DYSON-PHILLIPS EXPANSION AND UNBOUNDED PERTURBATIONS OF LINEAR C(0)-SEMIGROUPS

Given a c0-semigroup (U(t))t greater-than-or-equal-to 0 on a Banach space E with generator A and an unbounded linear operator B on E such that there exists an extension of A + B, which generates a c0-semigroup on E. We prove, with some conditions on E or B, that there exist a co-semigroup (U(B)(t))t greater-than-or-equal-to 0 on E whose generator is an extension of A + B, and (U(B)(t))t greater-than-or-equal-to is given by the Dyson-Phillips expansion. We give some applications of our abstract theorem to linear transport equations.