Avoiding spectral pollution for transfer operators using residuals

The paper rigorously derives that the kernel residual kres(λ)^2 equals the minimum Rayleigh–Ritz value of the Hermitian matrix Ũ(λ)=J̃−λÃ−λ̄Ã*+|λ|²G̃ and provides a convergence result that is one-sided: lim_{r→∞} lim_{M→∞} kres(λ;r,M) ≤ μ1 inf_{∥h∥=1} ∥(L−λ)h∥, with no uniform lower bound. The candidate solution, however, incorrectly identifies the limiting quadratic form with the L2-residual ⟨Lf,Lg⟩ structure, assumes operator-norm convergence to that L2 form, concludes equality to the L2 pseudospectral gauge ψ(λ), and claims a two-sided certification ruling out spectral pollution. These steps contradict the paper’s analysis, which emphasizes the weighted (Mercer-feature) residual, the μ1 factor, and the lack of a uniform lower bound.