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

TL;DR

This paper establishes a new limit law for cosine families in normed algebras, showing that if the limsup of their deviation from the identity is less than 2, then the family must be trivial.

Contribution

It generalizes existing boundedness results to unbounded cases using limsup conditions for cosine families, discrete families, and semigroups.

Findings

If limsup of  C(t) - I  < 2, then C(t) = I for all t.

The result extends to discrete cosine families and semigroups.

Provides a unifying limit law for cosine family triviality.

Abstract

For $\left(C(t)\right)_{t\in\mathbb R}$ being a cosine family on a unital normed algebra, we show that the estimate $\limsup_{t\to\infty^{+}}\|C(t) - I\| <2$ implies that $C(t)=I$ for all $t\in\mathbb R$. This generalizes the result that $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. We also state the corresponding result for discrete cosine families and for semigroups.