On the lattice property of the Koopman operator spectrum

The paper’s Theorem 1.1 asserts that for a deterministic discrete-time system with Koopman operator K on L2, the spectrum σ(K) is multiplicatively closed. This is directly contradicted by a simple finite deterministic counterexample: let T be a permutation with two disjoint cycles of lengths 2 and 3; then σ(K) = {1, −1, ω, ω²} and, for λ = −1 and η = ω, the product λη = −ω is not in σ(K). The paper’s proof hinges on a mis-stated Weyl criterion (treating approximate eigenvalues as characterizing the entire spectrum) and on an invalid truncation step; in particular, the key inequality that the truncation preserves the approximate-eigen relation (their passage to (3.8)) is not justified and fails in the finite example, where it produces an O(1) residual rather than the claimed o(1) bound. See the statement of Theorem 1.1 and its proof outline, including their Weyl-criterion setup and truncation scheme, in the paper’s Section 1 and Section 3 .