Online learning of Koopman operator using streaming data from different dynamical regimes ⋆

The paper states the key identities and algorithms (block Hankel relation, input projection, recursive update of Ξ, and Grassmannian-based acceptance) and cites prior work for derivations. In particular, it asserts YNΠ⊥UN = Γl ZliftΠ⊥UN and proceeds to extract Γl via SVD, then updates Ξ using the Oku–Kimura recursion ΞN+1 = ΞN + αee⊺ with α = (1 + u⊺Pu)−1 and e = y − Y U⊺Pu (Algorithm 1, eqs. (14a–e)), and uses dGr to accept data (Algorithm 2) . The candidate solution supplies the missing algebra: (i) the rank and projection conditions ensuring col(YΠ⊥U) = col(Γl) under s > lm + r and full-row-rank U; (ii) a complete RLS-based proof that the recursion reproduces Ξ′ from scratch; and (iii) a precise characterization of the acceptance rule and minimal r from the rank of Ξ. Aside from a minor wording slip (“left-multiply” vs. post-multiply), the model’s solution is consistent with the paper, adds the needed hypotheses, and gives rigorous derivations where the paper is concise and cites prior work. Hence both are correct, with the model providing a more explicit proof.