A characterization of norm continuity of propagators for second order abstract differential equations

In this paper, we obtain a concise characterization of norm continuity for t > 0 of propagators for the complete second order abstract differential equation on a Banach space E, u″(t)+Bu′(t)+Au(t)=0, t≥0 where B ε L(E). As a consequence, we discover that a strongly continuous cosine operator function or operator group is norm continuous for t > 0 if and only if its generator is bounded.