A note on Riccati matrix difference equations

The paper’s semigroup duality H(P, pΦ^n(Q)) = E_n(P)^T H(Φ^n(P), Q) E_n(P) + G_n(P) is stated and proved (Theorem 1.1), via a one-step Woodbury-based identity (their equation (30)) followed by an induction, exactly matching the model’s high-level plan for Part 1; this portion of the model’s solution agrees with the paper’s statements and methods . For the Floquet-type representation, the paper rigorously proves E_n(P) = E^n L_n(P)^{-1} with L_n(P) = I + (P − P∞) G_n, establishes invertibility of L_n(P) for n ≥ r using Gramian invertibility G_n ⪰ G_r > 0, and derives a uniform bound on ∥L_n(P)^{-1}∥ (Theorem 1.3 and Lemma 5.1) . By contrast, the model’s derivation of invertibility and a uniform conditioning bound for L_n(P) is flawed: (i) it incorrectly infers invertibility of L_n(P) from E_n(P) L_n(P) = E^n without establishing invertibility of E_n(P) or E^n, and (ii) it uses an unjustified lower bound on the symmetric part (L_n + L_n^T)/2 based on G_n ≤ H and P ≥ 0; this step ignores non-commutativity issues and the paper’s necessary restriction n ≥ r. The correct argument in the paper relies on a factorization of L_n and the positivity of G_n (for n ≥ r), not on the model’s spectral bound shortcut . Thus, while the model’s duality part is consistent with the paper, its Floquet invertibility/conditioning step is not justified under the paper’s assumptions.