Riccati Equations and Wiener-Hopf Factorization

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


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.


Operator Function Riccati Equation Continuous Semigroup Complex Hilbert Space Exponential Dichotomy 
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© Birkhäuser Verlag AG 2008

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