Perturbation of positive semigroups

Abstract

The purpose of this note is to study perturbations of generators of positive semigroups by positive operators. Let E be a complex Banach lattice and A be a linear operator on E with domain D (A). We say that A is resolvent positive if there exists w e P~ such that (2 - A): D (A) --~ E is bijective and (2 - A)- 1 is a positive operator on E for all 2 > w. Note that the generator of a positive semigroup is resolvent positive. Assume that A generates a positive semigroup (by which we always mean a C0-semi- group) and B: D(A)---* E is linear and positive such that A + B (with domain D (A + B) = D (A)) is resolvent positive. Then it was shown by Desch [8] that A + B generates a positive semigroup whenever E is a space L 1. A simple proof is given by Voigt [20]. If E is an LP-space, 1 < p < 0% then the assertion is false, in general (see [4]). However, we show in Section 1 that in the case where the semigroup generated by A is holomorphic, also A + B generates a holomorphic semigroup without any restriction on the space. Furthermore, we prove in Section 2 that A + B generates a semigroup whenever B is a positive rank-one perturbation of A. This is remarkable in view of a recent result of Desch-Schappacher [9]. If the semigroup generated by A is not holomorphic, there always exists a (necessarily non positive) rank-one perturbation B such that A + B is not a generator. In Section 3 we give a criterion for perturbation by multiplication operators which, in view of the Sobolev embedding theorems, is particularly useful for elliptic operators. As an illustrating example we consider Schr6dinger operators. In Section 4 the results are applied to systems of evolution equations, which obtained special attention recently (see [14]). Concerning terminology and basic results we follow [17] and [13]. A c k n o w 1 e d g e m e n t. We are indebted to J. Voigt for several valuable suggestions and comments. 1.