An extension of the special factorization with applications to Wiener- Hopf equations
type: Item
instance of: scholarly article
title: An extension of the special factorization with applications to Wiener- Hopf equations
zbMATH Open document ID: 0608.47024
DOI: 10.1016/0022-247X(85)90295-1
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: Let H be a separable Hilbert space, and \(E: \Sigma\to B(H)\) a resolution of the identity; \(\Sigma\) denotes the Borel sets of [0,1]; \(P^ t=E(]0,t])\), \(\forall t\in [0,1]\). \(T\in B(H)\) is said to be causal (anticausal) if \(P^ tTP^ t=P^ tT\) \((P^ tTP^ t=TP^ t)\), \(\forall t\in [0,1]\); T is memoryless if \(E(\omega)T=TE(\omega)\), \(\forall \omega \in \Sigma\). Then the author proves in particular the following factorization: Let \(T\in B(H)\) such that there exists \(\gamma\geq 0\), and a probability \(\mu\) on [0,1], absolutely continuous with respect to Lebesgue measure on [0,1]: \[ | E(\omega T| \leq \gamma \sqrt{\mu (\omega)},\quad | TE(\omega)| \leq \gamma \sqrt{\mu (\omega)},\quad \forall \omega \in \Sigma. \] Suppose \(T\geq 0\), \(D\geq 0\), D memoryless. Then we have the factorization: \(T=Z_++Z_-+Z_+DZ_-\), with \(Z_+\) causal, \(Z_-\) anticausal. And this factorization implies the solution of some abstract Wiener-Hopf equation.
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 3983922
zbMATH Keywords: resolution of the identity
zbMATH Keywords: factorization
zbMATH Keywords: memoryless
zbMATH Keywords: causal
zbMATH Keywords: anticausal
zbMATH Keywords: Wiener-Hopf equation
author: M. H. Milman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3983922