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Riccati Equations and Wiener-Hopf Factorization

Part of the Operator Theory: Advances and Applications book series (OT, volume 182)

Abstract

In this chapter we connect bounded additive exponentially dichotomous perturbations S of an exponentially dichotomous operator S0 to left and right canonical Wiener-Hopf factorizations of the fractional linear function
$$ W\left( \lambda \right) = \left( {\lambda - S_0 } \right)^{ - 1} \left( {\lambda - S} \right). $$
In fact, we prove the so-called triple equivalence of (i) canonical factorizability, (ii) a decomposition of the underlying Banach space X of the type
$$ \operatorname{Im} E\left( {0^ \pm ;S_0 } \right)\dot + \operatorname{Im} E\left( {0^ \pm ;S} \right) = X, $$
and (iii) the unique solvability of a vector-valued Wiener-Hopf equation with con|volution kernel ES0)Γ, where Γ=S-S0., In particular, if S0 and S are written in block matrix form with respect to the decomposition induced by the separating projection of S0 and the bounded additive perturbation Γ is off-diagonal with respect to this decomposition, we convert the equivalent statements derived into an existence result for certain solutions of Riccati equations in £(X). We conclude this chapter with perturbation results on the solutions of these Riccati equations.

Keywords

Operator Function Riccati Equation Continuous Semigroup Complex Hilbert Space Exponential Dichotomy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag AG 2008

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