DISCONTINUITIES CAUSE ESSENTIAL SPECTRUM

The paper proves Theorem 1: for a piecewise monotone T with a non‑trivial discontinuity a and a weight ϕ bounded and continuous on each branch, any Banach space B satisfying (1) C∞(I)⊆B, (2) Lϕ extends to B, (3) ∥h∥∞ ≤ ∥h∥B for all h in ⋃n Lnϕ(C∞(I)), must have essential spectral radius at least Λ(T,ϕ,a) (indeed EssSpec(Lϕ) contains the closed disk of radius Λ) . The core ingredients are the “jump calculus” J(Lg, ak)=γk−1ϕ(ak−1)J(g, ak−1) (Lemma 3.4) and a dual‑eigenfunctional construction using ℓλ(h)=∑k≥k0 λkαk J(h, ak), which is bounded on A by assumption (3); a rank‑one compact perturbation (M=L|A−K) then yields, for every |λ|<Λ, an eigenvalue of M∗, so Spec(L|A) contains the full disk |λ|≤Λ, and hence EssSpec(L) does too . By contrast, the model’s proof via a factor map onto a unilateral weighted shift on c0 has critical gaps. It asserts a bounded, surjective “jump vector” map Π:U→c0 from the B‑closure U of span{hj}, where hj are constructed via Lϕ, and then passes to a quotient to compare essential spectra. However, boundedness of the jump functionals on ⋃n Lnϕ(C∞(I)) (which follows from (3)) does not automatically imply the existence of a uniform bound |Δb(g)| ≤ C∥g∥B for all finite linear combinations g in U0 and hence does not ensure that Π is a bounded operator extending to U. This step is essential for the factorization but is not justified. Moreover, the solution claims hj∈C∞(I) is preserved by Lϕ, which the paper explicitly warns need not hold (Lϕ need not leave C∞(I) invariant) . Consequently, the model’s argument is incomplete, while the paper’s is complete and correct.