Unitary equivalence of one-parameter groups of Toeplitz and composition operators
The paper under review investigates unitary equivalence and similarity of two one-parameter groups of operators on the Hardy space \(H^2\) over the unit disk. The first group consists of composition operators \(C_{\varphi_t}\) for \(-\infty<t<\infty\), where each \[ \varphi_t(z) = \frac{(1+e^{-t})z+(1-e^{-t})}{(1-e^{-t})z+(1+e^{-t})} \] is a hyperbolic automorphism of the disk with fixed points \(\pm 1\). These composition operators form a strongly continuous one-parameter group, i.e., \(C_{\varphi_t}C_{\varphi_s} = C_{\varphi_{s+t}}\). The second group consists of Toeplitz operators \(T_{\overline{\psi}_t}\), where \[ \psi_{t}(z) = \Big(\frac{1-z}{1+z}\Big)^{ti/\pi}. \] These operators (which are, in fact, adjoints of the analytic Toeplitz operators \(T_{\psi_t}\)) also form a strongly continuous one-parameter group: \(T_{\overline{\psi}_t}T_{\overline{\psi}_s}=T_{\overline{\psi}_{s+t}}\). The authors first compute the infinitesimal generators of these two groups and their eigenvalues and eigenvectors. They show that all the eigenspaces are one-dimensional. They then use this fact, together with other properties of the eigenvectors, to establish connections between the composition and the Toeplitz operators. The main results in the paper are the following: (1) There is a bounded operator \(S\) from \(H^2\) onto itself that gives a similarity of \(\{C_{\varphi_{t}}\}_{t\in\mathbb{R}}\) and \(\{T_{\overline{\psi}_{t}}\}_{t\in\mathbb{R}}\). That is, for each real number \(t\), we have \(S^{-1}C_{\varphi_t}S = T_{\overline{\psi}_t}\). (2) There is \textit{no} unitary operator \(U\) on \(H^2\) such that \(U^{*}C_{\varphi_t}U=T_{\overline{\psi}_t}\) for all real numbers \(t\). (3) There is a unitary operator \(U\) of \(zH^2\) onto itself that gives a unitary equivalence of the one-parameter groups \(\{C^{*}_{\varphi_{t}}|_{zH^2}\}_{t\in\mathbb{R}}\) and \(\{T_{\psi_{t}}|_{zH^2}\}_{t\in\mathbb{R}}\). That is, for each real number \(t\), we have \(UC^{*}_{\varphi_t}|_{zH^2}=T_{\psi_t}|_{zH^2}U\). The paper is very well written and it is quite a pleasure to read. Even though it concerns only the above concrete one-parameter groups of operators, the paper contains several interesting ideas that may be useful in more general settings.
scientific article; zbMATH DE number 5978464