ALUTHGE TRANSFORMS OF GENERALIZED HYPERBOLIC OPERATORS

The paper claims the equivalence “invertible + shadowing and no nonzero r-homoclinic points for every r > 0 ⇔ hyperbolic” (Theorem 2.4) and reduces it to Theorem 2.3 via Lemma 3.1. However, Lemma 3.1 asserts Hr(T) ⊆ Ec(T) ⊆ Hr(T) for all r > 0 and its proof incorrectly concludes from ||x − y|| ≤ ε/2 that x ∈ Hr(T), without any control on ||T^n x − T^n y|| for |n| large (the key step is shown in the proof of Lemma 3.1). This gap undermines Theorems 2.3 and 2.4 as presented (see Lemma 3.1 and the subsequent uses of (3.1), and the “direct consequence” proof of Theorem 2.4). By contrast, the model’s solution provides an independent, correct proof: it uses shadowing to solve the cohomological equation y_{n+1} − T y_n = e_n for bounded e, then applies a dual-eigenfunctional argument to show (λI − T) is bijective for all |λ| = 1, yielding σ(T) ∩ 𝕋 = ∅.