On the Minimal Extension of C -Semigroups for 0 Second-Order Damped Equations

The present paper deals with a minimal extension of the classical semigroup theory for second-order damped differential equations in Banach spaces with closed, densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., semigroups. We present here generalizations of the Miyadera]Phillips]Feller theorem, the Hille type theorem, and the Trotter]Kato type theorem. The method is quite general and could be used for equations of any order. We focus our attention on a particular dynamical operator solution or main propagator and we assume some properties about it. From this we can obtain some information about the complementary basis operator solutions or secondary propagators