A $0-2$ law for cosine families with $\mathrm{limsup}$ to $\infty$

Felix L. Schwenninger, Hans Zwart

For ${\left(C(t)\right)}_{t\in \mathbb{R}}$ being a cosine family on a unital normed algebra, we show that the estimate ${\mathrm{limsup}}_{t\to {\infty }^{+}}\Vert C(t)-I\Vert <2$ implies that $C(t)=I$…