A remark on cosine families

Source abstract

Let C ( t ) , t ∈ R C(t),t \in R , be a strongly continuous cosine family and A its infinitesimal generator. Then the set E = d e f { x ∈ X : C ( t ) x E\stackrel {def}{=}\{ x \in X:C(t)x is once continuously differentiable in t on R } of the Banach space X is contained in the domain of ( − A ) α {( - A)^\alpha } for 0 ⩽ α > 1 / 2 0 \leqslant \alpha > 1/2 .