Unitary equivalence of one-parameter groups of Toeplitz and composition operators
type: Item
instance of: scholarly article
title: Unitary equivalence of one-parameter groups of Toeplitz and composition operators
zbMATH Open document ID: 1242.47022
author: Carl C. Cowen
author: Eva A. Gallardo Gutiérrez
published in: Journal of Functional Analysis
publication date: 2011-11-24T00:00:00Z
review text: 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…
reviewed by: Trieu Le
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B33
zbMATH DE Number: 5978464
zbMATH Keywords: composition operator
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: subnormal operator
zbMATH Keywords: unitary equivalence
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: j.jfa.2011.07.005
OpenAlex ID: W1992197811
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DOI: 10.1016/J.JFA.2011.07.005
description: scientific article; zbMATH DE number 5978464