On The Numerical Ranges of Composition Operators Induced By Mappings With The Denjoy-Wolff Point On The Boundary

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• 作者：William M. Higdon
• 刊名：Integral Equations and Operator Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：85
• 期：1
• 页码：127-135
• 全文大小：471 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8989
• 卷排序：85

文摘

Let D denote the open unit disk in the complex plane and let \({\phi}\) denote a holomorphic self-map of D. We show that if \({\phi}\) is not linear-fractional and has its Denjoy-Wolff point on the boundary of D, then 0 is an interior point of the numerical range of the composition operator \({C_{\phi}}\) on the Hardy space \({H^2({\mathbf D})}\). This answers a question raised by Bourdon and Shapiro (Integr Equations Oper Theory 44:410–441, 2002), and, together with their results, provides a complete description of when 0 belongs to the numerical range of a composition operator on \({H^2({\mathbf D})}\).KeywordsNumerical rangeComposition operatorHardy spaceDenjoy-Wolff point