On the abstract Cauchy problem for the operator \(\frac{d^ 2}{dt^ 2}- A\)
Let X be a Banach space and A a densely defined operator in X with a non- void resolvent set; denote by \(D_\infty =\cap^\infty_{n=0}D(A^ n)\) endowed with the Fréchet topology given by the seminorm system \(\{\| x\|_ n=\sum^{n}_{k=0}\| A^ kx\|,\quad n=0,1,2,...\}.\) It is proved that the Cauchy problem for the operator \(d^ 2/dt^ 2-A\) is well-posed in the sense of distributions iff \(A|_{D_\infty}\) generates a locally-equicontinuous cosine function of class \(C^\infty\) in \(L_ s(D_\infty)\). This is an extension of \textit{T. Ushijima}'s smoothness result of distribution semi- groups on \(D_\infty[\)J. Fac. Sci., Univ. Tokyo, Sect. IA 19, 65-127 (1972; Zbl 0239.47032)].
scientific article; zbMATH DE number 3844447
scientific article