Zero-two law for cosine families (Q748400)

Let \(C(t)\) be a strongly continuous cosine function on a Banach space \(X\). Then \(\limsup_{t\to 0+}\|C(t)-I\|<2\) implies that \(\lim_{t\to 0+}\|C(t)-I\|=0\) (zero-two law at origin for operator cosine function). Further, it is proved that \(\sup_{t\geq0}\|C(t)-I\|<2\) implies that \(C(t)\equiv I\). For both results, the constant 2 is optimal. For a strongly continuous semigroup \(T(t)\), it is proved that \(\sup_{t\geq0}\|T(t)-I\|<1\) implies that \(T(t)\equiv I\). An analogous result is obtained for a discrete cosine function with the constant \(\frac 32\).