The spectra of composition operators from linear fractional maps acting upon the Dirichlet space
type: Item
instance of: scholarly article
title: The spectra of composition operators from linear fractional maps acting upon the Dirichlet space
zbMATH Open document ID: 1072.47020
author: William M. Higdon
published in: Journal of Functional Analysis
publication date: 2005-03-08T00:00:00Z
review text: Let \(H(\mathbb{D})\) denote the space of all analytic functions in the open unit disk equipped with the topology of ``uniform converge on compact subsets''. The Dirichlet space, denoted by \(\mathcal{D}\), consists of all \(f\in H(\mathbb{D})\) for which \(\int_{\mathbb{D}} | f'| ^2 dA < \infty\), where \(dA\) is the Lebesgue area measure. The norm for \(\mathcal{D}\) is defined by \[ \| f\| ^2_{\mathcal{D}}=| f(0)| ^2+\frac{1}{\pi} \int_{\mathbb{D}} | f'| ^2 dA ,\quad f \in {\mathcal{D}}. \] \textit{E. A. Nordgren} determined the spectrum of the composition operator \(C_{\varphi}\) on the Hardy space \(H^2\) induced from an automorphism \(\varphi\) of \(\mathbb{D}\) [Can. J. Math. 20, 442--449 (1968; Zbl 0161.34703)]. The present author determines the spectrum of \(C_{\varphi}\) as an operator on the Dirichlet space when \(\varphi\) is an automorphism, and, more generally, a linear fractional map on \(\mathbb{D}\). Although eigenfunctions for the operator \(C_{\varphi}\) on \(H^2\) frequently do not belong to the space \(\mathcal{D}\), spectral results hold for the operator \(C_{\varphi}\) on \(\mathcal{D}\) much like those already known for the operator \(C_{\varphi}\) on \(H^2\).
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 2141636
zbMATH Keywords: Dirichlet space
zbMATH Keywords: composition operator
zbMATH Keywords: spectrum
zbMATH Keywords: eigenvalue
reviewed by: Juexian Li
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.10.015
OpenAlex ID: W1979675243
cites work: Q4741262
cites work: Spectra of some composition operators
cites work: Q4888816
cites work: Composition Operators
cites work: Q4340161
cites work: Composition operators and classical function theory
DOI: 10.1016/J.JFA.2004.10.015
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description: scientific article; zbMATH DE number 2141636