientific article; zbMATH DE number 7389220
scientific article