Isolated nonnegative solutions of infinite-dimensional algebraic Riccati equations

For an infinite-dimensional continuous (or discrete)-time linear system, based on the study of the representation of nonnegative solutions of the algebraic Riccati equation (ARE), we get some sufficient and necessary conditions for a nonnegative solution of (ARE) to be isolated in the set of all nonnegative solution of (ARE) with respect to the norm topology, the strong operator topology and weak operator topology, respectively.