Estimates for the perturbed Oseen semigroup

We present a sufficient condition for the perturbed Oseen semigroup which yields the same power-like decay rate as for the unperturbed one. We prove that this phenomenon takes place provided the perturbed Oseen operator has no eigenvalues inside the closed right half-plane. In particular, this result is valid for the perturbations that are small in some sense. The proof is based on the results on invertibility of elements of some Banach algebra of operator-valued functions, which results rests on the Allan --- Douglas principle.