Zero-two law for cosine families

For (C(t))t≥0 being a strongly continuous cosine family on a Banach space, we show that the estimate lim supt→0+ ‖C(t) − I‖ < 2 implies that C(t) converges to I in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of supt≥0 ‖C(t) − I‖ < 2 yields that C(t) = I for all t ≥ 0. For discrete cosine families, the assumption supn∈N ‖C(n)− I‖ ≤ r < 3 2 yields that C(n) = I for all n ∈ N. For r ≥ 3 2 , this assertion does no longer hold.