Co-isometric solutions of equation \(CU+U^*C=2D\) (Q1809881)

Let \(H\) be an infinite dimensional Hilbert space and \(B(H)\) the algebra of all bounded linear operators on \(H\). For \(C,D\in B(H)\) (\(C\) is invertible) consider the equation \((*)\) \(CU+U^*C=2D\). The authors show that its co-isometric solutions are in one-to-one correspondence with the self-adjoint solutions to the Riccati equation \(X^2-iDX+iXD+D^2-C^2=0\). All possible co-isometric solutions are given parametrically. (Recall that \(T\in B(H)\) is isometric if \(T^*T=I\) and co-isometric if \(T^*\) is isometric.) Thus the authors correct a statement made by \textit{M. Dobovisek} [Linear Algebra Appl. 296, 213-225 (1999; Zbl 0940.15016)] who claims that this is the case not of the co-isometric but of the unitary solutions to \((*)\). They show that the statement is true if \(D\) is positive, i.e. \(\langle Dx,x\rangle \geq 0\) for every vector \(x\).