On Riccati equations in Banach algebras

Let $R$ be a commutative complex Banach algebra with the involution $\cdot ^\star$ and suppose that $A\in R^{n\times n}$, $B\in R^{n\times m}$, $C\in R^{p\times n}$. The question of when the Riccati equation $PBB^\star P-PA-A^\star P-C^\star C=0$ has a solution $P\in R^{n\times n}$ is investigated. A counterexample to a previous result in the literature on this subject is given, followed by sufficient conditions on the data guaranteeing the existence of such a $P$. Finally, applications to spatially distributed systems are discussed.

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