Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

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• 作者：José Bonet ; Pawe? Domański
• 关键词：Primary 47B33 ; 46E10 ; 39B12 ; Secondary 47A15 ; 47A75 ; 39B22 ; 30D05 ; Spaces of real analytic functions ; composition operator ; eigenvalues and eigenvectors ; spectrum ; Abel’s functional equation ; iteration semigroup ; iteration root
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：81
• 期：4
• 页码：455-482
• 全文大小：385 KB
• 参考文献：1. Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989

文摘

We obtain full description of eigenvalues and eigenvectors of composition operators \({C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}\) for a real analytic self map \({\varphi:\mathbb{R} \rightarrow \mathbb{R} }\) as well as an isomorphic description of corresponding eigenspaces. We completely characterize those \({\varphi}\) for which Abel’s equation \({f\circ \varphi=f+1}\) has a real analytic solution on the real line. We find cases when the operator \({C_{\varphi}}\) has roots using a constructed embedding of \({\varphi}\) into the so-called real analytic iteration semigroups.