# Zero-two law for cosine families

9 Citations (Scopus)

## Abstract

Abstract. For $(C(t))_{t\geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\lim \sup_{t\rightarrow 0+} \|C(t) - I\| < 2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $sup_{t\geq 0} \|C(t) - I\| < 2$ yields that $C(t) = I$ for all $t \geq 0$. For discrete cosine families, the assumption $\sup_{n∈N} \|C(n) - I\| \leq r < \frac{3}{2}$ yields that $C(n) = I$ for all $n\in N$. For $r \geq \frac{3}{2}$, this assertion does no longer hold.
Original language Undefined 559-569 11 Journal of evolution equations 15 3 https://doi.org/10.1007/s00028-015-0272-8 Published - Sep 2015

## Keywords

• EWI-26531
• MSC-47D09 MSC-47D06
• Cosine families
• IR-98683
• Zero-two law
• Semigroup of operators
• METIS-315075
• Zero-one law