Perturbations of dissipative operators with relative bound one

Abstract

Let A be the generator of a ( C 0 ) ({C_0}) contraction semigroup on a Banach space. Let B be a dissipative operator with densely defined adjoint. Assume that the inequality ‖ B x ‖ ≦ ‖ A x ‖ + b ‖ x ‖ \left \| {Bx} \right \| \leqq \left \| {Ax} \right \| + b\left \| x \right \| holds on the domain of A. Then the closure of A + B A + B generates a ( C 0 ) ({C_0}) contraction semigroup.