# Riccati Equations and Wiener-Hopf Factorization

Chapter

## Abstract

In this chapter we connect bounded additive exponentially dichotomous perturbations In fact, we prove the so-called triple equivalence of (i) canonical factorizability, (ii) a decomposition of the underlying Banach space and (iii) the unique solvability of a vector-valued Wiener-Hopf equation with con|volution kernel

*S*of an exponentially dichotomous operator*S*_{0}to left and right canonical Wiener-Hopf factorizations of the fractional linear function$$
W\left( \lambda \right) = \left( {\lambda - S_0 } \right)^{ - 1} \left( {\lambda - S} \right).
$$

*X*of the type$$
\operatorname{Im} E\left( {0^ \pm ;S_0 } \right)\dot + \operatorname{Im} E\left( {0^ \pm ;S} \right) = X,
$$

*E*(·*S*_{0})Γ, where Γ=*S-S*_{0}., In particular, if*S*_{0}and*S*are written in block matrix form with respect to the decomposition induced by the separating projection of*S*_{0}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
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© Birkhäuser Verlag AG 2008