Evolution systems and perturbations of generators of strongly continuous groups (Q1773447)

The author considers a problem of unbounded perturbation of a \(C_0\)-group in a Banach space. The author proves firstly that if \(A\) generates a \(C_0\)-group in a Banach space and \(B\) is unbounded, then an extension of \(A+B\) generates a \(C_0\)-semigroup if and only if the family of operator \(\{e^{-rA}Be^{r A}\}_{r\geq 0}\) has an appropriate evolution system. From this result, the author then deduces some sufficient conditions for a perturbation of group generators to generate a semigroup. In particular, the Trotter product formula is obtained again.