On the abstract Cauchy problem for the operator\frac{{d^2 }}{{dt^2 }} - A.

LetX be a Banach space, A a closed operator with dense domainD(A) and non-void resolvent set; topologized by the semi-norm system. We prove that the Cauchy problem is well posed in the sense of distributions for the operator d²/dt²–A if and only if A restricted toD generates a locally — equicontinuous cosine function of class C inL _S D . This is an extension of Ushijima's smoothness result of distribution semi-groups 10].