Perturbations of Dissipative Operators with Relative Bound One1

Let A be the generator of a (C„) contraction semigroup on a Banach space. Let B be a dissipative operator with densely defined adjoint. Assume that the inequality H&cj;|^|M:r|| + holds on the domain of A. Then the closure of /f + B generates a (C0) contraction semigroup. Let A be the generator of a (C0) contraction semigroup on a Banach space X. Let B be a dissipative operator on X in the sense of Lumer and Phillips [3]. Assume that Q(B)^>5((A). Since A is closed it follows that there are constants a, Z>(A), (1) ||.Bx|| < a \\Ax\\ + b \\x\\. We say that B is bounded relative to A and refer to a as a relative bound. Gustafson [1], generalizing basic work of Rellich, Kato, and others (cf. [2]), showed that if the bound a in (1) can be taken strictly less than 1 it follows that A+B is the generator of a (C0) contraction semigroup. This is known to fail fora> 1. On the other hand. Wüst [4] recently showed that if A and B are symmetric operators on a Hilbert space with A selfadjoint then the validity of (1) with a — 1 implies that A + B is essentially selfadjoint, i.e., has selfadjoint closure. (Kato [2] had proved a slightly weaker result, starting from the analogue of (1) with norms replaced by their squares.) In this note we use a simplified version of Wiist's argument to extend the result to dissipative operators in a rather general Banach space setting. Theorem. Let X be a Banach space. Let A and B be as above with 2)(B)-(/(A). Assume that there is a constant b<oo such that, for all xe&(A), (2) l&cll < \\Ax\\ + b \\x\\. Suppose also that the adjoint B* has a dense domain in X*. Then the closure of A+B is the generator of a (C„) semigroup. Received by the editors July 22, 1971. AMS 1970 subject classifications. Primary 47A55, 47B44; Secondary 47D05.