Less than one implies zero

Abstract

In this paper we show that from the estimate ${\sup }_{t\ge 0}\Vert C(t)-\cos (at)I\Vert <1$ we can conclude that $C(t)$ equals $\cos (at)I$. Here ${\left(C(t)\right)}_{t\ge 0}$ is a strongly continuous cosine family on a Banach space.

arXiv commentCorrected the previous version (in particular, a mistake in Lemma 2.1), streamlined and added recent references (see Introduction). 6 pages