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 | English |
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Pages (from-to) | 559-569 |
Number of pages | 11 |
Journal | Journal of evolution equations |
Volume | 15 |
Issue number | 3 |
DOIs | |
Publication status | Published - Sept 2015 |
Keywords
- MSC-47D09 MSC-47D06
- Cosine families
- Zero-two law
- Semigroup of operators
- Zero-one law