On Riccati Equations in Banach Algebras

Let R be a commutative complex Banach algebra with the involution .* and suppose that A is an element of R(nxn), B is an element of R(nxm), C is an element of R(pxn). The question of when the Riccati equation PBB*P - PA - A*P - C*C = 0 has a solution P is an element of R(nxn) 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.