A Note on Closed-Range Composition Operators

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• 作者：John R. Akeroyd ; Arnab Dutta
• 关键词：Mathematics Subject ClassificationPrimary 47B33 ; 47B38 ; Secondary 30D55
• 刊名：Complex Analysis and Operator Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：11
• 期：1
• 页码：151-160
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general; Operator Theory; Analysis;
• 出版者：Springer International Publishing
• ISSN：1661-8262
• 卷排序：11

文摘

Let \(\varphi \) be an analytic mapping from the unit disk \({\mathbb {D}} := \{z: |z| < 1\}\) to itself. In the context of Banach spaces of analytic functions on \({\mathbb {D}}\), we address the question: Does closed-rangedness of the composition operator \(C_{\varphi }\) on one space confer closed-rangedness of \(C_{\varphi }\) on a smaller space? We find that the answer is quite often in the affirmative and provide conditions that guarantee this. We also show that, in the absence of these conditions, the answer is not always in the affirmative.