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 \({\sup_{t \geq 0} \|C(t) - I\| < 2}\) yields that C(t) = I for all \({t \geq 0}\). For discrete cosine families, the assumption \({\sup_{n \in \mathbb{N}} \|C(n) - I\| \leq r < \frac{3}{2}}\) yields that C(n) = I for all \({n \in \mathbb{N}}\). For \({r \geq \frac{3}{2}}\), this assertion does no longer hold.
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The first named author has been supported by the Netherlands Organisation for Scientific Research (NWO), Grant No. 613.001.004.
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Schwenninger, F.L., Zwart, H. Zero-two law for cosine families. J. Evol. Equ. 15, 559–569 (2015). https://doi.org/10.1007/s00028-015-0272-8
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DOI: https://doi.org/10.1007/s00028-015-0272-8