\({\mathcal A}\) -compact mappings

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• 作者：Pablo Turco
• 刊名：Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：110
• 期：2
• 页码：863-880
• 全文大小：522 KB
• 刊物类别：Mathematics and Statistics
• 出版者：Springer Milan
• ISSN：1579-1505
• 卷排序：110

文摘

For a fixed Banach operator ideal \({\mathcal A}\), we use the notion of \({\mathcal A}\)-compact sets of Carl and Stephani to study \({\mathcal A}\)-compact polynomials and \({\mathcal A}\)-compact holomorphic mappings. Namely, those mappings \(g:X\rightarrow Y\) such that every \(x \in X\) has a neighborhood \(V_x\) such that \(g(V_x)\) is relatively \({\mathcal A}\)-compact. We show that the behavior of \({\mathcal A}\)-compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of \({\mathcal A}\)-compact operators to \({\mathcal A}\)-compact polynomials. In order to study \({\mathcal A}\)-compact holomorphic functions, we appeal to the \({\mathcal A}\)-compact radius of convergence which allows us to characterize the functions in this class. Under certain hypothesis on the ideal \({\mathcal A}\), we give examples showing that our characterization is sharp.Keywords\({\mathcal A}\)-compact sets\({\mathcal A}\)-compact polynomialsHolomorphic mappings