Abstract
It is proved that for a cosine family \({\{c(t)\}_{t \in \mathbb{R}}}\) in a normed algebra with a unity e, the following assertions hold: (1) If \({\sup_{t \in \mathbb{R}}\| c(t) - e \| < 2}\), then c(t) = e for every \({t \in \mathbb{R}}\). (2) If \({\lim sup_{t \to 0}\| c(t) - e \| < 2}\), then \({\lim_{t \to 0} c(t) = e}\). It is also shown that the two respective results, each specific for one of the assertions, are equivalent.
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Chojnacki, W. Around Schwenninger and Zwart’s zero-two law for cosine families. Arch. Math. 106, 561–571 (2016). https://doi.org/10.1007/s00013-016-0898-y
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DOI: https://doi.org/10.1007/s00013-016-0898-y