Discrete time Riccati equation and the invariant subspaces of linear operators

Let H and Y be separable Hilbert spaces, U finite dimensional. Let A ∈ L(H), B ∈ L(U, H), C ∈ L(H,Y ), D ∈ L(U,Y ), and suppose that the open loop transfer function D(z) := D + zC(I − zA)B ∈ H∞(U, Y ). Let J ≥ 0 be a cost operator. We study a subset of self adjoint solutions P of the discrete time algebraic Riccati equation (DARE)