An extension of the special factorization with applications to Wiener- Hopf equations
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.
scientific article; zbMATH DE number 3983922