Wiener–Hopf Factorization Through an Intermediate Space

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• 作者：Frank-Olme Speck
• 关键词：Primary 47A68 ; Secondary 47B35 ; Wiener–Hopf operator ; Factorization ; Intermediate space ; Generalized inverse
• 刊名：Integral Equations and Operator Theory
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：82
• 期：3
• 页码：395-415
• 全文大小：648 KB
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  29.Penzel F., Speck F.-O.: Asy
• 作者单位：Frank-Olme Speck (1)
  
  1. Departamento de Matemática, Instituto Superior Técnico, Avenida Rovisco Pais, 1049-001, Lisboa, Portugal
  
• 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

An operator factorization conception is investigated for a general Wiener–Hopf operator \({W = P_2 A |_{P_1 X}}\) where X, Y are Banach spaces, \({P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)}\) are projectors and \({A \in \mathcal{L}(X,Y)}\) is boundedly invertible. Namely we study a particular factorization of \({A = A_- C A_+ \,\,{\rm where}\,\, A_+ : X \rightarrow Z \,\,{\rm and} \,\,A_- : Z \rightarrow Y}\) have certain invariance properties and \({C : Z \rightarrow Z}\) splits the “intermediate space-Z into complemented subspaces closely related to the kernel and cokernel of W, such that W is equivalent to a “simpler-operator, \({W \sim P C|_{P X}}\). The main result shows equivalence between the generalized invertibility of the Wiener–Hopf operator and this kind of factorization (provided \({P_1 \sim P_2}\)) which implies a formula for a generalized inverse of W. It embraces I.B. Simonenko’s generalized factorization of matrix measurable functions in L p spaces, is significantly different from the cross factorization theorem and more useful in numerous applications. Various connected theoretical questions are answered such as: How to transform different kinds of factorization into each other? When is W itself the truncation of a cross factor?