A Koopman Approach to Understanding Sequence Neural Models

Appendix B states Proposition 1 (invertible, measure-preserving ϕ ⇒ unitary Koopman operator) and shows: (i) integral invariance from indicators to general functions; (ii) multiplicativity Kϕ(fg)=Kϕf·Kϕg; and hence (iii) inner-product preservation and isometry. However, the final step claims “if ϕ is invertible then Kϕ* Kϕ = Kϕ Kϕ* … thus Kϕ is unitary,” without identifying the inverse/adjoint or establishing surjectivity; commutation does not imply unitarity. This leaves a gap. The candidate solution supplies the standard missing steps: Kϕ−1 is the two-sided inverse on L2, and Kϕ* = Kϕ−1, which completes the proof. See the paper’s proof sketch and its concluding line in App. B for the gap ; the paper earlier also appeals to the “well-known” fact without proof in the main text .