Optimal Extensions for pth Power Factorable Operators

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• 作者：O. Delgado ; E. A. Sánchez Pérez
• 关键词：Mathematics Subject Classification46E30 ; 46G10 ; 47B38
• 刊名：Mediterranean Journal of Mathematics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：13
• 期：6
• 页码：4281-4303
• 全文大小：671 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1660-5454
• 卷排序：13

文摘

Let \({X(\mu)}\) be a function space related to a measure space \({(\Omega,\Sigma,\mu)}\) with \({\chi_\Omega\in X(\mu)}\) and let \({T\colon X(\mu)\to E}\) be a Banach space-valued operator. It is known that if T is pth power factorable then the largest function space to which T can be extended preserving pth power factorability is given by the space Lp(m_T) of p-integrable functions with respect to m_T, where \({m_T\colon\Sigma\to E}\) is the vector measure associated to T via \({m_T(A)=T(\chi_A)}\). In this paper, we extend this result by removing the restriction \({\chi_\Omega\in X(\mu)}\). In this general case, by considering m_T defined on a certain \({\delta}\)-ring, we show that the optimal domain for T is the space \({L^p(m_T)\cap L^1(m_T)}\). We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.