A short proof of the zero-two law for cosine functions

摘要

Let (C(t))(t epsilon R) be a cosine function in a unital Banach algebra. We give a simple proof of the fact that if lim sup(t -> 0)vertical bar vertical bar C(t) - 1(A) vertical bar vertical bar < 2, then lim sup(t -> 0) vertical bar vertical bar C(t) - 1(A) vertical bar vertical bar = 0.