ADMISSIBILITY AND GENERALIZED NONUNIFORM DICHOTOMIES FOR DISCRETE DYNAMICS

The paper’s Theorem 3.4 is correctly stated and proved: assuming the block-growth condition L1 ≤ μ_{q_n}/μ_n ≤ L2 and the forward growth bound ||A_{m,n}x||_m ≤ M(μ_m/μ_n)^λ||x||_n, the invertibility of T^μ_Z implies a nonuniform μ-dichotomy with respect to (||·||_m). The paper defines T^μ_Z precisely and works via the invariant splitting X = X_n ⊕ Z_n (with Z_n = A_{n,1}Z), then establishes the dichotomy estimates through Lemmas 3.6–3.8, concluding the result (Theorem 3.4) . By contrast, the candidate solution follows a plausible Green-operator scheme but makes a crucial, unproven claim: property (iii) (bijectivity of A_m|_{ker P_m}: ker P_m → ker P_{m+1}). The proposed argument for injectivity/surjectivity on ker P_m is incomplete and, as written, incorrect (the construction using S ι_{m+1}(φ_{m+1}w) does not yield A_m u = w with u ∈ ker P_m). The backward (unstable) estimates then rely on this unjustified (iii). Hence, while the overall strategy is close in spirit to the admissibility ⇒ dichotomy route, the model’s proof misses an essential step and does not rigorously establish the μ-dichotomy.