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arXiv · 1402.1304

Zero-two law for cosine families

Felix Schwenninger·Hans Zwart
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Abstract

For $\left(C(t)\right)_{t \geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\limsup_{t\to 0^{+}}\|C(t) - I\| <2$ 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 $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. Additionally, we derive alternative proofs for similar results for $C_{0}$-semigroups.

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Felix Schwenninger, Hans Zwart. 2014-02-06. Zero-two law for cosine families. https://doi.org/10.1007/s00028-015-0272-8

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