Lyapunov exponents of renewal equations: numerical approximation and convergence analysis

The paper’s Section 3.3 establishes the key norm-convergence result by splitting TM,N − T via FM and TN, but the last step invokes Banach–Steinhaus to conclude ||FM − I||X←X → 0, which is false for orthogonal Fourier–Legendre projections on an infinite-dimensional Hilbert space (the operator norm of I − FM stays 1). This gap appears in the proof of Theorem 3.8 and needs a compactness argument instead; fortunately, it is available because TN has finite-dimensional range (h ≥ τ implies TN maps into polynomials of degree ≤ N−1), so TN is compact and ||TN(FM − I)|| → 0 follows from strong convergence of FM and compactness. The paper already notes w*N is a polynomial and that TN’s range is in polynomials; replacing the flawed step by the compactness argument repairs the proof. The model’s solution uses precisely this compactness mechanism (and a resolvent identity) to control the left/right outer projections and is therefore correct. See the paper’s construction of T via the fixed point w* and its discretization (equations (17)–(21) and (33)) and the stated convergence Theorem 3.8, along with the erroneous Banach–Steinhaus step at the end of that proof . The compactness of T(h,0) for h ≥ τ, established earlier (Theorem 3.1), aligns with the model’s compact-operator calculus and strengthens the fix .