Around Schwenninger and Zwart’s zero-two law for cosine families

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.