AN EPSILON-HYPERCYCLICITY CRITERION AND ITS APPLICATION ON CLASSICAL BANACH SPACES

The paper’s ε-Hypercyclicity Criterion (Theorem 1.2) explicitly requires the convergence conditions (2)–(3) only along an enumeration y_k of D2 and notes that this enumeration is essential; strengthening (2)–(3) to hold for each fixed y ∈ D2 recovers the usual Hypercyclicity Criterion (Proposition 5.1). The candidate solution misreads (2)–(3) as holding for every y ∈ D2 and builds a Baire-category argument on that stronger assumption. Under the paper’s actual hypotheses, the candidate’s Step 1 (density of ⋃_k T^{-n(k)}(B(y,r)) for each fixed y) is not justified. The paper’s constructive proof (via a fast-decaying series and a diagonal choice of indices) correctly yields δ-hypercyclicity for all δ>ε. See the statement of Theorem 1.2 and its proof, as well as the warning in Section 5 that the naive strengthening implies the Hypercyclicity Criterion .