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

Abstract

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 $C(t)=I$ for all $t\in \mathbb{R}$. This generalizes the result that ${\sup }_{t\ge 0}\Vert C(t)-I\Vert <2$ yields that $C(t)=I$ for all $t\ge 0$. We also state the corresponding result for discrete cosine families and for semigroups.

4 pages

References in corpus (1)

• Bounded cosine functions close to continuous scalar bounded cosine functions